Some Topics in the Theory of Decomposable Operators

Albrecht, Ernst; Eschmeier, Jörg; Neumann, Michael

doi:10.1007/978-3-0348-7698-8_1

Cited by 16 publications

(17 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Thus an operator T is decomposable if and only if T has both properties (β) and (δ) [3]. Albrecht and Eschmeier [2] have shown that the properties (β) and (δ) are completely dual; an operator T has one of these exactly when its adjoint has the other.…”

Section: Local Spectral Theorymentioning

confidence: 99%

Local spectral theory and orbits of operators

Miller

1999

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

Section: Local Spectral Theorymentioning

confidence: 99%

Local spectral theory and orbits of operators

Miller

1999

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…Recall from [8] [21,Proposition 4.3.6], there exists some non-zero xeX for which a T (x) is empty. This leads to an obvious obstruction to (2) and hence also to (3), whenever S has SVEP, A is injective, and K contains o C(S T) (A). To counteract the effects of the absence of SVEP, we need another class of spectral subspaces, the so-called glocal spectral subspaces X T {F), the definition of which simply bypasses this problem: for given TeL(X) and any closed F c C , let…”

Section: C (K) = {Ae L(x Y): Ax T (F) £ Y S (F+k) For All Closed Fmentioning

confidence: 99%

“…The Albrecht-Eschmeier description of quotients of decomposable operators is in terms of the glocal subspaces: the operator TeL(X) is similar to a quotient of a decomposable operator with respect to a closed invariant subspaces if and only if T has the decomposition property (3), which means that the splitting X = 3f T It will be useful to collect a few basic facts about the glocal subspaces in one place. Some notation will be needed.…”

Section: Preliminaries From Local Spectral Theorymentioning

confidence: 99%

Local spectral properties of commutators

Laursen

Miller

Neumann

1995

Proceedings of the Edinburgh Mathematical Society

Self Cite

View full text Add to dashboard Cite

For a pair of continuous linear operators T and S on complex Banach spaces X and Y, respectively, this paper studies the local spectral properties of the commutator C{S,T) given by C{S, T)( X, Y). Under suitable conditions on T and S, the main results provide the single valued extension property, a description of the local spectrum, and a characterization of the spectral subspaces of C{S, T), which encompasses the closedness of these subspaces. The strongest results are obtained for quotients and restrictions of decomposable operators. The theory is based on the recent characterization of such operators by Albrecht and Eschmeier and extends the classical results for decomposable operators due to Colojoara, Foia §, and Vasilescu to considerably larger classes of operators. Counterexamples from the theory of semishifts are included to illustrate that the assumptions are appropriate. Finally, it is shown that the commutator of two super-decomposable operators is decomposable.1991 Mathematics subject classification: 47A11, 47B40.

show abstract

“…It has been observed in [3] that an operator T ∈ L(X) is decomposable if and only if it has both properties (β) and (δ). In [2], Albrecht and Eschmeier proved that the property (β) and (δ) are dual to each other in the sense that an operator T ∈ L(X) satisfies (β) if and only if the adjoint operator T * on the dual space X * satisfies (δ) and that the corresponding statement remains valid if both properties are interchanged.…”

Section: Introductionmentioning

confidence: 99%