Abstract
Commutative symmetric operator families of the so-called -class are considered in Krein spaces. It is proved that the restriction of a family of this type on a special kind of invariant subspace is similar to a family of operators adjoint to multiplication operators by scalar functions acting on a suitable function space.
1. Introduction
In the well-known monograph of Colojoara and Foias [1], it is mentioned (in Chapter 5, Corollary 5.7) that -unitary and -self-adjoint operators in Pontrjagin spaces give examples of generalized spectral operators. Actually, it is clear that this observation can be applied also to many operator classes in Krein spaces, so Operator Theory in Krein spaces is a good laboratory where different methods of general Operator Theory can be tested. One of these methods is related with model spaces of function type.
The main goal of the paper is a problem of model representation for a commutative operator family acting on a separable Krein space and possessing a maximal nonnegative invariant subspace, presented as a direct sum of a neutral subspace with a finite dimension and a uniformly positive subspace. As it is known (see [2]), this family generates a spectral function with a peculiar spectral set that provides a resolution of the spectral type for the family. In particular, with every operator , one can associate a scalar function such thatwhere runs through the set of all closed intervals of the real line disjoint with .
The key results (Theorems 6.5, 6.19, and 6.29) say that there is a suitable function space (so-called basic model space) where generates a multiplication operator similar to a compression of on a subspace calculated through .
Section 2 gives a draft of problems considered in the paper, analyzing a simple case of self-adjoint operator in Pontryagin space. Section 3 contains definitions and well-known results used throughout the paper. Section 4 deals with a model representation of a resolution of the identity that is simultaneously -orthogonal and similar to an orthogonal resolution of the identity. In Section 5 a notion of unbounded elements conformed with a resolution of the identity is introduced and studied. Roughly speaking, a resolution sets a correspondence between a Hilbert space and a (vector-valued) function space . From this point of view, the unbounded elements correspond to measurable vector-valued functions outside of . The main results are presented in Section 6, where the notion of a basic model space is introduced. Also, a relation is established between multiplication operators by scalar functions acting in a basic model space, and operators of a commutative -symmetric algebra of -class. Here, emphasis is made on the problem of uniqueness of a basic model space for such algebras. Section 6.2 corresponds to resolutions of the identity with properties like those in Pontryagin spaces. Section 6.3 deals with general -orthogonal resolutions of the identity of -class, and Section 6.4 contains a theorem on a model representation of a commutative -symmetric operator family of -class. Historical and bibliographical remarks are presented in the last section.
It is assumed that the reader is familiar with elements of Krein space geometry and Operator theory (see [3–7]). In this paper, the terminology given in [8] will be used.
2. An Elementary Description of the Problem
Let be a separable Hilbert space with a scalar product . is said to be an indefinite metric space if it is equipped by a sesquilinear continuous Hermitian form (indefinite inner product) such that the corresponding quadratic form has an indefinite sign (i.e., takes positive, negative, and zero values). The indefinite inner product can be represented in the form , where is a so-called Gram operator. The operator is bounded and self-adjoint. If the Gram operator for an indefinite metric space is boundedly invertible and its invariant subspace that corresponds to the negative spectrum (or, alternatively, to the positive spectrum) of is finite dimensional, the space is called a Pontryagin space. In this section, we consider Pontryagin spaces only.
Now, let : be a bounded linear operator. This operator is said to be -self-adjoint (-s.a.) if for all . The properties of -s.a. operators differ from the properties of ordinary self-adjoint operators. For instance, a -s.a. can have a nonreal spectrum or/and a nontrivial Jordan chain. If has only a real spectrum, then there is, on , the eigen spectral function (e.s.f.) of with a finite set of critical points such that () Let . Then for every point of there are three options.(1)For an interval such that and , the spectral function is bounded and the representation takes place.(2) For an interval such that and , the spectral function is bounded but the representation does not take place.(3) For an interval such that and , the spectral function is unbounded.
The first item means that the operator is similar to a self-adjoint operator, that is, it is a scalar spectral operator. The second item corresponds to a spectral operator with a nontrivial finite-dimensional nilpotent summand. Thus only the third item represents a situation that is out of the ordinary theory of model representation for spectral operators with real spectrum. Below, we consider this exceptional case. Let us give an example of the case in discussion.Example 2.1. Let the system be an orthonormalized basis in a Hilbert space and let . Then this space is a Pontryagin space with . Let an operator be given by the conditions (i),(ii), ,(iii). These conditions define a bounded operator, so is naturally defined on and, moreover, is a -self-adjoint operator. The spectral function of can be described by the following conditions: so Put . Then . At the same time for . Thus, for and its spectral function , we have the following results:(i) is unbounded;(ii) belongs to the closure of the linear span generated by the subspaces , where , , so the kernel of the operator is not trivial;(iii) is degenerated, .
Let us note that the coexistence of these properties reflects a general situation (in particular, the first item yields the other two). Indeed, it is shown by the following theorem.
Theorem 2.2 (see [8, Theorem I.9.6]). Let a subspace be positive. Then there is a constant such that for every .
For simplicity everywhere below in this section, we assume thatThen by Theorem 2.2, the subspace generated as the closed linear span of the subspaces , where , , is not a positive subspace, thus it has the nontrivial isotropic part, . Let us denote, below, that . Since the subspace is evidently nonnegative, it can be presented in the formwhere is a positive subspace. Note that is not uniquely defined. Note also that, due to the general theory, the subspace is a part of a maximal nonnegative subspace invariant with respect to .
Theorem 2.3. Let be the e.s.f.
of a -self-adjoint
operator . Then there are a scalar Lebesgue measure , the Hilbert space associated with , and a collection of -measurable
scalar functions such that if , the operator is similar to
the operator acting by the formulae
on the formal linear span of and , where is a basis on , is the
indicator of , for , and Proof. According to Equality (2.5), the
restriction has the
following matrix representation:With no loss of generality (see
Proposition 3.3 below for details), one can assume that on and . Then the operator-valued function represents an
(orthogonal) resolution of the identity acting in the Hilbert space . From the classical theory, we know that there is a
Hilbert space of scalar
functions that is a model space for with some
corresponding operator : of similarity,
that is,Let be an
orthonormalized basis in . Then, for every , , and , the expression defines a
continuous linear functional in . Thus, for every , there exists the -measurable
function such that, for
an arbitrary , , and , the representationholds. Since the spectral
function is unbounded, for at least
one . It means that at least one function (maybe all) of
the collectionis not in . Then we just need to prove (2.7).
Let . Then the linear functional ()is bounded.
Next, by
definition, the space is the closure
of all vectors having the formwith vanishing near
zero; so, for every , there is a sequence such thatThe latter yieldsThus , therefore .Theorem 2.3 gives a
possibility to construct a partial model representation for in a sense
exposed below. With no loss of generality, one can assume that the Hilbert
scalar product on is such that
the corresponding Gram operator re-denoted as (i.e., ) has the
properties . Let (see (2.5))Then . Additionally, one can assume that the basis from Theorem
2.3 is orthonormalized, on and . Let and be
orthoprojections onto subspaces and , respectively.
