On Toeplitz Operators on the Weighted Harmonic Bergman Space on the Upper Half-Plane

Abstract

We study Toeplitz operators on the weighted harmonic Bergman space on the upper half-plane. Two classes of symbols are considered here: symbols that depend only on the vertical variable and symbols that depend only on the angular variable. For the first case, we prove that Toeplitz operators with such kind of symbols generate a commutative $C^*$-algebra in every weighted harmonic Bergman space. This algebra is isomorphic to the algebra of all very slowly oscillating functions. On the other hand, Toeplitz operators whose symbols depend only on the angular variable generate a non commutative $C^*$-algebra which is isomorphic to the $C^*$-algebra of all $2\times 2$ matrix-valued continuous functions $(f_{ij}(t))$ defined on $[-\infty ,\infty ]$ and such that they satisfy $f_{12}(\pm \infty )=f_{21}(\pm \infty )=0$ and $f_{11}(\pm \infty )=f_{22}(\mp \infty )$.