Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Convexity and Star-shapedness of Matricial Range   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 $I_p\otimes B_j$ 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.

8
The categories of this paper are: hep-ex
18
The categories of this paper are: econ
36
The categories of this paper are: cmp-lg
5
The categories of this paper are: gr-qc
19
The categories of this paper are: eess
1
The categories of this paper are: cs
3
The categories of this paper are: physics
35
The categories of this paper are: mtrl-th
27
The categories of this paper are: comp-gas
28
The categories of this paper are: alg-geom
29
The categories of this paper are: funct-an
10
The categories of this paper are: q-bio
26
The categories of this paper are: bayes-an
9
The categories of this paper are: nlin
0
The categories of this paper are: hep-ph
17
The categories of this paper are: q-fin