U-invariant kernels, defect operators, and graded submodules

摘要

Let κ be an U-invariant reproducing kernel and let H(κ) denote the reproducing kernel Hilbert C[z1, ..., zd]-module associated with the kernel κ. Let Mz denote the d-tuple of multiplication operators Mz1, ..., Mzd on H(κ).For a positive integer ν and d-tuple T=(T1, ..., Td), consider the defect operator DT*,ν:= ∑l=0ν (-1)l {ν choose l} {∑|p|=l(l!/p!){Tp}{T*}p}. The first main result of this paper describes all U-invariant kernels κ which admit finite rank defect operators DM*z, ν. These areU-invariant polynomial perturbations of R-linear combinations of the kernels κν, whereκν(z, w)=(1/(1-ınp{z){w})ν} for a positive integer ν.We then formulate a notion of pure row ν-hypercontraction, and use it to show that certain row ν-hypercontractions correspondto an A-morphism. This result enables us to obtain an analog of Arveson's Theorem F for graded submodules of H(κν). It turns out that for μ ν, there are no nonzero graded submodules M of H(κν) (ν ≧ 2) with finite rank defect D(Mz|M)*,μ).