Conditions Implying Self-adjointness and Normality of Operators

In this paper, we give new characterizations of self-adjoint and normal operators on a Hilbert space H . Among other results, we show that if H is a finite-dimensional Hilbert space and T ∈ B ( H ) , then T is self-adjoint if and only if there exists p > 0 such that | T | p ≤ | Re ( T ) | p . If...

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Bibliographic Details
Published in:Complex analysis and operator theory 2024-09, Vol.18 (6), Article 149
Main Author: Hranislav, Stanković
Format: Article
Language:English
Subjects:
Analysis
Apexes
Hilbert space
Mathematics
Mathematics and Statistics
Operator Theory
Operators
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Summary:In this paper, we give new characterizations of self-adjoint and normal operators on a Hilbert space H . Among other results, we show that if H is a finite-dimensional Hilbert space and T ∈ B ( H ) , then T is self-adjoint if and only if there exists p > 0 such that | T | p ≤ | Re ( T ) | p . If in addition, T and Re T are invertible, then T is self-adjoint if and only if log | T | ≤ log | Re ( T ) | . Considering the polar decomposition T = U | T | of T ∈ B ( H ) , we show that T is self-adjoint if and only if T is p -hyponormal (log-hyponormal) and U is self-adjoint. Also, if T = U | T | ∈ B ( H ) is a log-hyponormal operator and the spectrum of U is contained within the set of vertices of a regular polygon, then T is necessarily normal.
ISSN:1661-8254
1661-8262
DOI:10.1007/s11785-024-01596-0