Kernel decompositions for Schur functions on the polydisk (Q371621)

Let \({\mathbb D}^d\) be the unit polydisc in \({\mathbb C}^d\). As was shown by \textit{J. Agler} [``Some Interpolation Theorems of Nevanlinna-Pick type'', Unpublished manuscript (1988)], for every Schur function \(f\) on \({\mathbb D}^2\) there exist positive semi-definite kernels \(K\) and \(K'\) such that \(1-f(z)\overline{f(\zeta)}=(1-z_1\bar\zeta_1)K(z,\zeta)+(1-z_2\bar\zeta_2)K'(z,\zeta)\). A generalization of this statement to the case of a Schur function \(f\) on \({\mathbb D}^d\) \((d>2)\) was obtained recently in [\textit{A. Grinshpan} et al., J. Funct. Anal. 256, No. 9, 3035--3054 (2009; Zbl 1167.47009)]. In the present paper, a more general decomposition of the Pick kernel \(P(z,\zeta):=\frac{1-f(z)\overline{f(\zeta)}}{\prod_{i=1}^d(1-z_i\bar\zeta_i)}\) is obtained. Namely, for every Schur function \(f\) on \({\mathbb D}^d\) and nonempty disjoint subsets \(S\) and \(T\) of \(\{1,\dots,d\}\) such that \(S\cup T=\{1,\dots,d\}\), the Pick kernel \(P(z,\zeta)\) can be decomposed into the sum of two positive semi-definite kernels \(K_S\) and \(K_T\) so that the kernels \((1-z_j\bar\zeta_j)K_S(z,\zeta)\) and \((1-z_k\bar\zeta_k)K_T(z,\zeta)\) are positive semi-definite for every \(j\in S\) and \(k\in T\).