Abstract:

Let $H^{2}$ be the Hardy space on the unit disk ${\Bbb D}=\{\xi\in{\Bbb C}:|\xi|<1\}$. Suppose $u$ and $v$ are inner functions and at least one of them is nonconstant, the harmonic Hardy space $H_{u,v}^{2}$ is defined by $H_{u,v}^{2}=uH^{2}\oplus\overline{v}(H^{2})^{\bot}=uH^{2}\oplus\overline{vzH^{2}}.$ For any $x\in H_{u,v}^{2},$ define the Toeplitz operator on the $H_{u,v}^{2}$ by $\widehat{T}_{\varphi}x=Q_{u,v}(\varphi x),$ where $Q_{u,v}$ is the orthogonal projection from $L^{2}\rightarrow H_{u,v}^{2}.$ In this paper, the unitary equivalence of the harmonic Toeplitz operator and the dual truncated Toeplitz operator is obtained, moreover, the sufficient and necessary conditions for when two Toeplitz operators commute is given, and the properties of the harmonic Toeplitz algebra and the commutant of $\widehat{T}_{z}$ are described. Finally, the essential spectrum for the product of finitely many harmonic Toeplitz operators with continuous symbols is obtained in this paper.

Key words: Harmonic Hardy space, Harmonic Toeplitz operator, Unitary equivalence, Dual truncated Toeplitz operator, Essential spectrum