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

Abstract

AbstractThe 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 $$L_2$$ L 2 -spaces over the unit disk $${\mathbb {D}}$$ D or the complex plane $${\mathbb {C}}$$ 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 $$\frac{\partial ^m }{\partial z^m}\frac{\partial ^n }{\partial {{\overline{z}}}^n}$$ ∂ m ∂ z m ∂ n ∂ 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 $$L_2$$ L 2 -spaces.