On Certain Aspects of the Multidimensional S-Spectral Theory

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, S-decomposable, Sspectral, andA S -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.


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.
Colojoarȃ and Foias ¸in [5, Chapter 6, Proposition 5b] 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 C 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 B() be the algebra of all linear bounded operators on .
Furthermore, let S() be the family of all closed linear subspaces of , let  ⊂ C  be a compact set, and let F  be the family of all closed subsets  ⊂ C  that have the property: either  ∩  = Ø or  ⊃ .
In case that  = Ø, the -spectral capacity is said to be spectral capacity, and the system is decomposable [8].

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  = ( 1 ,  2 , . . .,   ) ⊂ B() be a commuting operators system and   ⊂ C  a compact minimal set having the property that  −1 (C ∞ (, ), ⊕) = 0 for any open  ⊂ C  with ∩  = Ø (minimal means that   is the intersection of all compact sets having the specified property).One will denote by (, ) the union of all open sets  ⊂ C  with the property that there exists a form  ∈ Λ −1 [ ∪ d, C ∞ (, )] 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 , sp(, ) will be said to be the spectrum of  related to , and   will be called the residual spectrum of .
By Corollary 5 there exists a form  1 defined in an open neighbourhood of  1 and satisfying the equality  = ( ⊕ ) * 1 on this neighbourhood.Since the space C ∞ ((, ), ) is invariant to multiplication by scalar functions of a C ∞ class ([3], 2.16.1) we can assume, without limiting the generality, that  * 1 is defined on (, ); indeed, by multiplying  * 1 by a suitable scalar function, the new form can be extended to (, ) and we will obtain a form  1 on (, ) verifying the equality  = ( ⊕ ) 1 on a neighbourhood of  1 .We will now suppose that the forms  1 ,  2 , . . .,   from the desired sequences were already determined and let us determine  +1 .According to the preceding corollary there exists a neighbourhood  +1 of the set  +1 and a form  +1 defined on this neighbourhood satisfying the equality  = ( ⊕ ) * +1 , and we are allowed to suppose moreover that  * +1 is defined on the whole (, ).But  = ( ⊕ )  on a vicinity   of   , and hence by subtraction we obtain ( ⊕ )( * +1 −   ) = 0 on   ∩  +1 ; since   ∩  +1 ⊂ C  \   , it will result that there exists a form   such that  * +1 −   = ( ⊕ )  on   ∩  +1 , and we may suppose that   is defined on (, ).We will put  +1 =  * +1 − ( ⊕ )  and obtain a form defined on (, ) equal to   on   ∩  +1 and satisfying the equality  = ( ⊕ ) +1 on the neighbourhood  +1 of  +1 .By this the demonstration ends.Let  = ( 1 ,  2 , . . .,   ) ⊂ () be a commuting operators system with the residual spectrum   and  an open neighbourhood of sp(, ); obviously  ⊃   .
We will prove that there exists a form  ∈ Λ  [ ∪ d, C ∞ 0 (C  , )] in the same cohomology class related to  ⊕  as  such that support () ∈ .
This form has the coefficients in C ∞ (C  , ) and satisfies the condition  = ( ⊕ ) ψ outside the relatively compact set  2 .Hence by setting  =  − ( ⊕ ) ψ we will obtain a form defined on C  with support () ⊂  2 ⊂  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.

Some Properties of 𝑆-Decomposable Systems
For the Banach space  and for an arbitrary open set  ⊂ C  , we denote by U(, ) the space of all -valued analytic functions on .
Proposition 7. Let  = ( 1 ,  2 , . . .,   ) ⊂ B() be an decomposable system, let  be an open polydisk with ∩ = Ø, let p be an integer, 0 ≤  ≤  − 1, and let  ∈ Λ  [, U(, )] such that  = 0 where  is defined by Then for any polydisk   ⊂  with   ⊂  there exists a form 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 C  , but a polydisk that does not cross .
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.

Corollary 5 .
Let {  }  =1 be a finite family of open sets from C  \   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.

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].
where  is any open polydisk from C  and  is any open set  ⊂ C  such that  ∩  = Ø,  ∩  = Ø; the proof is given in [8, Theorem 1.5.16.] for any ,  ⊂ C  .
)for any closed set  ⊂ C  ,  ⊃ .∘ )).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 C  , but  ⊃ .