Toeplitz Algebras on the Harmonic Fock Space

Abstract

We study Toeplitz operators acting on the harmonic Fock space and consider two classes of symbols: radial and horizontal. Toeplitz operators with radial symbols behave quite similar in both settings, namely, the Fock and the harmonic Fock spaces. In fact, these operators generate a commutative C∗-algebra which is isomorphic to the algebra of uniformly continuous sequences with respect to the square root metric. On the contrary, Toeplitz operators with horizontal symbols on the harmonic Fock space do not commute in general. Nevertheless, up to compact perturbation, they have a similar behavior to the corresponding Toeplitz operators acting on the Fock space. In fact, we prove that the Calkin algebra of the C∗-algebra generated by Toeplitz operators with horizontal symbols is isomorphic to the algebra consisting of bounded uniformly continuous functions with respect to the standard metric on \(\mathbb {R}\), which, at the same time, is isomorphic to the C∗-algebra generated by Toeplitz operators with horizontal symbols acting on the Fock space.