New York Journal of Mathematics Volume 22 (2016) 677-709
 Volume 22 Volume Index

U-invariant kernels, defect operators, and graded submodules

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 Published: July23, 2016 Keywords: U-invariant kernel, Graded submodule, A-morphism, Row ν-contraction Subject: Primary 47A13, 47A15, 46E22; Secondary 47B10, 47B32

Abstract

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 are U-invariant polynomial perturbations of R-linear combinations of the kernels κν, where κν(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 correspond to 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).

Author information

Sameer Chavan:
Indian Institute of Technology Kanpur, Kanpur- 208016, India
chavan@iitk.ac.in

Rani Kumari:
Indian Institute of Technology Kanpur, Kanpur- 208016, India
rani@iitk.ac.in