$\lambda$-Toeplitz operators with analytic symbols

Let $\lambda$ be a complex number in the closed unit disc $\overline{\Bbb D}$, and $\cal H$ be a separable Hilbert space with the orthonormal basis, say, ${\cal E}=\{e_n:n=0,1,2,\cdots\}$. A bounded operator $T$ on $\cal H$ is called a {\em $\lambda$-Toeplitz operator} if $\langle Te_{m+1},e_{n+1}\rangle=\lambda\langle Te_m,e_n\rangle$ (where $\langle\cdot,\cdot\rangle$ is the inner product on $\cal H$). The subject arises naturally as the "eigenoperators" of the map $$\phi(A)=S^*AS$$ on the ${\cal B}(\cal H)$, the space of bounded operators on $\cal H$, where $S$ is the unilateral shift on $Se_n=e_{n+1}$. 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.