Abstract
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 dimH 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 language | English |
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Pages (from-to) | 2497-2515 |
Number of pages | 19 |
Journal | Journal of Functional Analysis |
Volume | 275 |
Issue number | 9 |
DOIs | |
Publication status | Published - 1 Nov 2018 |
Keywords
- Bounded linear operators
- Higher rank numerical range
- Joint essential numerical range
- Joint matricial range
ASJC Scopus subject areas
- Analysis