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, S-spectral, and AS-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 S-decomposable restrictions and quotients. Subnormal, subscalar, and subdecomposable operators being restrictions and quotients of normal, scalar, and decomposable operators are thus S-decomposable (in fact S-normal, S-scalar, etc.). Eschmeier and Putinar [4] have shown that hyponormal operators are subscalar and therefore S-decomposable. Operators that admit scalar dilations (extensions) (C. Ionescu-Tulcea) or A-scalar (E. Stroescu) are S-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 S-decomposable or similar to an S-decomposable operator.

Colojoară and Foiaş 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 C. 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 X be a Banach space and let BX be the algebra of all linear bounded operators on X.

Furthermore, let SX be the family of all closed linear subspaces of X, let S⊂Cn be a compact set, and let FS be the family of all closed subsets F⊂Cn that have the property: either F∩S=Ø or F⊃S.

An S-spectral capacity is an application E:FS→SX verifying the following properties:

EØ=0, ECn=X;

E⋂i=1∞Fi=⋂i=1∞EFi, for any series Fii∈N⊂FS;

for any open finite S-covering {GS}∪{Gj}j=1m of Cn, we have X=E(G-S)+∑j=1mE(G-j).

A commuting system of operators a=a1,a2,…,an⊂BX is said to be S-decomposable if there is an S-spectral capacity such that

aiEF⊂EF, for any F∈FS and any i=1,2,…,n;

σa,EF⊂F, for any F∈FS.

In case that S=Ø, the S-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 S-decomposable systems in Section 3.

Definition 1.

Let a=a1,a2,…,an⊂BX be a commuting operators system and Sa⊂Cn a compact minimal set having the property that Hn-1C∞G,X,α⊕∂-=0 for any open G⊂Cn with G∩Sa=Ø (minimal means that Sa is the intersection of all compact sets having the specified property). One will denote by da,x the union of all open sets V⊂Cn with the property that there exists a form ψ∈Λn-1σ∪dz-,C∞V,X satisfying the equality sx=α⊕∂-ψ meaning that
(1)xs1∧s2∧⋯∧sn=z1-a1s1+⋯+zn-ansn∂∂z-nWW+∂∂z-1dz-1+⋯+∂∂z-ndz-n∧ψz
(we recall that there exist sets V with this property, e.g., the solving set ra,X). We will also denote
(2)ga,x=Cn∖da,x,ra,x=da,x∩Cn∖Sa,spa,x=Cn∖ra,x=ga,x∪Sa.
The set ra,x will be said to be the resolvent set of x related to a, spa,x will be said to be the spectrum of x related to a, and Sa will be called the residual spectrum of a.

We will call analytic resolvent set of x related to a and we will denote by ρa,x the set ρa,x=δa,x∩Cn∖Sa, where δa,x is the set of z∈Cn for which there exists an open neighbourhood V of z and n analytic function on V taking values in X, f1,f2,…,fn satisfying the identity
(3)x=ζ1-a1f1ζ+⋯+ζn-anfnζ,ζ∈V.
We will understand through the analytic spectrum of x related to a the set
(4)σa,x=Cn∖ρa,x=γa,x∪Sa,
where
(5)γa,x=Cn∖δa,x.
We will prove that for an operators system that admits a spectral S-capacity we have
(6)ga,x=γa,x,da,x=δa,x,ρa,x=ra,x,spa,x=σa,x.

Proposition 2.

For a commuting operators system a=a1,a2,…,an⊂BX one has the following:

x=0 implies ga,x=Ø, spa,x=Sa;

ga,x+y⊂ga,x∪ga,y, spa,x+y⊂spa,x∪spa,y∀x, y∈X;

ga,by⊂ga,x, spa,by⊂spa,x if bai=aib, b∈BX, x∈X;

ga,y⊂spa,y⊂σa,Y,

where Y is a (linear, closed) subspace of X invariant to all ai and y∈Y.Proof.

