Product of operators and numerical range

Mao Ting Chien; Hwa Long Gau; Chi Kwong Li; Ming Cheng Tsai; Kuo Zhong Wang

doi:10.1080/03081087.2015.1024195

Product of operators and numerical range

Mao Ting Chien, Hwa Long Gau, Chi Kwong Li, Ming Cheng Tsai, Kuo Zhong Wang

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

14 Scopus citations

Abstract

We show that a bounded linear operator (Formula presented.) is a multiple of a unitary operator if and only if (Formula presented.) and (Formula presented.) always have the same numerical radius or the same numerical range for all (rank one) (Formula presented.). More generally, for any bounded linear operators (Formula presented.) , we show that (Formula presented.) and (Formula presented.) always have the same numerical radius (resp., the same numerical range) for all (rank one) (Formula presented.) if and only if (Formula presented.) (resp., (Formula presented.)) is a multiple of a unitary operator for some (Formula presented.). We extend the result to other types of generalized numerical ranges including the (Formula presented.) -numerical range and the higher rank numerical range.

Original language	English
Pages (from-to)	58-67
Number of pages	10
Journal	Linear and Multilinear Algebra
Volume	64
Issue number	1
DOIs	https://doi.org/10.1080/03081087.2015.1024195
State	Published - 2 Jan 2016

Keywords

higher rank numerical radius
higher rank numerical range
k-numerical radius
k-numerical range
numerical radius
numerical range

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10.1080/03081087.2015.1024195

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abstract = "We show that a bounded linear operator (Formula presented.) is a multiple of a unitary operator if and only if (Formula presented.) and (Formula presented.) always have the same numerical radius or the same numerical range for all (rank one) (Formula presented.). More generally, for any bounded linear operators (Formula presented.) , we show that (Formula presented.) and (Formula presented.) always have the same numerical radius (resp., the same numerical range) for all (rank one) (Formula presented.) if and only if (Formula presented.) (resp., (Formula presented.)) is a multiple of a unitary operator for some (Formula presented.). We extend the result to other types of generalized numerical ranges including the (Formula presented.) -numerical range and the higher rank numerical range.",

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T1 - Product of operators and numerical range

AU - Chien, Mao Ting

AU - Gau, Hwa Long

AU - Li, Chi Kwong

AU - Tsai, Ming Cheng

AU - Wang, Kuo Zhong

PY - 2016/1/2

Y1 - 2016/1/2

N2 - We show that a bounded linear operator (Formula presented.) is a multiple of a unitary operator if and only if (Formula presented.) and (Formula presented.) always have the same numerical radius or the same numerical range for all (rank one) (Formula presented.). More generally, for any bounded linear operators (Formula presented.) , we show that (Formula presented.) and (Formula presented.) always have the same numerical radius (resp., the same numerical range) for all (rank one) (Formula presented.) if and only if (Formula presented.) (resp., (Formula presented.)) is a multiple of a unitary operator for some (Formula presented.). We extend the result to other types of generalized numerical ranges including the (Formula presented.) -numerical range and the higher rank numerical range.

AB - We show that a bounded linear operator (Formula presented.) is a multiple of a unitary operator if and only if (Formula presented.) and (Formula presented.) always have the same numerical radius or the same numerical range for all (rank one) (Formula presented.). More generally, for any bounded linear operators (Formula presented.) , we show that (Formula presented.) and (Formula presented.) always have the same numerical radius (resp., the same numerical range) for all (rank one) (Formula presented.) if and only if (Formula presented.) (resp., (Formula presented.)) is a multiple of a unitary operator for some (Formula presented.). We extend the result to other types of generalized numerical ranges including the (Formula presented.) -numerical range and the higher rank numerical range.

KW - higher rank numerical radius

KW - higher rank numerical range

KW - k-numerical radius

KW - k-numerical range

KW - numerical radius

KW - numerical range

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DO - 10.1080/03081087.2015.1024195

M3 - 期刊論文

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SN - 0308-1087

VL - 64

SP - 58

EP - 67

JO - Linear and Multilinear Algebra

JF - Linear and Multilinear Algebra

IS - 1

ER -

Product of operators and numerical range

Abstract

Keywords

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