A criterion for finite rank $λ$-Toeplitz operators

Abstract

Let $λ$ 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 $λ$-Toeplitz operator if $$ \langle Te_{m+1},e_{n+1}\rangle=λ\langle Te_m,e_n\rangle $$ (where $\langle\cdot,\cdot\rangle$ is the inner product on $\cal H$). The subject arises naturally from a special case of the operator equation \[ S^*AS=λA+B,\ \mbox{where $S$ is a shift on $\cal H$}, \] which plays an essential role in finding bounded matrix $(a_{ij})$ on $l^2(\Bbb Z)$ that solves the system of equations $$ \left\{\begin{array}{lcc} a_{2i,2j}&=&p_{ij}+aa_{ij}\\ a_{2i,2j-1}&=&q_{ij}+ba_{ij}\\ a_{2i-1,2j}&=&v_{ij}+ca_{ij}\\ a_{2i-1,2j-1}&=&w_{ij}+da_{ij} \end{array}\right. $$ for all $i,j\in\Bbb Z$, where $(p_{ij})$, $(q_{ij})$, $(v_{ij})$, $(w_{ij})$ are bounded matrices on $l^2(\Bbb Z)$ and $a,b,c,d\in\Bbb C$. It is also clear that the well-known Toeplitz operators are precisely the solutions of $S^*AS=A$, when $S$ is the unilateral shift. In this paper we verify some basic issues, such as boundedness and compactness, for $λ$-Toeplitz operators and, our main result is to give necessary and sufficient conditions for finite rank $λ$-Toeplitz operators.