Spherical contractions and interpolation problems on the unit ball (Q2769338)

A commuting tuple \(T=(T_1,\ldots,T_n)\) of Hilbert space operators is a spherical contraction if \(\sum_{j=1}^nT_j^*T_j\leq 1\). Let \(\mathbb B\) be the unit ball in \({\mathbb C}^n\). The reproducing kernel Hilbert space \(H(\mathbb B)\) is given by the kernel \(K(z,w)=(1-\langle z,w\rangle)^{-1}\), \(z,w\in\mathbb B\). For an arbitrary Hilbert space \(\mathcal E\), the symbol \(H(\mathcal E)\) stands for the tensor product \(H({\mathbb B})\otimes\mathcal E\). NEWLINENEWLINENEWLINEThe first section of this paper deals with the space of multipliers \(M(\mathcal E,E_*)\) consisting of all analytic \(L(\mathcal E,E_*)\)-functions \(f\) such that \(fH({\mathcal E})\subset H({\mathcal E}_*)\). The authors obtain realizations of the multipliers as fractional transforms. This leads, in particular, to a new proof of Arveson's version of the von Neumann inequality for \(n\)-contractions (i.e. tuples \(T\) for which \(T^*\) is a spherical contraction). NEWLINENEWLINENEWLINEIn the second section, using an abstract interpolation theorem proved in the first one, the authors deduce a criterion for the solvability of the corona problem within the Schur class. A factorization result, also proved in the first section, leads to a simplified definition of Arveson's curvature invariant. NEWLINENEWLINENEWLINEIn the third section, the authors extend an idea due to Koranyi and Pukanszki to interpolate by analytic functions with prescribed values on uniqueness sets in the polydisc to the case of vector-valued bounded analytic functions on the unit ball. NEWLINENEWLINENEWLINEOne should mention that a fractional representation of multipliers was independently obtained by \textit{J. A. Ball, T. T. Trent} and \textit{V. Vinnikov} [see Oper. Theory, Adv. Appl. 122, 89-138 (2001; Zbl 0983.47011)], a result of which the present authors learnt after the completion of their paper.