Toeplitz Operators on the Fock Space with Quasi-Radial Symbols

Abstract

The Fock space $${\mathcal {F}}({\mathbb {C}}^n)$$ is the space of holomorphic functions on $${\mathbb {C}}^n$$ that are square-integrable with respect to the Gaussian measure on $${\mathbb {C}}^n$$ . This space plays an important role in several subfields of analysis and representation theory. In particular, it has for a long time been a model to study Toeplitz operators. Esmeral and Maximenko showed in 2016 that radial Toeplitz operators on $${\mathcal {F}}({\mathbb {C}})$$ generate a commutative $$C^*$$ -algebra which is isometrically isomorphic to the $$C^*$$ -algebra $$C_{b,u}({\mathbb {N}}_0,\rho _1)$$ . In this article, we extend the result to k-quasi-radial symbols acting on the Fock space $${\mathcal {F}}({\mathbb {C}}^n)$$ . We calculate the spectra of the said Toeplitz operators and show that the set of all eigenvalue functions is dense in the $$C^*$$ -algebra $$C_{b,u}({\mathbb {N}}_0^k, \rho _k)$$ of bounded functions on $${\mathbb {N}}_0^k$$ which are uniformly continuous with respect to the square-root metric. In fact, the $$C^*$$ -algebra generated by Toeplitz operators with quasi-radial symbols is $$C_{b,u}({\mathbb {N}}_0^k, \rho _k)$$ .