The Carath\'eodory-Fej\'er Interpolation Problems and the von-Neumann Inequality

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 Carath'{e}odory-Fej'{e}r interpolation problem on the polydisc ${\mathbb{D}}^n\mathrm{.}$ 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 Carath'{e}odory-Fej'{e}r interpolation problem, in the special case of $n=2,$ adapting a theorem of Kor'{a}nyi and Puk'{a}nzsky is given. As a consequence, a class of polynomials are isolated for which a complete solution to the Carath'{e}odory-Fej'{e}r 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,thecomputationsaresomewhatcumbersomefor$n>2$ and are omitted. The inequality ${\lim }_{n\to \mathrm{\infty }}{C}_2(n)\le 2K_G^{\mathbb{C}}$, where $K_G^{\mathbb{C}}$ is the complex Grothendieck constant and $$C_2(n)=\sup \{\mathrm{\parallel }p(\mathit{T})\mathrm{\parallel }:\mathrm{\parallel }p{\mathrm{\parallel }}_{{\mathbb{D}}^n,\mathrm{\infty }}\le 1,\mathrm{\parallel }\mathit{T}{\mathrm{\parallel }}_{\mathrm{\infty }}\le 1\}$$ 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 $\mathit{T}:=(T_1,\dots ,T_n)$ of contractions. We show that $$\underset{n\to \mathrm{\infty }}{\lim }{C}_2(n)\le \frac{3\sqrt{3}}{4}{K}_G^{\mathbb{C}}$$ obtaining a slight improvement in the inequality of Varopoulos. We show that the normed linear space ${\mathrm{\ell }}^1(n),$ $n>1,$ 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.