A criterion for finite rank $\lambda$-Toeplitz operators

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 $\lambda$-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 from a special case of the operator equation \[ S^*AS=λA+B,\ \mbox{where $S$ is a shift on $\mathcal{H}$}, \] which plays an essential role in finding bounded matrix $(a_{ij})$ on $l^2(\mathbb{Z})$ that solves the system of equations $$\{\begin{array}{lcc}{a}_{2i,2j}& =& p_{ij}+a{a}_{ij}\\ a_{2i,2j-1}& =& q_{ij}+b{a}_{ij}\\ a_{2i-1,2j}& =& v_{ij}+c{a}_{ij}\\ a_{2i-1,2j-1}& =& w_{ij}+d{a}_{ij}\end{array}$$ for all $i,j\in \mathbb{Z}$, where $(p_{ij})$, $(q_{ij})$, $(v_{ij})$, $(w_{ij})$ are bounded matrices on $l^2(\mathbb{Z})$ and $a,b,c,d\in \mathbb{C}$. It is also clear that the well-known Toeplitz operators are precisely the solutions of $S^{\ast }AS=A$, 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.