Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Toeplitz operators, $\mathbb{T}^m$-invariance and quasi-homogeneous
  symbols   For a partition $\boldsymbol{k} = (k_1, \dots, k_m)$ of $n$ consider the
group $\mathrm{U}(\boldsymbol{k}) = \mathrm{U}(k_1) \times \dots \times
\mathrm{U}(k_m)$ block diagonally embedded in $\mathrm{U}(n)$ and the center
$\mathbb{T}^m$ of $\mathrm{U}(\boldsymbol{k})$. We study the Toeplitz operators
with $\mathbb{T}^m$-invariant symbols acting on the weighted Bergman spaces on
the unit ball $\mathbb{B}^n$. We introduce the
$(\boldsymbol{k},j)$-quasi-radial quasi-homogeneous symbols as those that are
invariant under the group $\mathrm{U}(\boldsymbol{k},j,\mathbb{T})$ obtained
from $\mathrm{U}(\boldsymbol{k})$ by replacing the factor $\mathrm{U}(k_j)$
with its center $\mathbb{T}$. These symbols are used to build commutative
Banach non-$C^*$ algebras generated by Toeplitz operators. These algebras
generalize those from the literature and show that they can be built using
groups. We describe the action of such Toeplitz operators on monomials through
explicit integral formulas involving the symbols. We prove that every Toeplitz
operator with $\mathbb{T}^m$-invariant symbol has an associated Toeplitz
operator with $\mathrm{U}(\boldsymbol{k})$-invariant symbol in terms of which
we can describe some properties.

21
The categories of this paper are: adap-org
7
The categories of this paper are: hep-th
12
The categories of this paper are: nucl-th
31
The categories of this paper are: ao-sci
24
The categories of this paper are: dg-ga
6
The categories of this paper are: astro-ph
25
The categories of this paper are: patt-sol
3
The categories of this paper are: physics
28
The categories of this paper are: alg-geom
22
The categories of this paper are: chao-dyn
16
The categories of this paper are: stat
37
The categories of this paper are: supr-con
33
The categories of this paper are: plasm-ph
8
The categories of this paper are: hep-ex
26
The categories of this paper are: bayes-an
34
The categories of this paper are: chem-ph