"Ando dilations, von Neumann inequality, and distinguished varieties" by B. Krishna Das and Jaydeb Sarkar
 

Ando dilations, von Neumann inequality, and distinguished varieties

Article Type

Research Article

Publication Title

Journal of Functional Analysis

Abstract

Let D denote the unit disc in the complex plane C and let D2=D×D be the unit bidisc in C2. Let (T1,T2) be a pair of commuting contractions on a Hilbert space H. Let dim⁡ran(IH−TjTj⁎)<∞, j=1,2, and let T1 be a pure contraction. Then there exists a variety V⊆D‾2 such that for any polynomial p∈C[z1,z2], the inequality ‖p(T1,T2)‖B(H)≤‖p‖V holds. If, in addition, T2 is pure, then V={(z1,z2)∈D2:det⁡(Ψ(z1)−z2ICn)=0} is a distinguished variety, where Ψ is a matrix-valued analytic function on D that is unitary-valued on ∂D. Our results comprise a new proof, as well as a generalization, of Agler and McCarthy's sharper von Neumann inequality for pairs of commuting and strictly contractive matrices.

First Page

2114

Last Page

2131

DOI

10.1016/j.jfa.2016.08.008

Publication Date

3-1-2017

Comments

Open Access, Green

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