Theorem 2.4. Let a space and functions be related with the e.s.f of a -s.a. operator as in Theorem 2.3, and let be a Hilbert space formed as the linear span of and , where the functions in are mutually orthogonal, have unit norm, and are orthogonal to the space which conserves its Hilbert structure. Then belongs to the linear span of for every , and the compression of is similar to the multiplication operator by acting on .
Theorem 2.4 is not new (see, e.g., a more general result in [9] and its applications in [10, 11]) but it opens some ways to generalization and generates some open questions and problems. For instance, (2.4)(c) is restrictive and must be dropped. This, in turn, implies a replacement of the space of scalar functions by a suitable space of vector-valued functions. Next, the passing from a single self-adjoint operator to a family of self-adjoint operators in the case of indefinite metric spaces has some difficulties which must be taken into account. Let us consider the following example.Example 2.5. Let the union of systems , , and give an orthonormalized basis in a Hilbert space . We define a fundamental symmetry on by formulaeThus becomes a Pontryagin space. Next, we define operators and by formulae It is easy to check that the operators and are -self-adjoint and commute. Next, the eigen spectral function (e.s.f.) of has the form where , . Moreover, for every , we have the representationsTherefore, can be considered as an “e.s.f.” of the family ; however,because both integrals are nonconvergent. Next, , but . Thus and , where means the weakly closed algebra generated by the identity operator and the corresponding -self-adjoint operator. The same reasoning shows that there is no -self-adjoint operator for which, simultaneously, and . Therefore, the description of the operator family in discussion cannot be reduced to a representation of one operator. At the same time the existence of the spectral function with (2.20) is not occasional (see Theorem 3.11 below) and gives a possibility to construct a partial model for the whole family.
So one of the problems is to generalize Theorem 2.4 to commutative families of self-adjoint operators in Pontryagin and Krein spaces. Next, it is clear that the space depends on the choice of (2.5), but the latter is not uniquely determined. Thus, another problem is a description of the ambiguity of or its analog with vector-valued functions. The present paper is devoted mainly to these two problems.
Of course, the direction marked by Theorem 2.4 is not unique in relation to the goal to construct a model representation of function type for operators in Pontryagin spaces. In particular, in [12], it was shown that a bounded cyclic self-adjoint operator in a Pontryagin space is unitary equivalent to the operator of multiplication by the independent variable in some space generated by a “distribution” . This distribution, in turn, is associated with a quasiintegral representation for the moment sequence , where is a cyclic vector. The model in consideration describes the behavior of on the whole Pontryagin space, and in this sense, it is complete. On the contrary, the model from Theorem 2.4 is a partial one; it describes only the part of that can be restored via its e.s.f.. It gives, indeed, an advantage to the model [12] but this advantage concerns mainly cyclic operators because in the case of a non-cyclic operator the corresponding space contains matrix-valued functions, and this space has a more or less complicated structure. The latter, however, is not the main obstacle. The experience with a canonic representation of normal operators in finite-dimensional spaces with an indefinite metric (see, e.g., [13, 14]) shows that any attempt to give an observable and, simultaneously, complete description for a commutative family of self-adjoint operators in spaces with indefinite metric with the rank of indefiniteness more than 2 has little chances to prosper. On the contrary, a partial model representation in the style of Theorem 2.4 can be generalized easily to commutative families of -self-adjoint operators and to commutative families of -self-adjoint operators of the so-called -class. A previous experience shows that, in spite of some incompleteness, this model was useful in some applications concerning a single -self-adjoint operator, see Section 7 for more details.
3. Preliminaries
3.1. Krein Spaces
Let be a Krein space with an indefinite sesquilinear form , let be its canonical decomposition, let and be canonical projections and , let be a canonical symmetry, and let be a canonical scalar product. Note that one of these canonical objects uniquely determines the others. Everywhere below, we fix, on , a unique form . At the same time, let us note that, in the question we consider, a concrete choice of Hilbert scalar product is not really essential. One needs only to fix the topology (defined by the above-mentioned scalar product) and the structure of . Let us mention the following result (see [15]) concerning some redefinitions of a Hilbert structure in Krein spaces.Proposition 3.1. Let be a canonical decomposition of the Krein space and let = be another canonical decomposition of the same space. Let the first decomposition define the canonical scalar product and let the second decomposition define the canonical scalar product , that is, . Then spectrum of the operator is strictly positive, and the operator , , has the properties Definition 3.2. The operator that we introduced in Proposition 3.1 is called the canonical isometry that maps the Krein space with the scalar product () on the same Krein space , but with the scalar product . If it is necessary to exactly indicate the corresponding scalar products or the canonical decompositions of the space , we replace with
Below nonnegative (especially maximal nonnegative) subspaces will play an important role. The set of all maximal nonnegative subspaces of the Krein space is denoted .
A subspace is called pseudoregular [16] if it can be presented in
the formwhere is a regular
subspace and is an isotropic
part of (i.e., ).Proposition 3.3. Let (i) and a
pseudoregular subspace,(ii) the isotropic
subspace of ,(iii) a scalar
product on such that the
norm is equivalent
to the original one,(iv),
and where and are uniformly
definite subspaces. Then one can define, on , a canonical scalar product such
that Definition 3.4. If a canonical scalar product
of Krein space has (3.5), it
is said to be compatible with
(3.4) and the choice of the scalar product on .Definition 3.5. Let be a
pseudoregular subspace and letbe two different decompositions of as a direct sum
of a regular subspace and the isotropic subspace. We will call the map , defined by the condition
the standard map associated with (3.6).Remark 3.6. The map is uniquely
determined by the condition (3.7) since the subspaces and are
projectionally complete and the subspace is neutral.
This map preserves the sesquilinear form (i.e., is -isometric).
Furthermore, the subspaces and can be
presented in the formwhere is uniformly
positive, is uniformly
negative andThen the union of (3.6) and (3.8) generates two decompositions of the
type (3.4).Define a special case of
pseudoregular subspaces: a nonnegative (non-positive) subspace is called a subspace of the class () if it is
pseudoregular and for as in (3.3). In
Pontryagin spaces, every subspace is pseudoregular, and every semidefinite
subspace belongs to class or .
Here, the term “operator” means “bounded linear operator.” By the symbol , we denote the operator -adjoint (-a.) to an operator . Thus, if , then is a -s.a. operator. For an operator , the symbol denotes its spectrum treated in the same way as in [17] or [8].
If an operator family is such that the condition implies , then this family is said to be -symmetric. An operator algebra is said to be a -algebra if it is closed in the weak operator topology, -symmetric, and contains the identity . The symbol means the minimal -algebra which contains .
One of the most important directions in the development of the operator theory is connected to the existence of invariant maximal semidefinite subspaces for certain operator sets (see [18] for references) and the study of the properties of the operators in such sets. A subspace is said to be -invariant (-invariant) if it is invariant with respect to the operator (operator family ).Proposition 3.7. Let be an operator that acts in and has an invariant pseudoregular subspace , and let = be also an -invariant. Consider two different decompositions of the space , as a direct sum of the isotropic part and a projectionally complete subspace. Let be two matrix representations of corresponding to these decompositions. Then where is the standard map associated with (3.10).Proposition 3.8. Let be two different canonical decompositions of the Krein space , and let and be invariant subspaces of a -s.a. operator . Then where are associated with Definition 3.2.Definition 3.9. A -symmetric operator family belongs to the class if there is a subspace in such that(i) is -invariant,(ii),(iii).Remark 3.10. If a -symmetric family and is a -invariant subspace corresponding to Definition 3.9, then the pseudoregular subspace is -invariant too.
