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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 890657, 14 pages
On a Class of Self-Adjoint Compact Operators in Hilbert Spaces and Their Relations with Their Finite-Range Truncations
Instituto de Investigacion y Desarrollo de Procesos, Universidad del Pais Vasco, Campus de Leioa, P.O. Box 644, 48080 Bilbao, Spain
Received 24 July 2013; Accepted 6 September 2013
Academic Editor: S. A. Mohiuddine
Copyright © 2013 M. De la Sen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
This paper investigates a class of self-adjoint compact operators in Hilbert spaces related to their truncated versions with finite-dimensional ranges. The comparisons are established in terms of worst-case norm errors of the composite operators generated from iterated computations. Some boundedness properties of the worst-case norms of the errors in their respective fixed points in which they exist are also given. The iterated sequences are expanded in separable Hilbert spaces through the use of numerable orthonormal bases.
Compact operators in infinite-dimensional separable Hilbert spaces are of relevance in the study of certain relevant applied problems in control theory and signal theory, . A natural property of such operators is that they can be represented with expansions using two orthogonal or orthonormal reciprocal bases of the separable Hilbert space. If the bases are orthonormal then both of them coincide so that this basis is autoreciprocal and then the formal study is facilitated [1, 2]. Many of the involved operators in mapping map an input space into an output space in the above problems are in addition self-adjoint. Another property of such operators is that they admit truncations using a finite number of the members of the orthonormal basis so that the truncated operators are also compact in a natural way, [1, 2]. The truncated operator describes a natural orthogonal projection of the involved vectors of the Hilbert space into a finite-dimensional space whose dimension is deceased as the number of members of the basis used for its representation decreases. On the other hand, important attention is being devoted to many aspects of fixed point theory in metric, Banach, and more general spaces including the study of mappings being contractive, nonexpansive, asymptotically contractive, asymptotically nonexpansive, quasi-nonexpansive, Kannan and Meir-Keeler and cyclic-type contractions, and so forth. Also, it has been studied the relevance of the theory in properties in both general theory and applications such as the existence and uniqueness of solutions in differential, difference, and hybrid equations as well as in continuous-time, discrete-time, and hybrid dynamic systems, stability theory in the above problems [3–7], the existence/uniqueness of fixed points and best proximity points, and the boundedness of iterated sequences being constructed through the maps and the convergence of such iterated calculations to limit points. See, for instance, [3–6, 8–15] and the references therein. The investigation of existence and uniqueness of common fixed points and best proximity points for several mappings and related properties is also important [10–12]. The study of fixed and best proximity points has also inherent study of convergence of sequences to such points. Other studies of properties of convergence of sequences and operator sequences have been described in different problems as, for instance, the research on approximating operators and approximation theorems that of sigma convergence of double sequences or that of lamda-statistical convergence and summability. See, for instance, [13–17] and the references therein.
This paper is devoted to the investigation of self-adjoint compact operators in separable Hilbert spaces, their finite-dimensional truncated counterparts, and the relations in-between the corresponding properties for the norms of the mutual errors end the errors in-between the corresponding fixed points and their respective convergence properties when iterated calculations through the operators are performed. Some examples of interest in signal theory and control theory are also given. The operators and the iterated sequences constructed through them are studied by using the expansions of the operators and their finite dimensional truncated versions by using a numerable orthonormal basis of the involved Hilbert space.
2. Preliminaries and Main Results
The following result includes some properties related to the approximations of and through orthonormal systems of different dimensions, complete orthonormal systems in , and orthonormal basis, that is, a maximal orthonormal system; that is, it is not a proper subset of any orthonormal system of , where and are an inner product space and a Hilbert space, respectively. Note that in the case where is separable, a complete orthonormal system is always an orthonormal basis and vice versa.
Lemma 1. Let be an inner product space of inner product (or ) endowed with a norm defined by for any , where , let and be a finite orthonormal system in and a given finite or numerable sequence of scalars, respectively, and let and be given integers fulfilling . If then is, in addition, assumed to be numerable. Then, the following properties hold for any .(i).(ii).(iii).(iv) any integers .(v)If is a finite-dimensional Hilbert space of dimension and , for all and , then (vi)If is a finite-dimensional Hilbert space of dimension and , for , then (vii)If is a separable infinite-dimensional Hilbert space and , for , then If, in addition, , then as. If, furthermore, there is some integer such that the real sequence converges to zero exponentially according to , for , then for any given with being some real constant and being a bounded constant dependent on satisfying .
