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dc.contributor.authorMarcoux, Laurent W.
dc.contributor.authorZhang, Yuanhang 18:50:03 (GMT) 18:50:03 (GMT)
dc.descriptionFirst published in Proceedings of the American Mathematical Society in volume 148 issue 5 in the year 2020, published by the American Mathematical Societyen
dc.description.abstractLet H be a complex, separable Hilbert space and B(H) denote the algebra of all bounded linear operators acting on H. Given a unitarily-invariant norm k · ku on B(H) and two linear operators A and B in B(H), we shall say that A and B are polynomially isometric relative to k · ku if kp(A)ku = kp(B)ku 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 k · ku is any unitarilyinvariant norm on Mn(C), if A, N ∈ Mn(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 ∈ B(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 non-normal 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.en
dc.description.sponsorshipThe 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).en
dc.publisherAmerican Mathematical Societyen
dc.subjectpolynomially isometricen
dc.subjectnormal operatorsen
dc.subjectunitarily-invariant normen
dc.subject(c, p)-normen
dc.subjectsingular valuesen
dc.subjectlavrentieff spectrumen
dcterms.bibliographicCitationMarcoux, L. W., & Zhang, Y. (2020). Operators polynomially isometric to a normal operator. Proceedings of the American Mathematical Society, 148(5), 2019–2033.
uws.contributor.affiliation1Faculty of Mathematicsen
uws.contributor.affiliation2Pure Mathematicsen

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