3.2. Spectral Functions with Peculiarities
The spectral resolution for different operator classes is one of the important problems in the operator theory. Let us start with the following definition.
Let = be a finite set of real numbers and let be the family of all Borel subsets of such that , where is the boundary of in . Let be a countably additive (in the weak topology) function, that maps to a commutative algebra of projections in a Hilbert space , where for every , and , and, moreover, . is called a spectral function on with the peculiar spectral set ; the mention of can be omitted. The symbol means the minimal closed subset of such that for every : and . Besides the symbol , we will use also, as a notation of a spectral function, the symbol , , where = .
A spectral function , that acts in a Krein space, is said to be -orthogonal or -s.a. if is a -orthoprojection for every .
Let us recall the definition of scalar spectral operator with real spectrum [19]. An operator acting in a Hilbert space is said to be a scalar spectral operator if there exists a spectral function with empty peculiar spectral set such that, for every , and = in the weak sense.
Now, let be a spectral function with peculiar spectral set . A scalar function is said to be defined almost everywhere (with respect to ), to have a finite value almost everywhere, and so on, if the corresponding property holds almost everywhere in the weak sense on an arbitrary set , . We will assume that the function is not defined at .
The following theorem was announced in [2] and proved in [20].
Theorem 3.11. Let be a commutative family of -s.a. operators with real spectra. Then there exists a J-orthogonal spectral function with a finite number of spectral peculiarities ( may be the empty set) such that the following conditions hold A spectral function with a peculiar spectral set , satisfying ((3.15)), is called an eigen spectral function (e.s.f.) of the operator family .Definition 3.12. Let be an e.s.f. of an operator family and let an operator and a function be connected by the system of equalities from (3.15)(c). Then the function is said to be the portrait of the operator , and the operator is said to be the original of in (with respect to ).Let a spectral function with a peculiar spectral set be an e.s.f. of . If , then will be called a peculiarity of . Let be a peculiarity. Fix a set : . The peculiarity is called regular if the operator family is bounded; otherwise, it is called singular. Note that the notion of regular and singular peculiarities is correctly defined since the boundedness of the family does not depend on .
3.3. Some Function Spaces
Assume that is a nondecreasing function defined on the segment , continuous in the points , , and , continuous (at least) from the left in all other points of the segment, and having an infinite number of growth points, where zero is one of these points. This function generates, on , a Lebesgue-Stieltjes measure and spaces (, , etc.) of complex-valued functions. At the same time we will consider also some spaces of vector-valued functions; so, from time to time, we will note, after a symbol of a space a symbol of a range for the functions forming this space, for instance, . Next, let us consider a slightly different construction. Let be a -measurable function defined a.e. on and such that a.e. for every , it is true that and , but = . Set The function is nondecreasing in both segments and , but it is unbounded in neighborhoods of zero. Define, for it, a corresponding function space. Let and be arbitrary functions continuous in and vanishing in some neighborhoods (different in the general case for and ) of zero. Then the integral is well defined and generates a structure of pre-Hilbert space on the set of all such functions. The completion of the space will be denoted (or ). Note that, due to (3.16), the spaces and form a Banach pair, so the space is well defined (for details, see [21]).
Let us pass to some notation relating to direct integrals of Hilbert spaces and corresponding model descriptions of self-adjoint operators (see [22, Section 41]; [23, Chapter 7]; [24, Chapter 4.4]; [25, Chapter VII]). We will use definitions close to the “coordinate notation” given in [22]. A difference between [22] and the definitions that follow is related to the fact that direct integrals, here, will be used not only for a resolution of Hilbert spaces but also for a resolution of Krein spaces. Let be some separable Hilbert space ( can be finite dimensional), let be an orthonormalized basis of this space, let be be the same as above. Let be a system of nonnegative -measurable functions defined almost everywhere (a.e) on the segment and such that every function of the system is the indicator of some set of nonzero measure and . DenoteIn this case the sum in the right part of the formula is considered as a formal expression without any suggestion of its convergence or divergence.
Here the space means the space of vector-valued functions defined a.e. (with respect to ) on the segment and taking values in under the conditionswhere runs the set of all -measurable a.e. finite scalar functions such thatThe topology on is introduced by a base for neighborhoods of zero, where any neighborhood of the base is defined by a couple of positive numbers and (the couples are different for the different neighborhoods) and contains all functions satisfying the condition . Next, the symbol means here a Hilbert space of functions such that .
The spaces and are said to be a standard space of measurable functions and a standard Hilbert space, respectively.
The choice of in the construction of spaces and is not essential ifSpaces of this type correspond, in particular, to model representations of self-adjoint operators with uniform multiplicity (see [25]). So and are said to be spaces of uniform multiplicity if conditions (3.20) are fulfilled.
If is not a space of uniform multiplicity, it can be represented as an orthogonal sum of spaces of uniform multiplicity. Note a special case of such representation.Definition 3.13. A space is said to be orderly decomposable on uniform components if (see (3.17))where (both and can occur), the above decomposition of is concordant with the choice of the basis in the sense that , is a space of uniform multiplicity , , = , is the indicator of some -measurable set , , , , , .Spaces orderly decomposable on uniform components play the key role in the theory of model representation for self-adjoint operators (for details, see, [25, Theorem VII.6]). Practically the same definition can be given for spaces . Definition 3.14. is said to be orderly decomposable on uniform components ifwhere is a space of uniform multiplicity, the rest of the elements in (3.22) are the same as in (3.21).We introduce some notation related to multiplication operators by scalar functions. Everywhere below we assume a scalar function to be defined a.e. on , -measurable, and a.e. bounded. For , setIt is clear that , so (3.23) defines on the continuous operator (= the multiplication operator by the function ). If satisfies some additional conditions, one can consider the operator as acting simultaneously on different spaces. If, for instance, is continuous, then the operator is well defined on every space independently of and . If , then can also be taken as a domain of . So, if it is necessary, we will mention, simultaneously, the operator and its domain using the notation , say, .
Let be the multiplication operator by the indicator of the set , , . Pass to the description of automorphisms acting on and commuting with operators , . Denote, by , the subspace of the space spanned by all vectors of the set such that (see (3.17)). The following result is well known (e.g., Birman, Solomjak [23, Chapter 7]; [24, Theorem 4.4.6]; [22, Proposition 1, Subsection 2; Section 41]).
Proposition 3.15. Let be a unitary operator commuting with for a.a. . Then there is an operator-valued weakly -measurable function defined a.e. on such that, for a.e. , the operator is unitary on and Remark 3.16. The space is a complete linear set in . So if initially an operator acts on and satisfies conditions (3.24), its domain can be extended to via the passage to the limit. Thus the operator is a continuous bijective mapping.