Proof. Properties (i)-(ii) follow from the best approximation lemma since Property (iii) is a direct consequence of subtracting both sides of the relations in Properties (i)-(ii). Property (iv) is Pythagoras theorem in inner product spaces. Property (v) (Bessel’s inequality) follows directly from Property (i) with the orthonormal system in the Hilbert space being a basis of . Property (vi) follows from Properties (ii)–(iii) with ; and the orthonormal system in being an orthonormal basis of since one gets from Property (i) and from (5), Property (ii), and , Hence, Property (vi). Property (vii) follows from the assumption that the infinite-dimensional Hilbert space is separable and Property (vi) leads to which holds under, perhaps, eventual reordering of the elements of the orthonormal basis of which is a complete orthonormal system for the separable Hilbert space . If there is some integer such that the real sequence converges to zero exponentially, then where , for all with being dependent on such that . Hence, Property (vii).
Note that Property (vi) of Lemma 1 quantifies an approximation of an element of a finite-dimensional Hilbert space via an orthonormal system in of smaller dimension than that of such a space. Property (vii) relies on the approximation of an element in an infinite-dimensional separable Hilbert space by using a numerable orthonormal basis of .
Lemma 2. Let be a linear, closed, and compact self-adjoint operator in an infinite-dimensional separable Hilbert space with a numerable orthonormal basis of generalized eigenvectors . Then, the following properties hold:(i),
for all for any , where ; the spectrum of the operator is defined by , for all and with as , for all .
If is the orthogonal projection operator of on the one-dimensional subspace generated by the eigenvector then If is the orthogonal projection operator of on the -dimensional eigensubspace , then with where , for all , for all .(ii)If, in addition, , then
Proof. Note that there is a numerable orthonormal basis for since is separable and infinite dimensional. Such a basis can be chosen as the set of generalized eigenvectors of the linear self-adjoint since it is closed and compact and then bounded
Also, since the linear operator is closed and compact, the spectrum of is a proper nonempty (since is infinite dimensional and bounded since it is compact) subset of and numerable and it satisfies , with , where , , and are the punctual, continuous, and residual spectra of , respectively. Note that is also an accumulation point of the spectrum since is infinite dimensional and is compact. Also, since is separable, the spectrum of is numerable, and ; for all , one gets where is an eigenvalue of ; that is, , associated with the eigenvector since so that so that, except perhaps for reordering, , for all with since is separable and is numerable. Assume that for any positive integer the following identity is true: Then, since is an orthonormal basis of generalized eigenvectors, where is the Kronecker delta. Then, . Furthermore, is compact as it follows by complete induction as follows. Assume that is compact, then it is bounded. Note also that is self-adjoint by construction and then normal. Thus, is compact since it is a composite operator of a bounded operator with a compact operator . Then, by complete induction, as , for any since is compact and is infinite dimensional. Also, where is the projection operator of on the one-dimensional subspace generated by the eigenvector so that as , for all . Thus, Property (i) has been proved. To prove Property (ii), take an orthonormal basis associated with the set of finite-dimensional eigenspaces of the respective eigenvalues. Note from Cauchy-Schwarz inequality that for some real constant , where is a nondecreasing sequence of finite nonnegative integers defined by being built such that each for accounts for the total of the dimensions of the eigenspaces associated with the set of eigenvalues previous to for after eventual reordering by decreasing moduli. Then, , for all , and where is now a set of linearly independent elements belonging to the orthonormal basis of that generate the eigenspace associated with with being an eigenvector and is a set of complex coefficients. Then, as , for all from (20), so that . If there are some multiple eigenvalues, with all being of finite multiplicity since the operator is compact, the above expression may be reformulated with projections on the finite-dimensional eigenspaces associated to each of the eventually repeated eigenvalues leading to , for all . Note that is the finite (1)-dimension of the eigenspace associated with , where is one-dimensional if is single. Finally, it follows from (19) that and Property (ii) has been proved.
Lemma 2 becomes modified for compact operators on a finite-dimensional Hilbert space as follows.
