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Operators polynomially isometric to a normal operator


Authors: Laurent W. Marcoux and Yuanhang Zhang
Journal: Proc. Amer. Math. Soc. 148 (2020), 2019-2033
MSC (2010): Primary 47B15, 15A60, 15A21
DOI: https://doi.org/10.1090/proc/14861
Published electronically: January 15, 2020
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Abstract: Let $ \mathcal {H}$ be a complex, separable Hilbert space and let $ \mathcal {B}(\mathcal {H})$ denote the algebra of all bounded linear operators acting on $ \mathcal {H}$. Given a unitarily-invariant norm $ \Vert \cdot \Vert _u$ on $ \mathcal {B}(\mathcal {H})$ and two linear operators $ A$ and $ B$ in $ \mathcal {B}(\mathcal {H})$, we shall say that $ A$ and $ B$ are polynomially isometric relative to $ \Vert \cdot \Vert _u$ if $ \Vert p(A) \Vert _u = \Vert p(B) \Vert _u$ for all polynomials $ p$. In this paper, we examine to what extent an operator $ A$ being polynomially isometric to a normal operator $ N$ implies that $ A$ is itself normal. More explicitly, we first show that if $ \Vert \cdot \Vert _u$ is any unitarily-invariant norm on $ \mathbb{M}_n(\mathbb{C})$, if $ A, N \in \mathbb{M}_n(\mathbb{C})$ are polynomially isometric and $ N$ is normal, then $ A$ is normal. We then extend this result to the infinite-dimensional setting by showing that if $ A, N \in \mathcal {B}(\mathcal {H})$ are polynomially isometric relative to the operator norm and $ N$ is a normal operator whose spectrum neither disconnects the plane nor has interior, then $ A$ is normal, while if the spectrum of $ N$ is not of this form, then there always exists a nonnormal operator $ B$ such that $ B$ and $ N$ are polynomially isometric. Finally, we show that if $ A$ and $ N$ are compact operators with $ N$ normal, and if $ A$ and $ N$ are polynomially isometric with respect to the $ (c,p)$-norm studied by Chan, Li, and Tu, then $ A$ is again normal.


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Additional Information

Laurent W. Marcoux
Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
Email: Laurent.Marcoux@uwaterloo.ca

Yuanhang Zhang
Affiliation: School of Mathematics, Jilin University, Changchun 130012, People’s Republic of China
Email: zhangyuanhang@jlu.edu.cn

DOI: https://doi.org/10.1090/proc/14861
Keywords: Polynomially isometric, normal operators, unitarily-invariant norm, $(c,p)$-norm, singular values, Lavrentieff spectrum
Received by editor(s): May 17, 2019
Received by editor(s) in revised form: September 5, 2019
Published electronically: January 15, 2020
Additional Notes: The first author’s research was supported in part by NSERC (Canada).
The second author’s research was supported by the Natural Science Foundation for Young Scientists of Jilin Province (No.: 20190103028JH), NNSF of China (No.: 11601104, 11671167, 11201171), and the China Scholarship Council (No.201806175122).
Communicated by: Stephan Ramon Garcia
Article copyright: © Copyright 2020 American Mathematical Society