HILBERT MODULES OVER THE SEGAL-BARGMANN SPACE AND p-ESSENTIAL NORMALITY

Different Hilbert module structures over the Segal-Bargmann space F² F 2 of Gaussian square-integrable entire functions are defined. These depend on classes of diagonal operators acting on F² F 2 with respect to decompositions into homogeneous subspaces. We first consider graded principal submodules generated by a single homogeneous polynomial. Generalizing results due to K. Guo and K. Wang, and under suitable conditions on the eigenvalue sequence defining the module structure, we prove p -essential normality for specific values of p . Starting from a specific commuting tuple of Toeplitz operators with homogeneous symbols in F² F 2 , we assign a decreasing scale of quotient modules to dilation-invariant subsets Ω ⊂ Cⁿ Ω ⊂ C n . Naturally, the question of essential or p -essential normality of such quotient modules arises.