Lemma 3. Let be a linear closed and compact self-adjoint operator in a finite-dimensional Hilbert space of finite dimension with a finite orthonormal basis of eigenvectors of . Then, the following properties hold.(i)
for any , where ; the spectrum of the operator is defined by , for all and , for all .(ii)If, in addition, for some real constants and , then
Outline of Proof. First note that the spectrum of is nonempty since the operator is self-adjoint. Note also that, since the Hilbert space is finite-dimensional Hilbert space, any set of normalized linearly independent eigenvectors of a self-adjoint operator is an orthonormal basis of such a Hilbert space . Property (i) is a direct counterpart of Property (i) of Lemma 2 except that can be a value of the punctual spectrum of but it is not an accumulation point of such a spectrum since the Hilbert space is finite-dimensional. Therefore, the result as of Lemma 1 does not hold. Then, Property (i) follows directly from the above considerations. Property (ii) follows from the relations
Remark 4. It turns out that Lemma 2 (ii) and Lemma 3 (ii) also hold if is not self-adjoint since the corresponding mathematical proofs are obtained by using an orthonormal basis formed by all linearly independent vectors generating each of the subspaces. However, if the operator is not self-adjoint or if it is infinite dimensional while being self-adjoint, the set of (nongeneralized) eigenvectors is not always an orthogonal basis of the Hilbert space.
In the following, we relate the properties of operators on with their degenerate versions obtained via truncations of their expanded expansions.
Theorem 5. Let be a separable Hilbert space and let be a linear degenerated -finite-dimensional approximating operator of the linear closed and compact self-adjoint operator . Then, the following properties hold.(i)Assume that , for all for some real constants and , where is a numerable orthonormal basis of generalized eigenvectors of . Then, (ii)Assume that there is a finite such that for some positive real constant . Thus, for any given positive real constant , there are nonnegative finite integers and such that for any finite (≥)-dimensional degenerated approximating operator of , the following inequality holds Furthermore, for any linear degenerated (≥)-finite-dimensional approximating operator of the linear closed and compact self-adjoint operator and some finite .(iii)If as for some such that , then as . Furthermore, such a is a fixed point of both and .
Proof. The operator is represented as follows:
The associated degenerated -finite-dimensional operator is
Thus, assume that . Then,
so that the assumption is true as it has been proved from (30) by complete induction. The following properties are also direct for any if for some real constants and ; for all and some finite , we have
Property (i) has been proved. On the other hand, if for some finite and some , then for any given real (1) , there is a positive finite integer such that for any and any (≥) , the following inequalities hold:
since , for all , as , , and , for all (≥) . Note that since exists such that for some finite , then, for any given (1) , (32) holds for any (≥) and some . Then, one gets via complete induction for any
and as if , for all (≥) . Thus, one gets from Lemma 1 (iv)
for any (≥) and for all (>) . Furthermore, note from (32) that and as and the function is nonincreasing. Also, a strictly monotone decreasing positive real sequence can be built with since there are infinite many values of the spectrum such that the inequality is strict since, otherwise, the convergence of the sequence to zero would be impossible. Then, from (34) and as if , for all (≥) , there are subsequences of positive real and positive integers and , respectively, as such that the following subsequent relation holds:
for all . Then,
and Property (ii) follows directly.
If and as for some , then which converges to zero such that and then . Also, is bounded, since it is compact, and it is then continuous since it is linear and bounded. Also, is of finite-dimensional and closed image, then compact, and then bounded and continuous since it is linear. Thus, , as leads to and Property (iii) has been proved.
Note that Theorem 5 (ii) cannot be generalized, in the general case, for the case of a finite dimensional approximating linear operator of smaller dimension to any linear degenerated operator of (finite) dimension . The reason is that the property that does not any longer hold, in general if is finite dimensional. On the other hand, a way of describing the operator and its approximating finite-dimensional counterpart is through the absolute error operator : . This is useful if either is finite dimensional of dimension where is the dimension of or if is nondegenerated. Another useful characterization is the use of the relative error operator satisfying the operator identity . Another alternative operator identity cannot be used properly if is infinite dimensional since is degenerated of finite dimension . We discuss some properties of the operator identity through the subsequent result.
Lemma 6. Let be a separable Hilbert space and let be a nonnull and nondegenerated (i.e., of infinite-dimensional image) linear closed and compact operator and let be the linear degenerated -finite-dimensional approximating operator of . Then, there is an operator such that can be represented by , , and with the following properties.(i)There exists an (in general, nonunique) operator , restricted to for each approximating of given dimension .(ii)The operator is nondegenerated, unique, and compact.(iii)The minimum modulus of is so that if it is invertible, its inverse is not bounded. If is degenerated, that is, finite dimensional of dimension , injective with closed image then its minimum modulus is positive and finite. If, furthermore, is invertible then is a compact operator with bounded minimum modulus .
