Subdifferential set of an operator

Document Type

Article

Publication Title

Monatshefte fur Mathematik

Abstract

Let X, Y be complex Banach spaces. Let L(X, Y) be the space of bounded operators. An important aspect of understanding differentiability is to study the subdifferential of the norm at a point, say x∈ X, this is the set, {f∈X∗:‖f‖=1,f(x)=‖x‖}. See page 7 in Deville et al. (Pitman Monographs and Surveys in Pure and Applied Mathematics. 64. Harlow: Longman Scientific and Technical. New York: John Wiley and Sons, Inc. 1993). Motivated by recent results of Singla (Singla in Linear Alg. Appl. 629:208–218, 2021) in the context of Hilbert spaces, for T∈ L(X, Y) , we determine the subdifferential of the operator norm at T, ∂T={Λ∈L(X,Y)∗:Λ(T)=‖T‖,‖Λ‖=1}. Our approach is based on the ‘position’ of the space of compact operators and the Calkin norm of T. Our ideas give a unified approach and extend several results from Singla (Linear Alg. Appl. 629:208–218, 2021) to the case of ℓp-spaces for 1 < p< ∞. We also investigate the converse, using the structure of the subdifferential set to decide when the Calkin norm is a strict contraction. As an application of these ideas, we partially solve the open problem of relating the subdifferential of the operator norm at a compact operator T to that of T(x) , where x is a unit vector where T attains its norm.

First Page

891

Last Page

898

DOI

10.1007/s00605-022-01739-5

Publication Date

12-1-2022

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