Convexity and star-shapedness of matricial range

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

Research output: Journal article publicationJournal articleAcademic researchpeer-review

6 Citations (Scopus)


Let A=(A 1,…,A m) be an m-tuple of bounded linear operators acting on a Hilbert space H. Their joint (p,q)-matricial range Λ p,q(A) is the collection of (B 1,…,B m)∈M q m, where I p⊗B j is a compression of A j on a pq-dimensional subspace. This definition covers various kinds of generalized numerical ranges for different values of p,q,m. In this paper, it is shown that Λ p,q(A) is star-shaped if the dimension of H is sufficiently large. If dim⁡H is infinite, we extend the definition of Λ p,q(A) to Λ ∞,q(A) consisting of (B 1,…,B m)∈M q m such that I ⊗B j is a compression of A j on a closed subspace of H, and consider the joint essential (p,q)-matricial range Λ p,q ess(A)=⋂{cl(Λ p,q(A 1+F 1,…,A m+F m)):F 1,…,F m are compact operators}. Both sets are shown to be convex, and the latter one is always non-empty and compact.

Original languageEnglish
Pages (from-to)2497-2515
Number of pages19
JournalJournal of Functional Analysis
Issue number9
Publication statusPublished - 1 Nov 2018


  • Bounded linear operators
  • Higher rank numerical range
  • Joint essential numerical range
  • Joint matricial range

ASJC Scopus subject areas

  • Analysis


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