 

Sameer Chavan and Rani Kumari
Uinvariant kernels, defect operators, and graded submodules view print


Published: 
July23, 2016 
Keywords: 
Uinvariant kernel, Graded submodule, Amorphism, Row νcontraction 
Subject: 
Primary 47A13, 47A15, 46E22; Secondary 47B10, 47B32 


Abstract
Let κ be an Uinvariant reproducing kernel and let H(κ) denote the reproducing kernel Hilbert C[z_{1}, ..., z_{d}]module associated with the kernel κ.
Let M_{z} denote the dtuple of multiplication operators M_{z1}, ..., M_{zd} on H(κ).
For a positive integer ν and dtuple T=(T_{1}, ..., T_{d}), consider the defect operator
D_{T*,
ν}:= ∑_{l=0}^{ν} (1)^{l} {ν \choose l} {∑_{p=l}(l!/p!){T^{p}}{T*}^{p}}.
The first main result of this paper describes all Uinvariant kernels κ which admit finite rank defect operators D_{M*z, ν}. These are
Uinvariant polynomial perturbations of Rlinear combinations of the kernels κ_{ν}, where
κ_{ν}(z, w)=(1/(1ınp{z){w})^{ν}} for a positive integer ν.
We then formulate a notion of pure
row νhypercontraction, and use it to show that certain row νhypercontractions correspond
to an Amorphism.
This result enables us to obtain an analog of Arveson's Theorem F for graded submodules of H(κ_{ν}).
It turns out that for μ < ν, there are no nonzero graded submodules M of H(κ_{ν}) (ν ≧ 2) with finite rank defect D_{(MzM)*,
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

