Toeplitz $C^*$-algebras on radially weighted Fock spaces: commutativity and spectral representation

Abstract

We study Toeplitz operators acting on radial weighted Fock spaces. We use tools from representation theory to construct commutative families of $C^*$-algebras that are generated by Toeplitz operators whose symbols are invariant under the action of $\U(n)$. For a partition $m=(b_1,...,b_k)$ of an integer $n$, we realize \[\mathbf{\U_m}:=\U(b_1)\times...\times\U(b_k)\] as a block diagonally subgroup of $\U(n)$ and describe the decomposition of the weighted Fock space into irreducible $\mathbf{U_m}$-modules. This allows us to study Toeplitz operators with $\mathbf{U_m}$-invariant, equivalently $k$-quasi-radial, symbols. Also, we provide an explicit integral representation of their eigenvalues. More generally, for an arbitrary compact subgroup $H\subseteq\U(n)$, we characterize the commutativity of the $C^*$-algebra generated by $H$-invariant Toeplitz operators in terms of the multiplicity-free property of the representation $\pi|_H$. Finally, for a logarithmically growing radial weight, we construct a bounded radial symbol for which the corresponding eigenvalue sequence is not uniformly continuous with respect to the square-root metric. Consequently, the uniform closure of the set of eigenvalue sequences does not coincide with the $C^*$-algebra of bounded sequences that are uniformly continuous with respect to the square-root metric.