Operator Semigroups, Flows, and Their Invariant Sets

Froelich, John; Marsalli, Michael

doi:10.1006/jfan.1993.1100

Cited by 4 publications

(5 citation statements)

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“…Rodríguez-Piazza and the first author [6] showed that the position operator on the space of square integrable functions with respect to a finite measure of compact support is strongly compact if and only if the restriction of the measure to the exterior boundary of its support is purely atomic. (A similar result was obtained earlier on by Froelich and Marsalli [4] within the framework of function algebras.) Further applications were derived, for instance, the weakly closed algebra generated by a strongly compact normal operator need not be a strongly compact algebra.…”

Section: Introductionsupporting

confidence: 88%

A local spectral condition for strong compactness with some applications to bilateral weighted shifts

Martín

Rosa²

2013

Proc. Amer. Math. Soc.

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An algebra of bounded linear operators on a Banach space is said to be strongly compact if its unit ball is precompact in the strong operator topology, and a bounded linear operator on a Banach space is said to be strongly compact if the algebra with identity generated by the operator is strongly compact. Our interest in this notion stems from the work of Lomonosov on the existence of invariant subspaces. We provide a local spectral condition that is sufficient for a bounded linear operator on a Banach space to be strongly compact. This condition is then applied to describe a large class of strongly compact, injective bilateral weighted shifts on Hilbert spaces, extending earlier work of Fernández-Valles and the first author. Further applications are also derived, for instance, a strongly compact, invertible bilateral weighted shift is constructed in such a way that its inverse fails to be a strongly compact operator.2000 Mathematics Subject Classification. Primary 47B07; Secondary 47B37, 47L10.

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Section: Introductionsupporting

confidence: 88%

A local spectral condition for strong compactness with some applications to bilateral weighted shifts

Martín

Rosa²

2013

Proc. Amer. Math. Soc.

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“…The following is a characterization of the subalgebras R ⊆ L ∞ (µ) with the property that the unit ball of R is precompact in the strong operator topology. Theorem 1.1 (Froelich and Marsalli [4]). Let R be a subalgebra of L ∞ (µ).…”

Section: Introductionmentioning

confidence: 99%

“…Froelich and Marsalli [4] considered the problem of characterizing among the above function algebras the ones that have a unit ball that is precompact in the strong operator topology. They proved the following Theorem 1.2.…”

Section: Introductionmentioning

confidence: 99%

“…Froelich and Marsalli [4] claimed that the sufficient conditions for the unit ball to be strongly precompact stated in parts (4) and (5) of Theorem 1.2 were also necessary.…”

Section: Introductionmentioning

confidence: 99%

“…Let {G n } ∞ n=1 denote the sequence of the bounded components of C\K. Froelich and Marsalli [4] showed that if the function algebra R(K) has a strongly precompact unit ball then the measure µ restricted to each ∂G n is discrete. Then, they concluded that the measure µ restricted to ∂K is also discrete by using the identity…”

Section: Introductionmentioning

confidence: 99%

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Function algebras with a strongly precompact unit ball

Lacruz

Rodrı́guez-Piazza

2013

Journal of Functional Analysis

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Let µ be a finite positive Borel measure with compact support K ⊆ C, and regard L ∞ (µ) as an algebra of multiplication operators on the Hilbert space L 2 (µ). Then consider the subalgebra A(K) of all continuous functions on K that are analytic on the interior of K, and the subalgebra R(K) defined as the uniform closure of the rational functions with poles outside K. Froelich and Marsalli showed that if the restriction of the measure µ to the boundary of K is discrete then the unit ball of A(K) is strongly precompact, and that if the unit ball of R(K) is strongly precompact then the restriction of the measure µ to the boundary of each component of C\K is discrete. The aim of this paper is to provide three examples that go to clarify the results of Froelich and Marsalli; in particular, it is shown that the converses to both statements are false.

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Operator Semigroups, Invariant Sets and Invariant Subspaces

Froelich

Marsalli

1994

Algebraic Methods in Operator Theory

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Operator Semigroups, Flows, and Their Invariant Sets

Cited by 4 publications

References 0 publications

A local spectral condition for strong compactness with some applications to bilateral weighted shifts

A local spectral condition for strong compactness with some applications to bilateral weighted shifts

Function algebras with a strongly precompact unit ball

Operator Semigroups, Invariant Sets and Invariant Subspaces

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