(1°) follows from the fact that, for x=0 and any neighbourhood V⊂Cn, the form ψ=0∈Λn-1[σ∪dz-,C∞(V,X)] verifies the relation sx=α⊕∂-ψ meaning that
(7)xs1∧s2∧⋯∧sn=z1-a1s1+⋯+zn-ansn∂∂z-nWW+∂∂z-1dz-1+⋯+∂∂z-ndz-n∧ψz.
Let z∈da,x∩da,y and z∈Vi such that there exist the forms
(8)ψi∈Λn-1σ∪dz-,C∞Vi,X,i=1,2
verifying the equalities sx=α⊕∂-ψ1, sy=α⊕∂-ψ2. Then the form
(9)ψ1+ψ2∈Λn-1σ∪dz-,C∞V1∪V2,X
verifies the equality
(10)sx+y=α⊕∂-ψ1+ψ2,
and hence (2°) is verified. The inclusions from (3°) result from the fact that, by considering the form ψ∈Λn-1σ∪dz-,C∞V,X such that sx=α⊕∂-ψ and by applying operator b to the coefficients of ψ, its commuting with each ai(i=1,2,…,n) implies
(11)α⊕∂-bψ=bα⊕∂-ψ=bxs
(admitting the equality on components). The inclusion (4°) and the remark on the resolvent set ra,Y lead to the equality xs=α⊕∂-ψ.

Lemma 3.

Let V1, V2 be two open sets in Cn such that V1∩V2≠Ø.

Then for any f∈C∞(V1∩V2,X) there exist fi∈C∞Vi,X(i=1,2) such that f=f1-f2 on V1∩V2 ([8], 1.2.1.).

Lemma 4.

Let a=a1,a2,…,an⊂BX be an operators system with the residual spectrum Sa and Vi(i=1,2) two open sets in Cn∖Sa such that there exist the forms ψi∈Λn-1σ∪dz-,C∞Vi,X with the property that sx=α⊕∂-ψi on Vi. Then there exists a form
(12)ψ∈Λn-1σ∪dz-,C∞V1∪V2,X
having the following property: sx=α⊕∂-ψ on V1∪V2.

Proof.

When V1∩V2=Ø we can consider ψz=ψiz for z∈Vi(i=1,2) and we have sx=α⊕∂-ψ on V1∪V2. If V1∩V2≠Ø we have α⊕∂-ψ2-ψ1=0 on V1∩V2. Since V1∩V2=G⊂Cn∖Sa, it results that there exists a form φ∈Λn-2σ∪dz-,C∞V1∩V2,X such that
(13)ψ2-ψ1=α⊕∂-φ.
Indeed, the nucleus of the cofrontier operator α⊕∂- is:
(14)Kerα⊕∂-:Λn-1σ∪dz-,C∞V1∩V2,XWWΛn-1⟶Λnσ∪dz-,C∞V1∩V2,XW=Imα⊕∂-:Λn-2σ∪dz-,C∞V1∩V2,XWWWW⟶Λn-1σ∪dz-,C∞V1∩V2,X.
By applying the preceding lemma to the coefficients of φ there follows φ=φ1-φ2 where φi∈Λn-2σ∪dz-,C∞Vi,X(i=1,2).

Consequently,
(15)α⊕∂-φ1-φ2=α⊕∂-φ=φ2-φ1,
where ψ1+α⊕∂-ψ1=ψ2+α⊕∂-ψ2 on V1∩V2. By putting ψi′=ψi+α⊕∂-φi(i=1,2) we will obtain sx=α⊕∂-ψi′ on Vi(i=1,2) and ψ1′=ψ2′ on V1∩V2. Hence by defining ψz=ψi′z for z∈Vi(i=1,2) one obtains a form as the one required in the text of the lemma. The lemma is proved.

Corollary 5.

Let Vii=1m be a finite family of open sets from Cn∖Sa such that the equation sx=α⊕∂-ψ has a solution ψ on each of them. If K⊂ra,x is a compact set, there exists an open neighbourhood V of K(V⊂ra,x) on which the equation sx=α⊕∂-ψ has a solution.

Proof.

Let Kνν=1∞ be a growing sequence of compact sets such that ra,x=⋃ν=1∞Kν. We will prove that there exists a corresponding sequence of forms ψν∈Λn-1σ∪dz-,C∞ra,x,X that verify the equality sx=α⊕∂-ψν on a neighbourhood of Kν. Then ψ=limν→∞ψν exists and it is a global solution. We will start with K1 (see [9, 10]).

