G, Ramesh
(2018)
Absolutely norm attaining paranormal operators.
Journal of Mathematical Analysis and Applications, 465 (1).
pp. 547556.
ISSN 0022247X
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Abstract
A bounded linear operator , where are Hilbert spaces is said to be norm attaining if there exists a unit vector such that . If for any closed subspace M of , the restriction of T to M is norm attaining, then T is called an absolutely norm attaining operator or operator. We prove the following characterization theorem: a positive operator T defined on an infinite dimensional Hilbert space H is an operator if and only if the essential spectrum of T is a single point and contains atmost finitely many points. Here and are the minimum modulus and essential minimum modulus of T, respectively. As a consequence we obtain a sufficient condition under which the property of an operator implies property of its adjoint. We also study the structure of paranormal operators and give a necessary and sufficient condition under which a paranormal operator is normal.
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