Properties of two variables Toeplitz type operators

Abstract

The investigation of properties of generalized Toeplitz operators with respect to the pairs of doubly commuting contractions (the abstract analogue of classical two variable Toeplitz operators) is proceeded. We especially concentrate on the condition of existence such a non-zero operator. There are also presented conditions of analyticity of such an operator. 1. Introduction Let L(H1, H2) denote the algebra of all bounded linear operators from H1 into H2, where H1, H2 are complex, separable Hilbert spaces. If H1 = H2 we will use the notation L(H1). The classical Toeplitz operators on the Hardy space on the unit disc are well known and they are fully characterized by the relation X = T z∗ XTz, where Tz is the shift operator – the multiplication operator by the independent variable on the Hardy space H2 on the circle T. This notion can be generalized when instead of the backward shift T z∗ in the equation above we will put arbitrary, possibly different, contractions. Namely, for given contractions S ∈L(H1) and T ∈L(H2), an operator X ∈L(H2, H1) is called generalized Toeplitz operator if X = SXT ∗. These type of operators were studied in [1, 6, 11]. The classical Toeplitz operators are also considered on the Hardy space on the torus H2(T2). The space H2(T2) can be seen as a subspace of L2(T2) = L2(T2, m ⊗m) (m denotes the normalized Lebesgue measure on T) and PH2(T2) is the appropriate projection. For any ϕ ∈L∞(T2) = L∞(T2, m ⊗m) we define the Toeplitz operator Tϕ ∈L(H2(T2)) by Tϕf = PH2(T2)(ϕf) (f ∈H2(T2)).