4. On a Model Representation for -Orthogonal Spectral Functions without Peculiarities
We define, on , an additional structure of a Krein space. Let be an operator acting on , being at the same time self-adjoint, unitary and commuting with for a.e. . By Proposition 3.15, the operator has the representationwhere is self-adjoint a.e. on . LetThen the inner productconverts the space to a Krein space. This Krein space is denoted and it is said to be a standard Krein space.
Remark 4.1. The product is a.e. well defined both for and .
A standard Krein space can be used for a model representation of a -spectral function (= -orth.sp.f.) without peculiarities. For simplicity everywhere below, we will assume thatDefinition 4.2. Let be a -orth.sp.f. with the empty set of peculiarities. A space is said to be a model space for if, for some canonical scalar product on , there is an isometric and -isometric operator such that, for every ,where, as above, is the multiplication operator by the indicator of the set , . The operator is said to be an operator of similarity.
Proposition 4.3. Every -orth.sp.f. with an empty set of peculiarities has a model space .Proof. By [19, Lemma XV.6.2], one can define, on , a new scalar product (noncanonical in the general case) , topologically equivalent to the initial scalar product and such that the spectral function , is orthogonal with respect to . Then , where the operators commute with for a.a. . Now one can introduce, on , a new canonical scalar product , where is the module of . The new scalar product generates the canonical decomposition . It easy to show that the subspaces and are -invariant for a.a. . Thus there are model spaces and for orth.sp.f. and , respectively. Without loss of generality one can assume that the scalar functions (see (3.17)) and are such that Put . The rest is straightforward.Remark 4.4. As is well known, a model representation for an orth.sp.f. is not uniquely determined. At the same time, all scalar measures for such representations in the case of the same orth.sp.f. are equivalent among themselves and the function (see (3.24)) does not depend on the choice of the model representation. This is the reason why a model representation for orth.sp.f., usually is taken in the class of Lebesgue spaces orderly decomposable on uniform components. In the case of -orth.sp.f., the situation is slightly more complicated. Specifically, for all model representations for a fixed -orth.sp.f., all admissible scalar measures are equivalent among themselves and two functions and do not depend among themselves and of the choice on the model representation. Conversely, if data include a class of equivalent measures, functions and , then a -orth.sp.f. with empty set of peculiarities can be restored up to -isometric equivalence. In particular, one can choose a suitable function , construct a space using and , construct a space using and , and put . Note that, even if the spaces and are orderly decomposable on uniform components, the spaces in the general case do not have this property.Proposition 4.5. Assume that is a -orth.sp.f. with empty set of peculiarities, and are two different standard Krein spaces such that each of them is a model space for and is derived from the same space by canonical symmetries and , respectively. Then, in , there is a unitary operator satisfying (3.24) and such that .Proof. In the general case the operators of similarity and correspond to different canonical scalar products on the same space , say, and , where and . On the other hand thanks to Definition 4.2, is self-adjoint simultaneously with respect to the scalar products and , hence it commutes with and . Thus, taking into account Proposition 3.1, we can assume that . Then the required operator can be defined by the formula . Indeed, first, this is unitary because the canonical scalar product on is the same for and ; second, for every function is true, ; third, . The rest follows from Proposition 3.15.The proof of the next proposition will be omitted because it contains the same well-known ideas as the proof of Proposition 3.15.Proposition 4.6. Let be a -unitary operator in a standard Krein space commuting with for a.a. . Then there is an operator-valued weakly measurable function defined a.e. on and taking values in the set of bounded operators acting in such that, for a.a. ,
5. Unbounded Elements in Hilbert Spaces
First, let us consider the notion of unbounded elements.
Assume that is a Hilbert space, is a resolution of the identity (= an orthogonal spectral function with the empty set of peculiar points) defined on the segment , continuous in zero (with respect to the -topology), andSet , where , .
Next, let be a mapping of the numerical set into (, ). The function is said to be conformed with if the following condition is fulfilled; for every , ; the equality holds, where , .
Note that has the following property:It is clear that the element from (5.2) is uniquely defined by and can be found by the formula .Definition 5.1. A function , which is conformed with , is said to be an unbounded element conformed with (or, if it cannot produce a misunderstanding or an unbounded element) if .Note that unbounded elements conformed with exist if and only if zero is a point of growth for , that is, for every 0 it is true that . Everywhere below in this section, this condition for is assumed to be fulfilled.
For brevity everywhere below unbounded elements will be denoted by symbols , , and so on. For , , we set .
Definition 5.2. Unbounded elements , conformed with a (common) resolution of the identity , are said to be linearly independent modulo if every nontrivial linear combination of them is an unbounded element from .Next, for every bounded -measurable function , one can define, on , the following operator :Using the previous notation, rewrite the last formula asRepresentation (5.4) allows a possibility to treat the operator in a more general sense; unbounded elements conformed with can be naturally included to the domain of . Note that the portrait of an unbounded element can be both a bounded element and an unbounded element. Moreover, in (5.4) could also be taken unbounded. In this case the portrait of a bounded element, optionally, is an unbounded element.Proposition 5.3. If a function is unbounded on every subset of complete -measure, then the vector space contains an infinite number of unbounded elements conformed with and a linearly independent modulo .The last proposition shows (see also below, Proposition 5.4) that if we need to operate with a finite number of unbounded elements, then only bounded functions are admissible in (5.4).
Until the end of the present section, the orthogonal resolution of the identity and a nondecreasing function , such that -measurability on coincides with the -measurability, will be fixed. Note that the existence of follows from the separability of (see [26, Section 76, Theorem 1]).
We give an additional notation. Let be a fixed family of unbounded elements conformed with and a linearly independent modulo . Both the unbounded elements and the ordinary vectors from can be considered as functions defined on and taking values in (see (5.2)). The linear span of vectors from and unbounded elements from , consistently taken as functions on , is denoted . Additionally will be considered as a Hilbert space, where is a subspace with the the same scalar product that was given on from the beginning, and unbounded elements from are mutually orthogonal and orthogonal to . The space is said to be an expansion of (generated by ).
Next, using the system , introduce the function ; The connection between and implies that the function introduced in (5.5) has (3.16). In this case the function from (3.16) can be calculated directly through and .Proposition 5.4. Let be a system of unbounded elements, forming together with the space , let be a -measurable function, let be the operator defined by (5.4). Then if and only if .Proof. Sufficiency of the formulated condition is clear, so we will consider its necessity only. Note that, by virtue of Proposition 5.3, condition implies the boundedness of the function . Next, since for , then, by (5.4) it is true that and, moreover, . Thus (5.5) implies .
For future applications, both the cases andare important.Proposition 5.5. Let be a system of unbounded elements, generating together with the space . Then there are no more than -measurable functions linearly independent modulo such that (5.6) is fulfilled.Proof. Due to (5.6), every element from the system can be uniquely represented (modulo ) as a linear combination of elements from , that is,Thus the operator generates the matrixin addition, if and only if . Hence functions comparable modulo have the same matrix (5.8).
