Abstract

This paper has two parts. The first one provides the preliminary notions introducing certain general concepts, in order to study, in the second part, the properties of some operator systems which admit spectral residual decompositions, -decomposable, -spectral, and -scalar systems, and so forth. The results obtained by Frunză, 1975, are generalized, taking the results of Foias, 1963, as a model and adopting them.

1. Introduction

Across this paper we will try to generalize for operators systems some of the results obtained by Vasilescu in [1] for a single operator: residual single valued extension property, analytic residuum, the problem of local spectra, and so forth.

Most of the proofs are adaptations of the ones from [2, 3] with minor changes.

All operators with a reasonable spectral decomposition have -decomposable restrictions and quotients. Subnormal, subscalar, and subdecomposable operators being restrictions and quotients of normal, scalar, and decomposable operators are thus -decomposable (in fact -normal, -scalar, etc.). Eschmeier and Putinar [4] have shown that hyponormal operators are subscalar and therefore -decomposable. Operators that admit scalar dilations (extensions) (C. Ionescu-Tulcea) or -scalar (E. Stroescu) are -decomposable. In fact, Eschmeier and Putinar [4] have shown that any operator is the quotient or restriction of a quotient of a decomposable operator and thus is -decomposable or similar to an -decomposable operator.

Colojoară and Foiaş in [5, Chapter 6, Proposition ] formulated an open problem: any decomposable operator is strongly decomposable (meaning are that the restrictions and quotients in relation to the spectral maximal space decomposable?). A partial answer was given in [6]: the operators with the spectrum of dimension ≤1 thus situated on a curve are strongly decomposable, more specifically if the spectrum is of dimension ≤1 and any of its subsets, in the relative topology, included, has a border ≤0, sets of class . There are subsets in the plane which do not have this property. The result was strengthened by the example given by Albrecht and Vasilescu [7]. Some of the results obtained for one operator were generalized for systems of operators.

Let be a Banach space and let be the algebra of all linear bounded operators on .

Furthermore, let be the family of all closed linear subspaces of , let be a compact set, and let be the family of all closed subsets that have the property: either or .

An -spectral capacity is an application verifying the following properties:(1) , ;(2) , for any series ;(3) for any open finite -covering of , we have .

A commuting system of operators is said to be -decomposable if there is an -spectral capacity such that(4) , for any and any ;(5) , for any .

In case that , the -spectral capacity is said to be spectral capacity, and the system is decomposable [8].

2. Systems of Commuting Operators

This section provides the preliminary notions introducing certain general concepts necessary for the study of the -decomposable systems in Section 3.

Definition 1. Let be a commuting operators system and a compact minimal set having the property that for any open with (minimal means that is the intersection of all compact sets having the specified property). One will denote by the union of all open sets with the property that there exists a form satisfying the equality meaning that (we recall that there exist sets with this property, e.g., the solving set ). We will also denote The set will be said to be the resolvent set of related to , will be said to be the spectrum of related to , and will be called the residual spectrum of .
We will call analytic resolvent set of related to and we will denote by the set , where is the set of for which there exists an open neighbourhood of and analytic function on taking values in , satisfying the identity We will understand through the analytic spectrum of related to the set where We will prove that for an operators system that admits a spectral -capacity we have

Proposition 2. For a commuting operators system one has the following:(1)  implies , ;(2) , , ;(3) , if , , ;(4) ,where is a (linear, closed) subspace of invariant to all and .

Proof. (1°) follows from the fact that, for and any neighbourhood , the form verifies the relation meaning that Let and such that there exist the forms verifying the equalities , . Then the form verifies the equality and hence (2°) is verified. The inclusions from (3°) result from the fact that, by considering the form such that and by applying operator to the coefficients of , its commuting with each    implies (admitting the equality on components). The inclusion (4°) and the remark on the resolvent set lead to the equality .

Lemma 3. Let , be two open sets in such that .
Then for any there exist such that on ([8], 1.2.1.).

Lemma 4. Let be an operators system with the residual spectrum and two open sets in such that there exist the forms with the property that on . Then there exists a form having the following property: on .

Proof. When we can consider for and we have on . If we have on . Since , it results that there exists a form such that Indeed, the nucleus of the cofrontier operator is: By applying the preceding lemma to the coefficients of there follows where .
Consequently, where on . By putting we will obtain on and on . Hence by defining for one obtains a form as the one required in the text of the lemma. The lemma is proved.

Corollary 5. Let be a finite family of open sets from such that the equation has a solution on each of them. If is a compact set, there exists an open neighbourhood of on which the equation has a solution.

