Joint Matricial Range and Joint Congruence Matricial Range of Operators

Pan Shun Lau; Chi Kwong Li; Yiu Tung Poon; Nung Sing Sze

doi:10.1007/s43036-019-00009-w

Joint Matricial Range and Joint Congruence Matricial Range of Operators

Pan Shun Lau, Chi Kwong Li, Yiu Tung Poon, Nung Sing Sze

Research output: Journal article publication › Journal article › Academic research › peer-review

Abstract

Let A= (A₁, … , A_m) , where A₁, … , A_m are n× n real matrices. The real joint (p, q)-matricial range of A, Λp,qR(A), is the set of m-tuple of q× q real matrices (B₁, … , B_m) such that (X^∗A₁X, … , X^∗A_mX) = (I_p⊗ B₁, … , I_p⊗ B_m) for some real n× pq matrix X satisfying X^∗X= I_pq. It is shown that if n is sufficiently large, then the set Λp,qR(A) is non-empty and star-shaped. The result is extended to bounded linear operators acting on a real Hilbert space H, and used to show that the joint essential (p, q)-matricial range of A is always compact, convex, and non-empty. Similar results for the joint congruence matricial ranges on complex operators are also obtained.

Original language	English
Pages (from-to)	609-626
Number of pages	18
Journal	Advances in Operator Theory
Volume	5
Issue number	3
DOIs	https://doi.org/10.1007/s43036-019-00009-w
Publication status	Published - 1 Jul 2020

Keywords

Compact perturbation
Congruence numerical range
Convex
Star-shaped

ASJC Scopus subject areas

Analysis
Algebra and Number Theory

More information

10.1007/s43036-019-00009-w

http://hdl.handle.net/10397/98542

Cite this

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title = "Joint Matricial Range and Joint Congruence Matricial Range of Operators",

abstract = "Let A= (A1, … , Am) , where A1, … , Am are n× n real matrices. The real joint (p, q)-matricial range of A, Λp,qR(A), is the set of m-tuple of q× q real matrices (B1, … , Bm) such that (X∗A1X, … , X∗AmX) = (Ip⊗ B1, … , Ip⊗ Bm) for some real n× pq matrix X satisfying X∗X= Ipq. It is shown that if n is sufficiently large, then the set Λp,qR(A) is non-empty and star-shaped. The result is extended to bounded linear operators acting on a real Hilbert space H, and used to show that the joint essential (p, q)-matricial range of A is always compact, convex, and non-empty. Similar results for the joint congruence matricial ranges on complex operators are also obtained.",

keywords = "Compact perturbation, Congruence numerical range, Convex, Star-shaped",

author = "Lau, {Pan Shun} and Li, {Chi Kwong} and Poon, {Yiu Tung} and Sze, {Nung Sing}",

year = "2020",

month = jul,

day = "1",

doi = "10.1007/s43036-019-00009-w",

language = "English",

volume = "5",

pages = "609--626",

journal = "Advances in Operator Theory",

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TY - JOUR

T1 - Joint Matricial Range and Joint Congruence Matricial Range of Operators

AU - Lau, Pan Shun

AU - Li, Chi Kwong

AU - Poon, Yiu Tung

AU - Sze, Nung Sing

PY - 2020/7/1

Y1 - 2020/7/1

N2 - Let A= (A1, … , Am) , where A1, … , Am are n× n real matrices. The real joint (p, q)-matricial range of A, Λp,qR(A), is the set of m-tuple of q× q real matrices (B1, … , Bm) such that (X∗A1X, … , X∗AmX) = (Ip⊗ B1, … , Ip⊗ Bm) for some real n× pq matrix X satisfying X∗X= Ipq. It is shown that if n is sufficiently large, then the set Λp,qR(A) is non-empty and star-shaped. The result is extended to bounded linear operators acting on a real Hilbert space H, and used to show that the joint essential (p, q)-matricial range of A is always compact, convex, and non-empty. Similar results for the joint congruence matricial ranges on complex operators are also obtained.

AB - Let A= (A1, … , Am) , where A1, … , Am are n× n real matrices. The real joint (p, q)-matricial range of A, Λp,qR(A), is the set of m-tuple of q× q real matrices (B1, … , Bm) such that (X∗A1X, … , X∗AmX) = (Ip⊗ B1, … , Ip⊗ Bm) for some real n× pq matrix X satisfying X∗X= Ipq. It is shown that if n is sufficiently large, then the set Λp,qR(A) is non-empty and star-shaped. The result is extended to bounded linear operators acting on a real Hilbert space H, and used to show that the joint essential (p, q)-matricial range of A is always compact, convex, and non-empty. Similar results for the joint congruence matricial ranges on complex operators are also obtained.

KW - Compact perturbation

KW - Congruence numerical range

KW - Convex

KW - Star-shaped

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U2 - 10.1007/s43036-019-00009-w

DO - 10.1007/s43036-019-00009-w

M3 - Journal article

AN - SCOPUS:85079815834

SN - 2538-225X

VL - 5

SP - 609

EP - 626

JO - Advances in Operator Theory

JF - Advances in Operator Theory

IS - 3

ER -

Joint Matricial Range and Joint Congruence Matricial Range of Operators

Abstract

Keywords

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