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The Caratheodory-Fejer Interpolation Problems and the Von-Neumann inequality

Electronic Theses of Indian Institute of Science

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Title The Caratheodory-Fejer Interpolation Problems and the Von-Neumann inequality
 
Creator Gupta, Rajeev
 
Subject Von-Neumann Algebras
Polynomial
Varopoulos Operators
Operator Space Structures
Korányi-Pukánszky Theorem
Nehari’s Theorem
Hankel Operator
Von-Neumann Inequality
Carathéodory-Fejér Interpolation Problem
Mathematics
 
Description The validity of the von-Neumann inequality for commuting $n$ - tuples of $3\times 3$ matrices remains open for $n\geq 3$. We give a partial answer to this question, which is used to obtain a necessary condition for the Carathéodory-Fejérinterpolation problem on the polydisc$\D^n. $ in the special case of $n=2$ (which follows from Ando's theorem as well), this necessary condition is made explicit.
We discuss an alternative approach to the Carathéodory-Fejérinterpolation problem, in the special case of $n=2$, adapting a theorem of Korányi and Pukánzsky. As a consequence, a class of polynomials are isolated for which a complete solution to the Carathéodory-Fejér interpolation problem is easily obtained.
Many of our results remain valid for any $n\in \mathbb N$, however the computations are somewhat cumbersome.
Recall the well known inequality due to Varopoulos, namely, $\lim{n\to \infty}C_2(n)\leq 2 K^\C_G$, where $K^\C_G$ is the complex Grothendieck constant and
\[C_2(n)=sup\{\|p(\boldsymbolT)\|:\|p\|_{\D^n,\infty}\leq 1, \|\boldsymbol T\|_{\infty} \leq 1\}.\]
Here the supremum is taken over all complex polynomials $p$ in $n$ variables of degree at most $2$ and commuting $n$ - tuples$\boldsymbolT:=(T_1,\ldots,T_n)$ of contractions. We show that
\[\lim_{n\to \infty} C_2 (n)\leq \frac{3\sqrt{3}}{4} K^\C_G\] obtaining a slight improvement in the inequality of Varopoulos.
We also discuss several finite and infinite dimensional operator space structures on $\ell^1(n) $, $n>1. $
 
Contributor Misra, Gadadhar
 
Date 2018-05-30T04:55:04Z
2018-05-30T04:55:04Z
2018-05-30
2015
 
Type Thesis
 
Identifier http://etd.iisc.ernet.in/2005/3640
http://etd.iisc.ernet.in/abstracts/4510/G26910-Abs.pdf
 
Language en_US
 
Relation G26910