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 publicationJournal articleAcademic researchpeer-review


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 (XA1X, … , XAmX) = (Ip⊗ B1, … , Ip⊗ Bm) for some real n× pq matrix X satisfying XX= 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.

Original languageEnglish
Pages (from-to)609-626
Number of pages18
JournalAdvances in Operator Theory
Issue number3
Publication statusPublished - 1 Jul 2020


  • Compact perturbation
  • Congruence numerical range
  • Convex
  • Star-shaped

ASJC Scopus subject areas

  • Analysis
  • Algebra and Number Theory


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