2022/08/13 by G. Ramesh, Ramesh, G., Shanola S. Sequeira +1
Mathematics · #Holomorphic and Operator Theory #Advanced Banach Space Theory #Advanced Topics in Algebra
paper · pdf · doi:10.48550/arxiv.2208.06574
Let H1, H2 be complex Hilbert spaces. A bounded linear operator T : H1 → H2 is said to be norm attaining if there exists a unit vector x ∈ H1 such that ‖Tx‖ = ‖T‖. If T|M : M → H2 is norm attaining for every closed subspace M of H1, then we say that T is an absolutely norm attaining (AN-operator). If the norm of the operator is replaced by the minimum modulus m(T) = inf\‖Tx‖ : x ∈ H1, ‖x‖ =1\, then T is said to be a minimum attaining and an absolutely minimum attaining operator (AM-operator), respectively. In this article, we give representations of quasinormal AN, AM-operators and the operators in the closure of these two classes. Later we extend these results to the class of hyponormal operators in the closure of AN-operators and a further look at some sufficient conditions under which these operators become normal.