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

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The validity of the von-Neumann inequality for commuting n - tuples of 3× 3 matrices remains open for n≥ 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 Dⁿ. 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 H²(T²) then becomes apparent. Many of our results remain valid for any n∈ N, however, the computations are somewhat cumbersome for n>2 and are omitted. The inequality ₙ→ ∞C₂(n)≤ 2 KᶜG, where KGᶜ is the complex Grothendieck constant and C₂(n)={p(T):pDₙ,∞≤ 1, T∞ ≤ 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 T:=(T₁,,Tₙ) of contractions. We show that ₙ→ ∞C₂(n)≤ 3√34 KᶜG obtaining a slight improvement in the inequality of Varopoulos. We show that the normed linear space ℓ¹(n), n>1, has no isometric embedding into k× k complex matrices for any k∈ N and discuss several infinite dimensional operator space structures on it.