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- W2766615242 abstract "Let ${bf A} = (A_1, dots, A_m)$ be an $m$-tuple of bounded linear operators acting on a Hilbert space ${cal H}$. Their joint $(p,q)$-matricial range $Lambda_{p,q}({bf A})$ is the collection of $(B_1, dots, B_m) in {bf M}_q^m$, where $I_potimes B_j$ 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 $Lambda_{p,q}({bf A})$ is star-shaped if the dimension of $cal H$ is sufficiently large. If $dim {cal H}$ is infinite, we extend the definition of $Lambda_{p,q}({bf A})$ to $Lambda_{infty,q}({bf A})$ consisting of $(B_1, dots, B_m) in {bf M}_q^m$ such that $I_infty otimes B_j$ is a compression of $A_j$ on a closed subspace of ${cal H}$, and consider the joint essential $(p,q)$-matricial range $$Lambda^{ess}_{p,q}({bf A}) = bigcap { {bf cl}(Lambda_{p,q}(A_1+F_1, dots, A_m+F_m)): F_1, dots, F_m hbox{ are compact operators}}.$$ Both sets are shown to be convex, and the latter one is always non-empty and compact." @default.
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- W2766615242 date "2017-10-26" @default.
- W2766615242 modified "2023-09-27" @default.
- W2766615242 title "Convexity and Star-shapedness of Matricial Range" @default.
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