Absolutely norm attaining paranormal operators

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:465

Article

Ramesh, G.¹ 

[1] IIT Hyderabad, Dept Math, E Block,305,Kandi 5, Sangareddy 502285, Telangana, India

关键词: Compact operator;    Norm attaining operator;    AN-operator;    Wcyl's theorem;    Paranormal operator;    Reducing subspace;   

DOI  :  10.1016/j.jmaa.2018.05.024

来源: Elsevier

【 摘 要 】

A bounded linear operator T : H-1 -> H-2, where H-1, H-2 are Hilbert spaces is said to be norm attaining if there exists a unit vector x is an element of H-1 such that parallel to T-x parallel to = parallel to T parallel to. If for any closed subspace M of H-1, the restriction T vertical bar(M) : M -> H-2 of T to M is norm attaining, then T is called an absolutely norm attaining operator or AN-operator. We prove the following characterization theorem: a positive operator T defined on an infinite dimensional Hilbert space H is an AN-operator if and only if the essential spectrum of T is a single point and [m(T), m(e)(T)) contains atmost finitely many points. Here m(T) and m(e)(T) are the minimum modulus and essential minimum modulus of T, respectively. As a consequence we obtain a sufficient condition under which the AN-property of an operator implies NI-property of its adjoint. We also study the structure of paranormal AN-operators and give a necessary and sufficient condition under which a paranormal AN-operator is normal. (C) 2018 Elsevier Inc. All rights reserved.

【 授权许可】

Free