Remark 5.6. For the function , (5.7) is always fulfilled. At the same time the example of the unbounded elements and from the space shows that the case, where there are no functions different from a constant modulo and simultaneously satisfying (5.6), is possible. Moreover, the estimation mentioned in Proposition 5.5 is excessive for every because two arbitrary matrices of the form (3.19) generated by (5.7) commute. For a discursion of the linear dimension of a group of commutative matrices, see, for instance, [27, Part 2, Section 10].
6. A Function Model for a -Symmetric Family of the Class
6.1. Some Remarks
In this section a function model of -symmetric family with real spectrum will be discussed. This model is defined with the help of an e.s.f. of (see Section 3). It is incomplete because the model describes the operator family restricted on some important subspaces (in particular, on the subspace (see (3.15)(d))) and not this family itself. By virtue of Theorem 3.11, it is clear that the general situation can be reduced to the case of -orth.sp.f. with a unique spectral peculiarity in zero. Furthermore, the case of a regular peculiarity is trivial because, under this condition, all operators from are spectral in the sense of Dunford and have a finite-dimensional nilpotent part. Thus is such thatDuring the first stage, assume additionally that, for every closed interval , the following condition holds:Note that (6.2) does not necessarily mean that is a Pontryagin space because the subspace can contain both positive and negative subspaces of infinite dimension. We introduce some notation. LetNote that, by virtue of (6.1)(c), the equality holds, and due to (6.2), the subspace is uniformly positive (recall that the subspace is finite dimensional). Without loss of generality one can assume that, on ,Now, we pass to a detailed analysis of the structure of . First, let andThenthus it is enough to study the behavior of on the subspaceAssume that a canonical scalar product on is such that . In this case, (6.7) is an invariant subspace for .
A direct verification shows that, if (6.5) holds, the operator has the following matrix realization:with respect to (6.7), whereand the operator corresponds to the representationLet be a family of all sets under Condition (6.5). If , then, for the operators and , the condition of the form (6.8) holds simultaneously. Since , thenNow, we discuss the spectrum multiplicity of the family . Recall that a subspace is said to be cyclic with respect to if .Definition 6.1. In what follows, a nonsingular multiplicity of -orth.sp.f. means the minimal dimension of all cyclic subspaces with respect to .Remark 6.2. Due to the choice of one can assume that a cyclic subspace is taken such that , so the nonsingular multiplicity of coincides with the multiplicity (in the ordinary sense) of the (orthogonal) spectral function .Proposition 6.3. If (6.1) and (6.2) are fulfilled and the nonsingular multiplicity of is greater than , then there is a decomposition such that , , for every , the subspace is uniformly positive and the -orth.sp.f. has nonsingular multiplicity less than or equal to .
Proof. Since and (6.2) holds, there is a sequence of disjoint subsets such that a condition of the type (6.5) holds for all , , and for every , the equality is true. It is clear that . Let be some orthonormalized basis in , . Set, . We show that the subspaces and are as desired. First, we prove that, for every set , the equalityholds. Indeed (see (6.11)), if and , then Next, it is clear that is an invariant subspace with respect to the operators , therefore, the subspace has the same property. Thus, taking into account (6.13), one can conclude that is invariant with respect to .
Remark 6.4. Proposition 6.3 shows that, in some problems one can assume that the nonsingular multiplicity of is finite. At the same time, this hypothesis is not convenient in many cases because the decomposition is not uniquely defined and, moreover, the subspace can be always extended saving all properties enumerated in Proposition 6.3. Basically, Proposition 6.3 gives a possibility to illustrate peculiarities of -orth.sp.f. in question using -orth.sp.f. with finite nonsingular multiplicity.In addition to (6.3), set Now, let and be some standard Hilbert space and some standard space of measurable vector-functions, respectively, and let be a finite collection of vector functions such thatBelow is the linear span formed by and the collection , where (see Section 5) the functions from are assumed to be normalized, pairwise orthogonal, and orthogonal to , that is, on , a structure of Hilbert space is defined. Note that, sometimes, will be considered without its Hilbert structure as a vector subspace from but this case will be noted explicitly.
6.2. Functional Model of (A Special Case)
Theorem 6.5. If, for -orth.sp.f. , (6.1) and (6.2) hold, and for a canonical scalar
product , (6.4) holds, then there exist a standard Hilbert
space , a collection satisfying
(6.16) which generates the expansion of , and an isometric operator : , such that, for
every , the following representations take place (see
(6.10)): where , Proof. First, is an
orthogonal (in the ordinary Hilbert sense) spectral function, so there is a
space such that the
operator is similar to
the operator . Let a space be already
chosen and let an isometric operator : be such that . Now, it is necessary to find a collection of
unbounded elements , so that the corresponding space and the
operator would be as
desired. Choose, in , some orthonormalized basis . Let be a closed
interval. Consider the expression , where . It can be considered as a continuous linear
functional acting in or,
equivalently, in . By the well-known theorem of Riesz on the general
representation of continuous linear functional, there is a function such thatwhere or (thanks to
properties of the spectral function)Condition (6.20) means that there is a function such that, for
every interval as defined
above,where is the
indicator of the interval . The collection is as desired.
We show this. First, we prove that the set is linearly
independent modulo . Suppose the contrary, that is, suppose the existence
of a collection of coefficients such
thatLet . By the definition of the space , there is a sequence such that for . Set . Note that in some
neighborhood of zero, therefore, the representationis valid, hence ,These equalities and (6.22) imply
which is a contradiction, that is, (6.22) is impossible.
Second, form according to
the above procedure using the space and the
collection of unbounded elements . Next, setIn this case the dual operator is defined by
the equalitiesNow, pass to (6.17). Assume (as above) that for a closed
interval the condition holds. Then, in
concordance with (6.8) and (6.19), one has , where , , so , . The last equalities, (3.21), and (3.23) imply (see
the notation introduced in (6.15)) , . Due to the choice of , the rest follows from (6.9) and (6.27).Definition 6.6. If -orth.sp.f. satisfies (6.1)
and (6.2), and the space has (6.17),
then is said to be a basic model space for (compatible
with (6.3) and (6.15)), and the operator is said to be
an operator of similarity (generated by ).
Theorem 6.7. Let a space be the
expansion of a standard Hilbert space generated by a
collection satisfying
(6.16). Then there exist a Pontryagin space and a -orth.sp.f. on such that is a basic
model space for .Proof. Let be the set of
triples with and . For and , put
Next, let be the
canonical basis in . Then, for intervals , where and , we put
Direct verification shows that generates a -orth.sp.f.
with (6.1) and (6.2). Let us check thatWe denote, by the symbol , the set of all functions from vanishing near
zero, and for every , we set . Then the space is the closure
of the setLet a system be an
orthonormalized basis in . Thenand (6.31) can be rewritten aswhere, by (6.32), Note that
system also has
(6.16).
Initially, the
basis was arbitrary
but now we pass to choose it in a special way. Since the elements from are unbounded,
there is a sequence such that and for every . Let . Since , one can assume that there exists , and put . Now, we apply the same scheme for defining . Let . Then for every . Since the elements from are unbounded
for every choice of , there exists a (new) sequence such that and for every . Let . Since , one can assume that there exists , and put . Then . The rest of the proof of (6.30) is now evident.
