Toeplitz-Hankel type operators on Dirichlet spaces

Abstract

LetLα,2 be the function space defined by the (weighted) Dirichlet energy integrals on the unit disk of complex plane. By constructing new orthogonal polynomials we give an orthogonal decomposition\(L^{\alpha ,2} = \oplus _{k = 0}^\infty (A_k \oplus \bar A_k )\) such thatA0α is just the (weighted) Dirichlet space. We define three kinds of Toeplitz and Hankel type operators, develop their boundedness andSp-criteria, and reveal cut-off phenomena and Wu' phenomena.