Abstract |
Let λ be a complex number in the closed unit disc, and H be a separable Hilbert space with the orthonormal basis, say, ε={e_n:n=0,1,2,…}. A bounded operator T on H is called a λ-Toeplitz operator if < Te_{m+1},Te_{n+1} >=λ< Te_m,Te_n > (where <•,•> is the inner product on H). The subject arises just recently from a special case of the operator equation S*AS = λA + B, where S is a shift on H, which plays an essential role in finding bounded matrix (a_{ij}) on l^2(Z) that solves the system of equations 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} for all i, j ∈ Z, where (p_{ij}), (q_{ij}), (v_{ij}), (w_{ij}) are bounded matrices on l^2(Z) and a, b, c, d ∈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. The purpose of this paper is to discuss some basic topics, such as boundedness and compactness, of the λ-Toeplitz operators, and study the similarities and the differences with the corresponding results for the Toeplitz operators. |