2015/08/28 by Rajeev Gupta, Gupta, Rajeev · 1 citation
Computer Science · Mathematics · #Algebraic and Geometric Analysis #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Matrix Theory and Algorithms #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.1508.07199
openalex publication_date 2015/08/28 · openalex created_date 2022/09/04 · openalex updated_date 2026/07/28
The validity of the von-Neumann inequality for commuting n - tuples of\n3\× 3 matrices remains open for n\≥ 3. We give a partial answer to\nthis question, which is used to obtain a necessary condition for the\nCarath 'eodory-Fej 'er interpolation problem on the polydisc mathbb Dn.\nIn the special case of n=2 (which follows from Ando's theorem as well), this\nnecessary condition is made explicit. An alternative approach to the\nCarath 'eodory-Fej 'er interpolation problem, in the special case of n=2,\nadapting a theorem of Kor 'anyi and Puk 'anzsky is given. As a consequence,\na class of polynomials are isolated for which a complete solution to the\nCarath 'eodory-Fej 'er interpolation problem is easily obtained. A natural\ngeneralization of the Hankel operators on the Hardy space of H2( mathbb T2)\nthen becomes apparent. Many of our results remain valid for any n\∈ mathbb\nN, however, the computations are somewhat cumbersome for n>2 and are\nomitted. The inequality \limn\→ \∞C2(n)\≤ 2 K^ mathbb CG, where\nKG^ mathbb C is the complex Grothendieck constant and\n\C2(n)=
sup
big
|p(
boldsymbol T)
|:
|p
|
mathbb Dn,
infty
leq 1,\n
|
boldsymbol T
|
infty
leq 1
big
is due to Varopoulos. Here the\nsupremum is taken over all complex polynomials p in n variables of degree\nat most 2 and commuting n - tuples boldsymbol T:=(T1,\…,Tn) of\ncontractions. We show that \
limn
to
inftyC2(n)
leq
frac3
sqrt34\nK^
mathbb CG obtaining a slight improvement in the inequality of Varopoulos.\nWe show that the normed linear space \ℓ1(n), n>1, has no isometric\nembedding into k\× k complex matrices for any k\∈ mathbb N and\ndiscuss several infinite dimensional operator space structures on it.\n