Next, we put for every , and for . The rest is straightforward.We note that a basic model space for a given -orth.sp.f. is
not uniquely defined and discusses the arbitrariness for the choice of a
concrete model. The construction of the basic model space was started from
(6.3) and these depend on the choice of a canonical scalar product. So the
first question is the following: does the basic model space depend on a choice
of canonical scalar product? In order to answer this question, note, first,
that does not depend
on any canonical scalar product and is calculated using only the Hermitian
sesquilinear form and -orth.sp.f. . Second, one can consider the factor space /. Then, by virtue of (6.2), the sesquilinear form generates in it
the structure of Hilbert spaces, and the spectral function induced on / by the operator
function is orthogonal.
Third, by Definition 4.2 and Proposition 4.3, this induced function is similar
to the spectral function . Summarizing all mentioned arguments, we obtain the
following proposition.Proposition 6.8. If is a basic
model space for -orth.sp.f. satisfying
(6.1) and (6.2), and is the
corresponding operator of similarity, then the orth.sp.f. , induced on the Hilbert factor space by the -orth.sp.f. , is similar to the operator family acting in , and the corresponding operator of similarity is induced by
the operator .Thus the choice of as a source for
constructing a basic model space is realized in
the same class of spaces as for the orth.sp.f. and does not
depend on a canonical scalar product on .Proposition 6.9. Let spaces and be expansions
of a space generating,
respectively, by collections and with (6.16),
let both and be a basic
model space for the same -orth.sp.f. with (6.1) and
(6.2), and let and be the
corresponding operators of similarity. If for every function , the relation holds, then the spaces and coincide as
vector subspaces in .Proof. The
complete proof will be divided in several stages. First, consider the
caseThis equality means that, in particular, the basic model spaces and were
constructed according to the procedure mentioned during the proof of Theorem
6.5 using the same decomposition , but, generally speaking, a different choice of
scalar products , , and (or) basis in . PutSince the Hilbert structure of the spaces and is not in
question, only the choice of the bases and in is important
(if scalar products are different then corresponding bases are necessarily
different). LetThen, by (6.19) and (6.21)where . But and are the
coefficients of decomposition for the same vector with respect to bases and , respectively. Thus (6.37) gives the collection of
equalities , , henceSince the segment is arbitrary
and runs through
the set of all functions from , this equality givesThus
. Since the spaces and have an equal
status, the inverse inclusion is also true. So and coincide as
subsets of . The proposition is proved under (6.35).
For the second
stage assume that the model spaces and are constructed
using different decompositions of the space , that is, , , but the basis in and, moreover,
the scalar product in for both spaces
are the same. ThusNext, the space can be
represented in the formwhere is a linear
operator. It is clear that , where is a vector set
from . Note that the equalityholds for all . Indeed, if , then by (6.42), , where . This implies , and thanks to (6.34), one has . Since is a one-to-one
mapping, (6.43) is proved.
Now, let . Then by (6.15), the representation implies , where (cf., (6.19))so if (, thenThe last equality can be rewritten as where , . On the other
hand, by (6.15) and (6.43), the representation and the
equality implyThe comparison between (6.46) and (6.47) givesSince the function is subjected
only to the condition and the closed
segment is arbitrary,
then , . The proposition is proved under Hypothesis (6.41).
The general case can be reduced to the two particular cases considered above,
so the rest is straightforward.
The following proposition is a partial inverse of Proposition 6.9.Proposition 6.10. Let spaces and be the expansions of a space generated, respectively, by collections and satisfying (6.16). If and coincide as vector subspaces in and the space is a basic model space for some -orth.sp.f. with (6.1) and (6.2), then the space is also a basic model space for .
Proof. Let be the operator
of similarity for that
corresponds to the basic model space . The scheme of the proof will be the same as for
Proposition 6.9; two particular cases will be considered and the superposition
of these will give the general case.
So let a system be such
thatSet . The corresponding decomposition (generated by ) will be taken
also for the construction of an operator of similarity corresponding
to . First, set for .
Next, for , , one hasthis, jointly with (6.49) implieswhere , . The vector system is taken as the
new orthonormalized basis in (so, generally
speaking, the Hilbert structure on is redefined).
Set , . Case (6.49) has been finished.
Now let , , . Putwhere and , that is, . Note also that the Hilbert structure of does not
change. It is easy to check that the operator and the
decomposition are as
required.Remark 6.11. The proof of Proposition 6.10 shows
that, if is a basic
model space for -orth.sp.f. with (6.1) and
(6.2), is a
corresponding operator of similarity, and and are conformed
with the decomposition , where , then for every different decomposition it is possible
to find a new Hilbert structure of the vector set (evidently this
change does not touch ) and a new
isometric map (an operator of similarity) : such that and (e.g.,
(6.34)) (. An analogous proposition can be formulated for a
change of the given Hilbert structure on .Remark 6.12. Propositions 6.9 and 6.10 show that the Hilbert structure of the basic model
space introduced
during an expansion of is not, in some
sense, really important. On the other hand, an example mentioned below
demonstrates that spaces and obtained as
different expansions of a common space can be basic
model spaces for the same -orth.sp.f. and be
different vector subspaces in . Thus (6.34) in Proposition 6.9 cannot be omitted.Example 6.13. Let be the totality
of triples , where and are some fixed
abstract vectors , , so the measure defining the space is the standard
Lebesgue measure. For and , put
Next, letfor , and let for .Direct verification
shows that is a -orth.sp.f.
with (6.1), (6.2). In the capacity of a basic model space for , one can take, first, the space generated by and , or, second, the space generated by and . In the first case, and in the second
case, ( and are the
corresponding operators of similarity). At the same time, the spaces and form different
vector subsets in because and , that is, unbounded elements and are linearly
independent modulo .
Theorem 6.14. Let spaces and be the
expansions of a space generated,
respectively, by collections and with (6.16).
Then and are basic model
spaces for a common -orth.sp.f. with (6.1) and
(6.2) if and only if there is a unitary-valued function of the type
(3.23) such that the sets and coincide as
vector spaces in .Proof. Necessity By virtue of Remark 6.11, one can assume that, for model spaces , , and corresponding operators of similarity and , there exist the same representation and the same
Hilbert structure on . In this case, the operator , is well
defined. Direct verification gives , which implies . Since the subspace is invariant
both for and , these two operators commute on . Moreover, for every closed interval , the condition holds for all , . The rest follows from Proposition 3.15.Sufficiency
By
Theorem 6.7, one can construct a Pontryagin space and a -orth.sp.f. such that is a basic
model space for . The rest follows from (3.23) and Proposition 6.10.
The results obtained above show that the choice of a basic model space for -orth.sp.f. with (6.1) and (6.2) is reduced (up to a finite number of functions) to the choice of a standard Hilbert space . The last choice is not uniquely defined. The arbitrariness of the choice of can be diminished if one takes in consideration only the standard Hilbert spaces orderly decomposable on uniform components. In the latter case, the choice is reduced (see [25, Theorem VII.6]) to the choice of scalar measure in the corresponding class of equivalent measures. Note that, if is orderly decomposable on uniform components, then it is naturally embedded to the space and, therefore, this space is also orderly decomposable on uniform components. Moreover, the space does not depend on but on the corresponding class of equivalent measures. This reasoning leads to the following theorem.