Proof. The existence of such an operator is proved by construction. Let be an orthonormal basis of generalized eigenvectors of and an orthonormal basis of , respectively. Then, one gets for some sequences of complex coefficients , for all ,
Then, is a unique nondegenerated compact operator from its representation (40). It follows that the operator identity holds on if and only if ; for all and, equivalently, since and are linear,
Since the vectors in form an orthonormal basis, (41), if the following constraints defining the operator , restricted as , hold for a nonnull operator so that (42) holds if and only if since the elements of are linearly independent. Then (43) holds under infinitely many combinations of constraints on the spectrum of . In particular, (43) holds if
Equations (43) are also satisfied with , for all , for all , and for which holds, for instance, if for all . Thus, is then non-unique, in general. Properties (i)-(ii) have been proved.
Now, let be the minimum modulus of the linear operator . If , then if is injective with closed image (this implies that such an image is finite dimensional), then and since are both bounded since they are compact, one gets If is infinite dimensional, then and it cannot then have bounded inverse. If is degenerated of dimension , then is the null operator with . If is degenerated of dimension and invertible, then and so that is bounded and compact since it is a composite operator of a compact operator on and a bounded operator on . Property (iii) has been proved.
Example 7. Assume that are two degenerated finite-dimensional operators on a separable Hilbert space of, respectively, dimensions two and one defined by ; for all and ; for all . Thus, the constraints (42) hold for an incremental operator of spectrum defined by , with and . Then,
Remark 8. If is infinite dimensional and invertible, then is not compact, since is unbounded, since .
Hilbert spaces for the formulation of equilibrium points, stability, controllability [16, 18, 19], boundedness, and square integrability (or summability in the discrete formalism) of the solution in the framework of square-integrable (or square-summable) control and output functions are of relevant importance in signal processing and control theory and in general formulations of dynamic systems, in general. See, for instance, [1, 2, 7, 9, 16, 17, 19, 20] and the references therein. Two examples with the use of the above formalism to dynamic systems and control issues are now discussed in detail.
Example 1. Consider the forced linear time-invariant differential system of real coefficients and th as
under a piecewise continuous square-integrable forcing function ; that is, , with . The unique solution for any given initial conditions for is
where the superscript stands for transposition, are Euclidean vectors of, respectively, first and last components being unity and the remaining ones being zero , and
The matrix function is a -semigroup generated by the infinitesimal generators , respectively [17, 19]. Using a sampling period of length , we can write from (48) for time instants being integer multiples of the sampling period where and provided that the input is , for all . The matrix function can be expanded as follows: where is the spectrum of , that is, set of distinct eigenvalues of with respective multiplicities for in the minimal polynomial of where is the degree of the minimal polynomial of , and then and are complex constants. The above ; are everywhere continuous and linearly independent time-differentiable functions on . Then, the unique solution (or output) of (47) for zero initial conditions is with provided that , guaranteed from (51) if and only if ; for all and is a convolution operator and is a convolution integral operator since the differential system is time-invariant where , for all . Thus, such an operator is normal, since it is time invariant , and then self-adjoint. Now, define the sequence of samples for a sampling period as with the operator being defined from on the space of square-summable sequences , where , for all . Assume that the forcing input is piecewise constant, for all , for all . Note that if , then , the unilateral Laplace transform of , is strictly proper; that is, it has more poles than zeros. In the case that , is proper by not strictly proper; that is, it has the same number of poles and zeros. It turns out that we can define an operator sequence : for all , with the second one being a natural projection on of an operator on so that, by using ; for all , one gets: with ; for all , , , with being the identity operator. One has from (51) that and as if ; . Some particular cases are discussed below under the assumption and ; implying , , so that and , since is bounded.
Proposition 2 (constant piecewise constant open-loop control). Assume that , for all , and consider a constant open-loop control , for all . The following properties hold.(i)The sequence satisfies , subject to , for all , where the operator is defined as the sequence of scalar gains , for all in the Banach space which is the Euclidean Hilbert space for the product of real numbers being an inner product. Furthermore, .(ii)Assume that for some given , and . Then, and as , for all , for all (≥) for some finite .(iii)There is for each given and such that , for all . Also, for each given satisfying , for all , it follows that
Proof. Property (i) follows from , or equivalently, , for all subject to an initial condition . Since is bounded, as , and , then . Thus, the sequence is generated as , subject to , for all , where the operator is defined in the Banach space as the sequence of scalar gains , for all which is the Eucl