A Bishop-Phelps-Bollobás type property for minimum attaining operators

Bala, Neeru and Ramesh, G. (2021) A Bishop-Phelps-Bollobás type property for minimum attaining operators. Operators and Matrices (2). pp. 497-513. ISSN 1846-3886

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Abstract

t. In this article, we study the Bishop-Phelps-Bollob´as type theorem for minimum attaining operators. More explicitly, if we consider a bounded linear operator T on a Hilbert space H and a unit vector x0 ∈ H such that T x0 is very close to the minimum modulus of T , then T and x0 are simultaneously approximated by a minimum attaining operator S on H and a unit vector y ∈ H for which Sy is equal to the minimum modulus of S . Further, we extend this result to a more general class of densely defined closed operators (need not be bounded) in Hilbert space. As a consequence, we get the denseness of the set of minimum attaining operators in the class of densely defined closed operators with respect to the gap metric.

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