Theorem 6.15. Let be a basic
model space for -orth.sp.f. with (6.1) and
(6.2), and let the space be orderly
decomposable on uniform components. Then every basic model space for constructed as
an expansion of a standard Hilbert space orderly decomposable on uniform
components has the following form; where is a fixed
scalar function satisfying the conditions 0 a.e. on , and , and the operator-valued function is subject to
(3.24). Conversely, every functional vector space of the form (6.56) is a basic
model space for .Remark 6.16. Let us
consider an interpretation of Theorem 6.15 for the case of -orth.sp.f. with simple
nonsingular multiplicity (in particular, can be cyclic
although this condition is not necessary). In this case, converts to the
space of scalar
functions, and converts to the
multiplication operator by a function , where is a -measurable
function. Next, let be a collection
of unbounded elements generating the expansion that is a basic
model space for . Then every basic model space for constructed as
an expansion of a standard Hilbert space orderly decomposable on uniform
components has, as a subset of , the following form:Concluding this part, let us consider an estimation of the norm of a -orthogonal
projection , where the set satisfies (6.5)
and the conventions given above on the choice of the canonical scalar product
on are
preserved.Proposition 6.17. If, under the mentioned above
conditions, , then .Proof. If , when the desired formula directly follows from
(6.8), (6.9), so we assume that . LetThen . On the other hand taking into account (6.9), (6.11),
we have . A first step is an estimation of . Since is an
orthogonal projection, it is clear that it is enough to consider the case . Next, . ThusThe proof will be finished if vectors and , satisfying (6.58) and converting (6.59) to an
equality, can be found.
Since the
operator is finite
dimensional, there is a vector , such
thatLetSince (see (6.9), )is well
defined. Then
Finally (see (6.60)), .
Proposition 6.18. Let be a basic model space for -orth.sp.f. with (6.1) and (6.2), and let be the corresponding system with (6.16). Then Proof. Let be the orthonormalized basis in corresponding to the system . Then the operator has, with respect to this basis the matrix representation ( with the elements . The rest is a usual estimation for the norm of a positive matrix.
6.3. A Functional Model for (The General Case)
In the previous subsection, we studied -orth.sp.f. with (6.1), (6.2). Now, we turn to a more general case dropping (6.2). It implies that (6.9) and (6.11) must be substituted by some new ones. Let us conserve (6.3), (6.15). However, it is necessary to take into account that, now, generally speaking, the subspace is indefinite. Recall that -orth.sp.f. belongs to the class , so there is an -invariant pair of -orthogonal maximal semi-definite pseudoregular subspaces and with finite-dimensional isotropic part; moreover, due to (3.15)(b), we can assume that, for every closed interval , the subspace is positive and the subspace is negative. Thanks to the hypothesis, the following subspaces are well defined:Setand assume that a fundamental scalar product on is simultaneously canonical for the subspace and, on the last space, compatible (see Definition 3.4) with the given decomposition of the subspacesThuswhere the operator is the same as in (6.10), and is a canonical symmetry of the form on . Then, for satisfying (6.5), the elements of the matrix realizationsatisfy (cf. (6.9)) the following relationsHere, is the operator -adjoint to the operator (note that now is a Krein space). We note also that (6.11) remains valid.
Now, let be a standard Krein space (see Section 3) and let be a system of unbounded elements conformed with the operator-valued function and linearly independent modulo (i.e., satisfies (6.16)). Denote, by , the linear span generated by the space and the system . Define, on , structures of Hilbert and Krein spaces in the following way: on both structures coincide with the original structures, and functions of the system are, by definition, positive (as elements of the Krein space), mutually orthogonal and -orthogonal, normalized and -normalized, and orthogonal and -orthogonal to . The space is said to be the expansion of (generated by the collection ).Theorem 6.19. If a -orth.sp.f. satisfies (6.1) (but not (6.2)) and a scalar product on is compatible with (6.66), then there are, first, a subspace and a system with (6.16) forming together the space , and, second, an isometric -isometric operator : , , such that, for every , where , Proof. Since the scalar product on is canonical and conformed with the decomposition , the spectral function is simultaneously orthogonal and -orthogonal. Then for , by Proposition 4.3 and in concordance with Definition 4.2, there exist a model space and an operator of similarity . The rest of the proof is analogous to the corresponding stage of the proof for Theorem 6.5 taking into account that (6.9) must be substituted by (6.69).
Remark 6.20. Under the conditions of Theorem 6.19, the operator function is similar not only to the operator function but also to the operator-function , where the space is formed by the standard Hilbert space (it is the same as for -) and the system of unbounded elements . Thus the properties of the operator-valued function (as well as for do not depend on either (6.2) is fulfilled or not for the operator-valued function . However this dependence exists for the -orth.sp.f. , and this fact is connected with the different structure of the operator in (6.9) and (6.69). Note, in particular, that, if (6.2) is not fulfilled, then the estimation for the norm of operator given in Proposition 6.18 remains true, but at the same time, the equality for from Proposition 6.17 is, generally speaking, incorrect.Example 6.21. Take the space coinciding with , denote the canonical basis of this space, that is, , , and define the canonical symmetry by the equalities , , , , , and the spectral function by the relations(i) for every ;(ii)if , then , , , for , and for .For , one has and . Note that, here, the values and have the same order of growth for . Note also that for every closed segment .Definition 6.22. If, for (6.3) and (6.65), a relation between a -orth.sp.f. satisfying (6.1) and a space is given by (6.70), then is said to be a basic model space for (compatible with (6.3), (6.15), and (6.66)) and the operator is said to be an operator of similarity corresponding to this space.Note that Definition 6.22 does not contradict Definition 6.6, but amplifies it; if, for -orth.sp.f. , (6.2) is fulfilled, then , so the canonical scalar product is uniquely defined on and coincides with .
Now, we show that every space can be considered as a basic model space for a suitable -orth.sp.f. .
Theorem 6.23. Let a space be the
expansion of a standard Krein space generated by a
collection satisfying
(6.16). Then there exist a Krein space and a -orth.sp.f. on such that and is a basic
model space for .Proof. Let be the set of
triples with and . For and , put (see
(4.3))
Next, let be the
canonical basis in . For the intervals , where and , we put
Direct verification shows that generates a -orth.sp.f.
satisfying (6.1). The rest of this proof looks like the corresponding part of
the proof of Theorem 6.7 with some evident modifications.Next, the basic model space is constructed not only via the -orth.sp.f. itself and via
the subspaces and naturally
generated by (note that, in
this case, it is convenient to consider the mentioned subspaces as linear
topological spaces), but also via some other subspaces and via the Hilbert
structure introduced on and that can be
defined with some ambiguity.Proposition 6.24. Let a space (i.e., the expansion
of standard Krein space generated by a
collection with (6.16)) be
a basic model space for a -orth.sp.f. compatible with
(6.3), (6.15), and (6.66). Then for every different
decomposition where and are,
respectively, uniformly positive and uniformly negative subspace, and and are -invariant,
there is a collection with (6.16)
such that the expansion - of generated by is also a basic
model space for compatible with
(6.3), (6.15), and (6.73).Proof. Note
that a collection of unbounded elements is uniquely
defined by a set composed by the following components: a choice of a normalized
basis in , a decomposition , and an isometric -isometric map : . This collection always exists for the mentioned set,
so it is enough to show the existence of . Consider two cases.
