$\lambda$-Toeplitz operators with analytic symbols

Abstract

Let $\lambda$ be a complex number in the closed unit disc $\overline{\mathbb{D}}$, and $\mathcal{H}$ be a separable Hilbert space with the orthonormal basis, say, $\mathcal{E}=\{e_n:n=0,1,2,\cdots \}$. A bounded operator $T$ on $\mathcal{H}$ is called a {\em $\lambda$-Toeplitz operator} if $⟨T{e}_{m+1},e_{n+1}⟩=\lambda ⟨T{e}_m,e_n⟩$ (where $⟨\cdot ,\cdot ⟩$ is the inner product on $\mathcal{H}$). The subject arises naturally as the "eigenoperators" of the map \[ ϕ(A)=S^*AS \] on the $\mathcal{B}(\mathcal{H})$, the space of bounded operators on $\mathcal{H}$, where $S$ is the unilateral shift on $S{e}_n=e_{n+1}$. In this paper, we study the essential spectra for $\lambda$-Toeplitz operators when $|\lambda |=1$, and we will use the results to determine the spectra of certain weighted composition operators on Hardy spaces.