Spectral Decompositions on Banach Spaces

Erdelyi, Ivan; Lange, Ridgley

doi:10.1007/bfb0065476

Cited by 32 publications

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“…By (8) the last manifold is closed, so by the closed graph theorem, there is R > 0 such that for each a we have u' a , υ' a in X T We mention finally that R. Evans [9] has studied "boundedly decomposable" operators. Each such operator satisfies (8) and is therefore strongly bi-decomposable by the last remark. Details are left to the reader.…”

Section: (Iii) T Is (Sqd) and For Each Open G And Closed F Subspaces mentioning

confidence: 87%

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Duality and asymptotic spectral decompositions

Lange¹

1986

Pacific J. Math.

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Asymptotic spectral decomposition for an operator on a Banach space is studied in light of the well-known theory of decomposable operators of Foias type. It is proved that adjoints of strongly quasidecomposable operators have the single-valued extension property. Duality theorems for strongly decomposable operators are given, for example, an operator has strongly decomposable adjoint iff it has a rich supply of strongly analytic subspaces. For reflexive spaces sharper results are obtained. Decomposable operators are characterized as those quasi-decomposable operators satisfying an additional duality property. Also an asymptotic spectral decomposition with strongly analytic subspaces implies decomposability. Strongly bi-decomposable operators are also studied.1. Introduction. The theory of decomposable operators, initiated by Foias [11], and given substantial development by him and others in the period 1963-1975, can now be said to have reached a satisfactory maturity. It is known, in particular, that decomposable operators enjoy a "completely symmetric duality theory," i.e., an operator is decomposable exactly when its adjoint is. It has also been proven by the author There has also been parallel development of spectral theory along the separate but related path of "asymptotic decompositions." To be precise, let T be a bounded linear operator on the Banach space X. We say that T has an asymptotic spectral decomposition if, for each finite open cover {Gf. 1 < / < n) of the complex plane, there is a system of Γ-invariant subspaces {M l9 M 29 ... 9 M n } such that

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Section: (Iii) T Is (Sqd) and For Each Open G And Closed F Subspaces mentioning

confidence: 87%

“…We collect these results in For completeness, we now discuss a condition sufficient for an operator to be strongly bi-decomposable (see (8) below). We first show that this condition is also sufficient for the following criterion due to Wang [27].…”

Section: (Iii) T Is (Sqd) and For Each Open G And Closed F Subspaces mentioning

confidence: 99%

Duality and asymptotic spectral decompositions

Lange¹

1986

Pacific J. Math.

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“…The main ingredients in the proof of power-regularity of decomposable operators are Theorem 2.1 in the preceding section and Theorem 12.15 in [7].…”

Section: Decomposable Operatorsmentioning

confidence: 99%

“…Let 0 < t\ < t2 < oo, and consider the disc G = {X £ C; \X\ < t2). Since T is decomposable, it follows from [7,Theorem 12.15] that there exists a subspace M in Lat(T), such that o(TM) c G and a(TM) f)G = 0. This is equivalent to the conditions r(TM) < t2 < q(TM).…”

Section: Decomposable Operatorsmentioning

confidence: 99%

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Power Regular Operators

Atzmon¹

1995

Transactions of the American Mathematical Society

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Multipliers with Natural Local Spectra on Commutative Banach Algebras

Eschmeier

Laursen

Neumann

1996

Journal of Functional Analysis

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Several notions from the abstract spectral theory of bounded linear operators on Banach spaces are investigated and characterized in the context of multipliers on a semi-simple commutative Banach algebra. Particular emphasis is given to the determination of the local spectra of such multipliers in connection with Dunford's property (C), Bishop's property ( ;), and decomposability in the sense of FoiasÂ . The strongest results are obtained for regular Tauberian Banach algebras with approximate units and for multipliers whose Gelfand transforms on the spectrum of the Banach algebra vanish at infinity. The general theory is then applied to convolution operators induced by measures on a locally compact abelian group G. Our results give new insight into the spectral theory of convolution operators on the group algebra L 1 (G) and on the measure algebra M(G). In particular, we identify large classes of measures for which the corresponding convolution operators have excellent spectral properties. We also obtain a number of negative results such as examples of convolution operators on L 1 (G) without natural local spectra, but with natural spectrum in the sense of Zafran.

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Spectral Decompositions on Banach Spaces

Cited by 32 publications

References 0 publications

Duality and asymptotic spectral decompositions

Duality and asymptotic spectral decompositions

Power Regular Operators

Multipliers with Natural Local Spectra on Commutative Banach Algebras

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