Local spectral properties of commutators

Laursen, Kjeld; Miller, V. G.; Neumann, Michael

doi:10.1017/s0013091500019106

Cited by 3 publications

(2 citation statements)

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“…Thus, finite dimensionality of X T ({λ}) is necessary for λ / ∈ σ sF (T ), for a rather large class of operators, those with property (C); but it is not sufficient, for this class: for non-isolated points of the spectrum it is not true that finite dimensionality of X T ({λ}) implies that λ / ∈ σ sF (T ), as the example of the right shift R on 2 (N) shows: σ sF (R) = T, but 2 (N) R ({λ}) = {0} for every λ ∈ C, [11,Proposition 2]. If λ is an isolated point of the spectrum, however, finite dimensionality of X T ({λ}) is a characterization of non-membership of the semi-Fredholm spectrum, as well as of the essential spectrum.…”

Section: General Local and Global Spectral Theorymentioning

confidence: 99%

Essential spectra through local spectral theory

Laursen¹

1997

Proc. Amer. Math. Soc.

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Abstract. Based on a nice observation of Eschmeier, this is a study of the use of local spectral theory in investigations of the semi-Fredholm spectrum of a continuous linear operator. We also examine the retention of the semiFredholm spectrum under weak intertwining relations; it is shown, inter alias, that if two decomposable operators are intertwined asymptotically by a quasiaffinity then they have identical semi-Fredholm spectra. The results are applied to multipliers on commutative semisimple Banach algebras.

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Section: General Local and Global Spectral Theorymentioning

confidence: 99%

Essential spectra through local spectral theory

Laursen¹

1997

Proc. Amer. Math. Soc.

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“…Albrecht and Eschmeier [2] showed that property (fi) completely characterizes the restrictions of decomposable operators to invariant subspaces and their analytic functional model shows that every Banach space operator is similar to the quotient of an operator with property (/3); see [5]. Moreover, Albrecht and Eschmeier prove properties (P) and (S) to be completely dual; an operator T has one of these precisely when T* has the other.…”

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An operator satisfying Dunford's condition (C) but without bishop's property (β)

Miller

1998

Glasgow Math. J.

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1. Introduction. For X a complex Banach space and U an open subset of the complex plane C, let O{U, X) denote the space of analytic X-valued functions defined on U. This is a Frechet space when endowed with the topology of uniform convergence on compact subsets, and the space X may be viewed as simply the constants in O(U, X). Every bounded operator T on X induces a continuous mapping , X) and X e U. Corresponding to each closed F c C there is also an associated analytic subspace X T (F) = XD ran(7c//r). For an arbitrary Te C(X), the spaces XT(F) are T-invariant, generally non-closed linear manifolds in X.An operator T e C(X) has the decomposition property (8) provided that the space X decomposes asis an open cover of the complex plane. T e C(X) is decomposable in the sense of Foias provided that T has property (<5) and that the analytic subspaces XT(F) are closed whenever F is a closed subset of the plane; see [1], [3].The local resolvent of Tata vector x e X is the set PT(X) consisting of all X e C for which there is a open neighborhood U such that x e TyO{U, X). The local spectrum of T at x is a-rix) = C\PT(X).If T is such that for every closed F c C the linear manifold An operator T has Bishop's property (ft) provided that for every open U C C the mapping Tu is injective and has closed range. Albrecht and Eschmeier [2] showed that property (fi) completely characterizes the restrictions of decomposable operators to invariant subspaces and their analytic functional model shows that every Banach space operator is similar to the quotient of an operator with property (/3); see [5]. Moreover, Albrecht and Eschmeier prove properties (P) and (S) to be completely dual; an operator T has one of these precisely when T* has the other.Property {p) implies ( Q , and ( Q in turn implies the single-valued extension property, [6, Proposition 1.2]. Therefore T e C(X) has property (C) if and only if the analytic subspaces XT(F) are closed whenever F is a closed subset of the plane, and T is decomposable if and only if T has both properties (C) and (8); equivalently, if and only if T has both properties (0) and (8). Thus it is a natural question whether property ( Q is strictly weaker than property (()). Laursen and Neumann mention it explicitly in [6], but this question has circulated informally for some time.

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Polaroid-Type Operators

Aiena

2018

Fredholm and Local Spectral Theory II

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Local spectral properties of commutators

Cited by 3 publications

References 15 publications

Essential spectra through local spectral theory

Essential spectra through local spectral theory

An operator satisfying Dunford's condition (C) but without bishop's property (β)

Polaroid-Type Operators

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