Quasinormality and Numerical Ranges of Certain Classes of Dual Toeplitz Operators

The study of dual Toeplitz operators was elaborated by Stroethoff and Zheng (2002), where various corresponding algebraic and spectral properties were established. In this paper, we characterize numerical ranges of certain classes of dual Toeplitz operators. Moreover, we introduce the analog of Halmos' fifth classification problem for quasinormal dual Toeplitz operators. In particular, we show that there are no quasinormal dual Toeplitz operators with bounded analytic or coanalytic symbols which are not normal. 1. Introduction Let be the unit disk of the complex plane , and let be the Lebesgue measure on . The Lebesgue space of (classes of) square summable complex-valued functions is denoted by . The Bergman space is the Hilbert subspace of consisting of analytic functions. The orthogonal complement of in is denoted by . The Hilbert space is readily seen to be not a reproducing kernel Hilbert space. This is one of the major difficulties that occurs when dealing with this space. A second one is the fact that its elements have no standard common qualities such as analyticity harmonicity, while a lesser difficulty is the complicated form of the corresponding basis. Despite the difficulties just listed, Stroethoff and Zheng in [1, 2] have adopted new techniques to investigate various properties of a class of operators acting on , namely, dual Toeplitz operators. A dual Toeplitz operator is defined on to be a multiplication (by the symbol) followed by a projection onto . Although dual Toeplitz operators are different from Toeplitz operators in many respects, they do share some properties with them. But surprisingly, dual Toeplitz operators on resemble much more Hardy space Toeplitz operators than Bergman space Toeplitz operators. Lu in [3] and Cheng and Yu in [4] considered dual Toeplitz operators in higher dimensions; while Yu and Wu in [5] considered dual Toeplitz operators in the framework of Dirichlet spaces. The study of the numerical ranges of Hardy space Toeplitz operators goes back to Brown and Halmos [6]. Subsequent treatment was reconsidered in Halmos' book [7]. Later on, Klein [8] showed that the numerical range depends only on the spectrum of the given Hardy space Toeplitz operator. The Bergman space case was successfully considered only twenty years later by Thukral [9] in case of bounded harmonic symbols. More recently Choe and Lee [10],as well as Gu[11], treat higher-dimensional Bergman space analogs. The case of Bergman space Toeplitz operators with bounded radial symbols has been considered very recently by Wang and Wu [12]. The connection