Toeplitz $C^*$-algebras on radially weighted Fock spaces: commutativity and spectral representation
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 allow us to study Toeplitz operators with symbols that are invariant under $\mathbf{U_m}$ and $k$-quasi-radial symbols, and we provide and 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 $π|_H$. Finally, for a logarithmically growing radial weight, we construct a bounded radial symbol for which the corresponding eigenvalue sequence 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.