The Carathéodory-Fejér Interpolation Problems and the von-Neumann Inequality

Abstract

The validity of the von-Neumann inequality for commuting $n$ - tuples of $3\times 3$ matrices remains open for $n\ge 3$. We give a partial answer to this question, which is used to obtain a necessary condition for the Carathéodory-Fejér interpolation problem on the polydisc ${\mathbb{D}}^n.$ In the special case of $n=2$ (which follows from Ando's theorem as well), this necessary condition is made explicit. An alternative approach to the Carathéodory-Fejér interpolation problem, in the special case of $n=2,$ adapting a theorem of Korányi and Pukánzsky is given. As a consequence, a class of polynomials are isolated for which a complete solution to the Carathéodory-Fejér interpolation problem is easily obtained. A natural generalization of the Hankel operators on the Hardy space of $H^2({\mathbb{T}}^2)$ then becomes apparent. Many of our results remain valid for any $n\in \mathbb{N},$ however, the computations are somewhat cumbersome for $n>2$ and are omitted. The inequality ${\lim }_{n\to \infty }{C}_2(n)\le 2K_G^{\mathbb{C}}$, where $K_G^{\mathbb{C}}$ is the complex Grothendieck constant and \[C_2(n)=\sup\big\{\|p(\boldsymbol T)\|:\|p\|_{\mathbb D^n,\infty}\leq 1, \|\boldsymbol T\|_{\infty} \leq 1 \big\}\] is due to Varopoulos. Here the supremum is taken over all complex polynomials $p$ in $n$ variables of degree at most $2$ and commuting $n$ - tuples $\mathbf{T}:=(T_1,\dots ,T_n)$ of contractions. We show that \[\lim_{n\to \infty}C_2(n)\leq \frac{3\sqrt{3}}{4} K^\mathbb C_G\] obtaining a slight improvement in the inequality of Varopoulos. We show that the normed linear space ${\ell }^1(n),$ $n>1,$ has no isometric embedding into $k\times k$ complex matrices for any $k\in \mathbb{N}$ and discuss several infinite dimensional operator space structures on it.

This is my thesis submitted to Indian Institute of Science, Bangalore on 20th July, 2015

Cited by in corpus (1)

• The Carathéodory-Fejér interpolation problem for the polydisc