By Corollary 5 there exists a form ψ1 defined in an open neighbourhood of K1 and satisfying the equality sx=α⊕∂-ψ1* on this neighbourhood. Since the space C∞ra,x,X is invariant to multiplication by scalar functions of a C∞ class ([3], 2.16.1) we can assume, without limiting the generality, that ψ1* is defined on ra,x; indeed, by multiplying ψ1* by a suitable scalar function, the new form can be extended to ra,x and we will obtain a form ψ1 on ra,x verifying the equality sxs=α⊕∂-ψ1 on a neighbourhood of K1. We will now suppose that the forms ψ1,ψ2,…,ψi from the desired sequences were already determined and let us determine ψi+1.

According to the preceding corollary there exists a neighbourhood Vi+1 of the set Ki+1 and a form ψi+1 defined on this neighbourhood satisfying the equality sx=α⊕∂-ψi+1*, and we are allowed to suppose moreover that ψi+1* is defined on the whole ra,x. But sx=α⊕∂-ψi on a vicinity Vi of Ki, and hence by subtraction we obtain α⊕∂-ψi+1*-ψi=0 on Vi∩Vi+1; since Vi∩Vi+1⊂Cn∖Sa, it will result that there exists a form φ′ such that ψi+1*-ψi=α⊕∂-φ′ on Vi∩Vi+1, and we may suppose that φ′ is defined on ra,x. We will put ψi+1=ψi+1*-α⊕∂-φ′ and obtain a form defined on ra,x equal to ψi on Vi∩Vi+1 and satisfying the equality sx=α⊕∂-φi+1 on the neighbourhood Vi+1 of Ki+1. 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 a=a1,a2,…,an⊂BX be a commuting operators system with the residual spectrum Sa and U an open neighbourhood of spa,x; obviously U⊃Sa.

We will prove that there exists a form χ∈Λn[σ∪dz-,C0∞(Cn,X)] in the same cohomology class related to α⊕∂- as sx such that support (χ)∈U.

It follows that there exists a form ψ∈Λn-1σ∪dz-,C∞ra,x,X such that sx=α⊕∂-ψ. Let U1 and U2 be two open neighbourhoods relatively compact of spa,x, such that spa,x⊂U1⊂U-1⊂U2⊂U-⊂U, and let us consider scalar C∞-function h on C∞, h=1 outside U2, and h=0 on U1. By using h let us define the form ψ~ by ψ~=hψ on ra,x and ψ~=0 on U1.

This form has the coefficients in C∞Cn,X and satisfies the condition sx=α⊕∂-ψ~ outside the relatively compact set U2. Hence by setting χ=sx-α⊕∂-ψ~ we will obtain a form defined on Cn with support χ⊂U-2⊂U that is precisely the form having the specified properties. Considering formula [8, 1.2.4.] and using form χ above we can write
(16)x=12πin∫U-1nπχ∧dz1∧⋯∧dzn
which will yield the local version of Cauchy-Weil formula.

3. Some Properties of <inline-formula>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M261">
<mml:mrow>
<mml:mi>S</mml:mi></mml:mrow>
</mml:math></inline-formula>-Decomposable Systems

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

Proposition 7.

Let a=a1,a2,…,an⊂BX be an S-decomposable system, let D be an open polydisk with D∩S=Ø, let p be an integer, 0≤p≤n-1, and let ψ∈Λpσ,UD,X such that σψ=0 where α is defined by
(17)αψz=z1-a1s1+z2-a2s2WW+⋯+zn-ansn∧ψz.
Then for any polydisk D′⊂D with D-′⊂D there exists a form ψ∈Λp-1σ,UD′,X such that ψ=αφ on D′.

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

Theorem 8.

If a=a1,a2,…,an⊂BX is S-decomposable, then S⊃Sa.

Proof.

With minor differences, the proof is identical with the one for the decomposable systems (S=Ø) ([8, Proposition 2.1.4.]) where Sa=Ø is called property L. It will have to show that for any polydisk U⊂Cn such that U∩S=Ø we have HiUU,X,α=0(0≤i≤n-1).