(1) Let . Thus, on the Krein space , there are two different -invariant
canonical decompositions. Then, thanks to Proposition 3.8 and (6.70)so one can take .
(2) Let and let the
pairs and be related by a
condition of the type (3.9). Then by Remark 3.6 and
Proposition 3.7, one can take .
Now, let us
pass to the general case. Let and . Then , and one can takeCorollary 6.25. Assume that is a -orth.sp.f.
satisfying (6.1), (6.3) is fixed, and is a standard
Krein space. Then, independently of any realization of (6.73), an expansion of , that is a basic model space for compatible with
(6.73), exists if and only if there is a -isometric
operator such that Remark 6.26 (cf., with Proposition 6.8). Let be the
factor-space of generated by . Then is a Krein
space and induces, on
this space, the -orth.sp.f. . Note that the unique peculiarity of is regular and
can be removed, so, as a slight abuse of terminology, one can say that has no
peculiarities. It is clear that in (6.76)
exists if and only if there exists a -isometric
operator such that . Taking this reasoning and Corollary 6.25 into
account, one can say that an arbitrariness in the choice of a basic model space
for -orth.sp.f. satisfying
(6.1) does not depend on the choice of but essentially
depends on an arbitrariness in the choice of a basic model space for -orth.sp.f. with the empty
set of peculiarities. The last problem was considered in Section 4.Proposition 6.27. Assume that spaces and are constructed
on the base of the same space passing by the
following way: first, introduce on it two (maybe different) sesquilinear forms
by fundamental symmetries and ; second, expand these two standard Krein spaces using
collections of unbounded elements and , respectively. Assume also that, first, and are basic model
spaces for a -orth.sp.f. , second, and are
corresponding operators of similarity, and, third, every function satisfies
(6.34). Then and the spaces , and coincide as
subsets in ).Proof. Condition (6.34) implies that is true for
every , so . Hence . The rest of the reasoning is similar to that of
Proposition 6.9.Comparing the last proposition
with Proposition 6.10, note that the fulfilment of (6.34) is essential for the simultaneous
coincidence with as subsets in , and with as standard
Krein spaces, but at the same time is not necessary.Theorem 6.28. Assume
that spaces and are constructed
on the base of the same space passing by the
following way: first, introduce on it two (maybe different) sesquilinear forms
by fundamental symmetries and , and second, expand these two standard Krein spaces
using collections of unbounded elements and , respectively. Then and are basic model
spaces for a common -orth.sp.f. satisfying
(6.1) if and only if there exist operator functions and , , with (3.24) and (4.7), respectively, such that the
sets and coincide as
subsets of .The proof of this
theorem is similar to that of Theorem 6.14.
6.4. The Passage to -Symmetric Operator Families
Up to this point, we discussed model representations not for a family but for a -orth.sp.f. with the unique spectral peculiarity in zero subjected to the conditionNow, let be a -symmetric commutative operator family. Then its linear span contains a family (not uniquely defined) of -s.a. operators such that the linear span of coincides with the linear span of . In [20], it was proved that there is the decomposition , where and are -invariant, all operators from have real spectra, , and, at least, one -s.a. operator from has no real points in its spectrum. Let be an e.s.f. for . Extend the action of for all setting . This extended function is said to be an e.s.f. of . Comparing Theorem 3.11 and Theorem 6.19, it is easy to obtain the following result.Theorem 6.29. If is a commutative -symmetric family, its e.s.f. satisfies (6.1), a canonical scalar product on is compatible with (6.66), is a basic model space for , and is a corresponding operator of similarity, then, for every operator , there is a function such that where , the space and the operator are defined via (6.15) and (6.70), , and is the multiplication operator by also acting in the space .Remark 6.30. If, under the conditions of Theorem 6.29, -orth.sp.f. satisfies (6.2), then the space must be substituted by the space , and, respectively, the operator must be substituted by the operator .Definition 6.31. If is a commutative -symmetric family and (satisfying (6.1)) is its e.s.f., then a basic model space for is said to be also a basic model space for .Remark 6.32. Assume that is a commutative -symmetric family, its e.s.f. satisfies (6.1), and and are, respectively, a basic model space and a corresponding operator of similarity for . If, under these conditions, an operator and a function are related by (6.78), then, according to Definition 3.12, the function is a portrait of the operator (with respect to , and the operator is an original (not unique in the general case) of in . It is clear that the portrait of an operator does not depend on the choice of the basic model space and can be found via (3.15)(c), but at the same time depends on the choice of .Remark 6.33. Theorem 6.29 brings a natural problem concerning a characterization of functions that can be portraits for operators from a given commutative -symmetric operator family with a fixed choice of -orth.sp.f. . A partial answer to the problem is contained by Propositions 5.4 and 5.5. Indeed, under the present conditions, the function from (5.5) has the form
7. Closing Remarks
Spectral functions are a traditional source for constructions of model spaces for different classes of normal operators in Hilbert spaces, so it is natural to use the same approach in Krein spaces. The first theorem concerning the existence of spectral functions for -s.a. operators was published by Krein and Langer in [28] (see also [6] for detailed proofs). The spectral functions introduced in Section 3.2 are a particular case of generalized spectral functions [1]. Our definition is inspired by spectral measures arising in the operator theory in indefinite metric spaces (see [3, 6, 8, 29]) but formally, independent (see [30]) of this Theory. Another development related with generalized spectral functions can be found in [31–34]. There are some works on model representations for self-adjoint operators and algebras in Pontryagin spaces (the majority of them consider the case with the rank of indefiniteness 1) [35–40], (see also [41] for more references). In [12], the case of a self-adjoint cyclic operator acting in an arbitrary Pontryagin space was considered. The class that is investigated in the present paper differs from the well-known class of definitizable operators (see [42] for discussion) and represents another natural generalization of the class of self-adjoint operators in Pontryagin spaces. The notion of unbounded elements was given by the author in [43], thereupon, this notion was used in [9], where the existence of a basic model space for a single -self-adjoint operator was proved. The latter result was applied in [10] (devoted to a description of a broad functional calculus for -self-adjoint operators) and [11], where the bicommutant problem for a -self-adjoint operator was studied. Some remarks concerning a model representation of -unitary operators were given in [44]. A connection between the spaces of the type and operator algebras was pointed out for the first time in [45] (for Pontryagin spaces with the rank of indefiniteness 1) and [46]. On this subject, see also [47, 48]. Although the concept of basic model spaces for commutative families of -self-adjoint or -self-adjoint operators of the -class was used by the author in some previous papers, its complete description and corresponding proofs as well as the discussion of the ambiguity of basic model spaces are given here for the first time.
Acknowledgments
This work was completed with the support of Project FONACIT (Venezuela) G-97000668. The author is very grateful to Tom Berry and Luís Mata Lorenzo for their assistance in making this paper more readable. The author would like also to thank the referee of this paper for his/her very useful comments.