Operators polynomially isometric to a normal operator
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- by Laurent W. Marcoux and Yuanhang Zhang PDF
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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 $\| \cdot \|_u$ 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 $\| \cdot \|_u$ if $\| p(A) \|_u = \| p(B) \|_u$ 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 $\| \cdot \|_u$ 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.References
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Additional Information
- Laurent W. Marcoux
- Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
- MR Author ID: 288388
- Email: Laurent.Marcoux@uwaterloo.ca
- Yuanhang Zhang
- Affiliation: School of Mathematics, Jilin University, Changchun 130012, People’s Republic of China
- MR Author ID: 931652
- Email: zhangyuanhang@jlu.edu.cn
- 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
- © Copyright 2020 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 148 (2020), 2019-2033
- MSC (2010): Primary 47B15, 15A60, 15A21
- DOI: https://doi.org/10.1090/proc/14861
- MathSciNet review: 4078086