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 $\lambda$-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 from a special case of the operator equation \[ S^*AS=\lambda A+B,\ \mbox{ where $S$ is a shift on $\cal H$}, \] 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 $\lambda$-Toeplitz operators and, our main result is to give necessary and sufficient conditions for finite rank $\lambda$-Toeplitz operators.
Categories: math.FA