We note that HiUU,X,α=0 implies HiC∞G,X,α⊕∂-=0(0≤i≤n-1) where U is any open polydisk from Cn and G is any open set G⊂Cn such that U∩S=Ø, G∩S=Ø; the proof is given in [8, Theorem 1.5.16.] for any U, G⊂Cn.

One motivates this through induction on i, beginning with i=0. Let f∈UU,X such that αf=0; according to the preceding proposition we will have f=0 on any polydisk D′ with D′⊂U and f=0 on U. Suppose that for any open polydisk D⊂Cn with D∩S=Ø we have Hi-1UD,X,α=0 with i fixed, 0≤i≤n-1, and let us prove that HiUU,X,α=0.

Let Dν be a sequence of polydisks, Dν∩S=Ø, such that Dν⊂D-ν+1 for any ν with ⋃ν=1∞Dν=U and ψ∈Λiσ,UU,X such that αψ=0. By applying the preceding proposition for D2, we infer that there exists a form φ1∈Λi-1σ,UD2,X such that ψ=αφ1 on D2; analogously we can find a form φ2′ on D3 with ψ=αφ2′ on D3. One obtains αφ1-φ2′=0 on D2 where, by applying the inductive hypothesis, we infer that there exists a form χ∈Λi-2σ,UD2,X, such that φ1-φ2′=αψ. We will retain from Taylor’s decomposition of χ on D2 a sufficient number of terms, such that χ′ (the retained part) verifies αχ-αχ′<1/2 on D-1. Thinking analogously, we can define a sequence of forms φν, φν∈Λi-1σ,UDν+1,X, enjoying the properties
(18)ψ=αφνWonDν+1,φν+1-φν<12ν+1WonD.

The sequence φν obviously converges to a form having the analytic coefficients on U and satisfying ψ=αφ on U.

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

Let a be a commuting system of operators on the space X, a⊂BX, with the residual spectrum Sa. If H is an arbitrary set from Cn such that H⊃Sa, we will put XaH=x,x∈X,spa,x⊂H and XaH=x,x∈X,σa,x⊂H; XaH and XaH are linear subspaces of X and XaH⊂XaH.

Theorem 9.

If a=a1,a2,…,an⊂BX is an S-decomposable system and E is its S-spectral capacity, then
(19)EF=XaF,
for any closed set F⊂Cn, F⊃S.

Proof.

According to Theorem 8, S⊃Sa; hence F⊃Sa and XaF makes sense. The inclusion EF⊂XaF follows by the fact that spa,x⊂σa,EF (Proposition 2 (4°)). In the same manner as for [8, Theorem 2.2.1], one proves the inverse inclusion, with the observation that F is not an arbitrary subset of Cn, but F⊃S.

Corollary 10.

Let a be an S-decomposable system. Then for any closed F⊃S, the subspace XaF is spectral maximal space of a; more precisely, for any subspace Z invariant to a such that σa,Z⊂F, one has Z⊂Xa(F); moreover σa,XaF⊂F.

Proof.

The inclusion σa,XaF⊂F follows by the preceding theorem, since σa,EF⊂F.

If Z is invariant to a with σa,Z⊂F then any z∈Zspa,z⊂σa,Z⊂F; hence z∈XaF, meaning Z⊂Xa(F).

Proposition 11.

If a is S-decomposable then, for any x∈X, one has spa,x=σa,x.

Proof.

Let us prove first that spa,x⊂σa,x or its equivalent σa,x⊂ra,x. Let z∈δa,x and according to [8, Theorem 1.1.3.] let us consider an open neighbourhood V of z and n analytic functions defined on V taking values in X, f1,f2,…,fn that verify the equality
(20)x=ζ1-a1f1ζ+⋯+ζn-anfnζ,ζ∈V. We consider the n-1 degree form defined on V,
(21)ψζ=∑i=1n-1i-1fiζs1∧s2∧⋯∧si∧⋯∧sn.

This form can be considered as an element of Λn-1σ∪dz-,C∞V,X and it easily verifies the equality sx=α⊕∂-ψ on V taking into account the analyticity of the functions fi(δfi=0); hence it results that V⊂da,x; that is, δa,x⊂da,x or ga,x⊂γa,x hence ga,x∪Sa=spa,x⊂γa,x∪Sa=σa,x. For the inverse inclusion σa,x⊂spa,x, let z∈ra,x and let D be an open polydisk with its centre in z such that D⊂ra,x.

