Sub-Bergman Hilbert Spaces on the Unit Disk

If the contraction T is a Toeplitz operator on H2 or A2 induced by an analytic function φ, we then denote the resulting space by H(φ). Similarly, if T is the Toeplitz operator on H2 or A2 induced by a conjugate analytic symbol φ, then we denote the resulting space by H(φ). In the context of Hardy spaces, H(φ) and H(φ) are called sub-Hardy Hilbert spaces by Sarason in [6]. We thus arrive at the title of the present paper. The theory of sub-Hardy Hilbert spaces was developped by de Branges, Rovnyak, Sarason, and some of their students and collaborators. Sarason’s recent monograph [6] presents most of the main developments in this area. The purpose of this paper is to examine some of the problems considered in [6] in the context of Bergman spaces. Our approach will be via the general theory of reproducing kernels. As a by-product of our analysis we shall obtain a sharper version of a result of Hedenmalm’s about extremal functions for invariant subspaces of the Bergman space.