Open Access
July 2018 Reflexive sets of operators
Janko Bračič, Cristina Diogo, Michal Zajac
Banach J. Math. Anal. 12(3): 751-771 (July 2018). DOI: 10.1215/17358787-2018-0002


For a set M of operators on a complex Banach space X, the reflexive cover of M is the set Ref(M) of all those operators T satisfying TxMx¯ for every xX. Set M is reflexive if Ref(M)=M. The notion is well known, especially for Banach algebras or closed spaces of operators, because it is related to the problem of invariant subspaces. We study reflexivity for general sets of operators. We are interested in how the reflexive cover behaves towards basic operations between sets of operators. It is easily seen that the intersection of an arbitrary family of reflexive sets is reflexive, as well. However this does not hold for unions, since the union of two reflexive sets of operators is not necessarily a reflexive set. We give some sufficient conditions under which the union of reflexive sets is reflexive. We explore how the reflexive cover of the sum (resp., the product) of two sets is related to the reflexive covers of summands (resp., factors). We also study the relation between reflexivity and convexity, with special interest in the question: under which conditions is the convex hull of a reflexive set reflexive?


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Janko Bračič. Cristina Diogo. Michal Zajac. "Reflexive sets of operators." Banach J. Math. Anal. 12 (3) 751 - 771, July 2018.


Received: 21 September 2017; Accepted: 24 January 2018; Published: July 2018
First available in Project Euclid: 18 May 2018

zbMATH: 06946080
MathSciNet: MR3824750
Digital Object Identifier: 10.1215/17358787-2018-0002

Primary: 47A99
Secondary: 47L05 , ‎47L07

Keywords: convex set of operators , locally linearly dependent operators , reflexive set of operators

Rights: Copyright © 2018 Tusi Mathematical Research Group

Vol.12 • No. 3 • July 2018
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