Isometries, direct sum decompositions, analytic type function spaces, and radial operators

Abstract

The paper connects rather general statements about the direct sum decomposition of a Hilbert space generated by isometries with the study of analytic type subspaces in weighted L_2-spaces over the unit disk {mathbb {D}} or the complex plane {mathbb {C}}, and with characterization of the so-called radial operators acting on these spaces. We characterize the properties of the so-called (m, n)-analytic functions, those that are annihilated by frac{partial ^m }{partial z^m}frac{partial ^n }{partial {{overline{z}}}^n}, which as particular cases include analytic, anti-analytic, harmonic, and various related to them functions. We characterize also the so-called radial operators, with a particular attention to Toeplitz ones, that act on different subspaces of our weighted L_2-spaces.