Since x∈Xa(spa,x=E(spa,x)), hence by [8, Theorem 1.1.3.] there exist the analytic functions f1,f2,…,fn defined on D and taking values in X, such that
(22)x=∑i=1nζi-aifiζ,ζ∈D.

That means that z∈ρa,x; hence ra,x⊂ρa,x hence σa,x⊂spa,x.

Corollary 12.

If a is an S-decomposable system, then for any H⊂Cn with H⊃S one has XaH=XaH.

Proof.

It easily follows by the preceding proposition.

Proposition 13.

If a is an arbitrary system of operators, then
(23)σa,X=⋃x∈Xspa,x.

Proof.

The inclusion ⋃x∈Xspa,x⊂σa,X results from Proposition 2 (4°), spa,x⊂σa,X.

Conversely, if z∈⋂x∈Xra,x, then HnUz,X,α=0; since z∈Sa, there exists an open polydisk D, with D∩Sa=Ø, for which, according to Theorem 8, we have HiUD,X,α=0(i=0,1,…,n-1).

Then by [8, Corollary 1.4.3.] it follows that z∈ra,x; hence ⋃x∈Xspa,x⊃σa,X.

Definition 14.

The support of an S-spectral capacity E is the set
(24)supp E=∩F;Fclosed,EF=X.

Proposition 15.

If a=a1,a2,…,an⊂BX is an S-decomposable system and E is its S-spectral capacity, then suppE=σa,X.

Proof.

The inclusion σa,X⊂supp E results from the fact that, for any closed F such that EF=X, we have σa,X=σa,EF⊂F, where σa,X⊂∩F,Fclosed,EF=X=supp E. For the inverse inclusion, let z0∈ra,X and let us prove that z0∉supp E. Let V be an open neighborhood of z0 such that V-⊂ra,X and let F be a closed set under the conditions z0∉F, F⊃S and X=EF+EV-; these conditions are possible because z0∈S(S⊂σa,X).

Let x∈EV-; since V⊂ra,X it results that sx=α⊕∂-ψ in a neighborhood Cn∖V of the spectrum spa,X; hence by applying [8, formula 1.2.4], we find x=0. Therefore EV-=0; that is, EF=X; hence from z0∉F it follows that z0∉supp E.

Corollary 16.

If a=a1,a2,…,an⊂BX is an S-decomposable system, then one has σa,EF⊂σa,X, for any closed set F⊂Cn.

Proof.

We have
(25)EF∩σa,X=EF∩Eσa,X=EF∩X=EF,
hence
(26)σa,EF=σa,EF∩σa,X⊂F∩σa,X⊂σa,X.

4. Conclusions

The originality consists in the adaptation of the results obtained for the spectral capacities by Frunză to the decomposable systems of operators to the S-spectral capacities which are appropriate to the S-decomposable systems of operators.

These theorems remain valid also for a special class of S-decomposable systems, namely, S-spectral systems. An important result was obtained in the particular case when the topological dimension of S is zero and the S-decomposable (S-spectral) systems become decomposable (spectral systems).

We will underline the relevance, importance, and necessity of studying the S-decomposable operators showing the consistence of this class, in the sense of how many and how substantial its subfamilies are. Relations with this same subject can also be found in the “regularities” of Müller [11]. The S-decomposable operators appear firstly as restrictions and quotients of decomposable operators, particularly restrictions and quotients of spectral operators, and therefore AS-scalar or S-spectral operators (generalized S-spectral, generalized S-scalar, etc.).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

VasilescuF. H.FrunzăŞt.An axiomatic theory of spectral decompositions for systems of operators IFrunzăŞ.The Taylor spectrum and spectral decompositionsEschmeierJ.PutinarM.ColojoarăI.FoiaşC.BacaluI.Some properties of decomposable operatorsAlbrechtE. J.VasilescuF. H.On spectral capacitiesFoiasC.Spectral maximal spaces and decomposable operators in Banach spaceTaylorJ. L.A joint spectrum for several commuting operatorsTaylorJ. L.The analytic-functional calculus for several commuting operatorsMüllerV.