Proof. Let be a growing sequence of compact sets such that . We will prove that there exists a corresponding sequence of forms that verify the equality on a neighbourhood of . Then exists and it is a global solution. We will start with (see [9, 10]).
By Corollary 5 there exists a form defined in an open neighbourhood of and satisfying the equality on this neighbourhood. Since the space is invariant to multiplication by scalar functions of a class ([3], 2.16.1) we can assume, without limiting the generality, that is defined on ; indeed, by multiplying by a suitable scalar function, the new form can be extended to and we will obtain a form on verifying the equality on a neighbourhood of . We will now suppose that the forms from the desired sequences were already determined and let us determine .
According to the preceding corollary there exists a neighbourhood of the set and a form defined on this neighbourhood satisfying the equality , and we are allowed to suppose moreover that is defined on the whole . But on a vicinity of , and hence by subtraction we obtain on ; since , it will result that there exists a form such that on , and we may suppose that is defined on . We will put and obtain a form defined on equal to on and satisfying the equality on the neighbourhood of . By this the demonstration ends.

Remark 6. A local version of the Cauchy-Weil formula ([8, 1.2.4]) can be established on the same way as in [8, formula 1.5.1].
Let be a commuting operators system with the residual spectrum and an open neighbourhood of ; obviously .
We will prove that there exists a form in the same cohomology class related to as such that support .
It follows that there exists a form such that . Let and be two open neighbourhoods relatively compact of , such that , and let us consider scalar -function on , outside , and on . By using let us define the form by on and on .
This form has the coefficients in and satisfies the condition outside the relatively compact set . Hence by setting we will obtain a form defined on with support that is precisely the form having the specified properties. Considering formula [8, 1.2.4.] and using form above we can write which will yield the local version of Cauchy-Weil formula.

3. Some Properties of -Decomposable Systems

For the Banach space and for an arbitrary open set , we denote by the space of all -valued analytic functions on .

Proposition 7. Let be an -decomposable system, let be an open polydisk with , let p be an integer, , and let such that where is defined by Then for any polydisk with there exists a form such that on .

The proof of Proposition 2.1.3. presented in [8] is also true in this case, with a single comment that is not anymore any polydisk of , but a polydisk that does not cross .

Theorem 8. If is -decomposable, then .

Proof. With minor differences, the proof is identical with the one for the decomposable systems ([8, Proposition 2.1.4.]) where is called property . It will have to show that for any polydisk such that we have .
We note that implies where is any open polydisk from and is any open set such that , ; the proof is given in [8, Theorem 1.5.16.] for any , .
One motivates this through induction on , beginning with . Let such that ; according to the preceding proposition we will have on any polydisk with and on . Suppose that for any open polydisk with we have with fixed, , and let us prove that .
Let be a sequence of polydisks, , such that for any with and such that . By applying the preceding proposition for , we infer that there exists a form such that on ; analogously we can find a form on with on . One obtains on where, by applying the inductive hypothesis, we infer that there exists a form , such that . We will retain from Taylor’s decomposition of on a sufficient number of terms, such that (the retained part) verifies on . Thinking analogously, we can define a sequence of forms , , enjoying the properties
The sequence obviously converges to a form having the analytic coefficients on and satisfying on .

The uniqueness of the spectral -capacities for -decomposable operators systems can be proved. We will now prove this on other ways, emphasizing the connection between the spectral -capacity related to an operator and certain linear subspaces, described using the local spectrum, which is most useful.

Let be a commuting system of operators on the space , , with the residual spectrum . If is an arbitrary set from such that , we will put and ; and are linear subspaces of and .

Theorem 9. If is an -decomposable system and is its -spectral capacity, then for any closed set , .

Proof. According to Theorem 8, ; hence and makes sense. The inclusion follows by the fact that (Proposition 2 (4°)). In the same manner as for [8, Theorem 2.2.1], one proves the inverse inclusion, with the observation that is not an arbitrary subset of , but .

Corollary 10. Let a be an -decomposable system. Then for any closed , the subspace is spectral maximal space of ; more precisely, for any subspace invariant to a such that , one has ; moreover .

Proof. The inclusion follows by the preceding theorem, since .
If is invariant to with then any ; hence , meaning .

Proposition 11. If a is -decomposable then, for any , one has .

Proof. Let us prove first that or its equivalent . Let and according to [8, Theorem  1.1.3.] let us consider an open neighbourhood of and analytic functions defined on taking values in , that verify the equality . We consider the degree form defined on ,
This form can be considered as an element of and it easily verifies the equality on taking into account the analyticity of the functions ; hence it results that ; that is, or hence . For the inverse inclusion , let and let be an open polydisk with its centre in such that .
Since , hence by [8, Theorem 1.1.3.] there exist the analytic functions defined on and taking values in , such that
That means that ; hence hence .