An operator-valued kernel associated with a commuting tuple of Hilbert space operators

We associate a one parameter family of positive definite E-valued kernels K-a,K-T with any commuting d-tuple T of bounded linear operators on a Hilbert space H, where a is a multi-sequence of non-zero complex numbers and E is an auxiliary Hilbert space. If H-a,H-T, denotes the reproducing kernel Hilbert space associated with K-a,K-T, then there exists an isometry U-a,U-T from H-a,H-T, into H. It turns out that U-a,U-T is surjective if and only E is a cyclic subspace for T. We apply the above scheme to the commuting toral Cauchy dual d-tuple S-t and the constant multi-sequence a(t) with value 1 (resp. commuting spherical Cauchy dual d-tuple S-S and the multi-sequence a(s,alpha) := (d+vertical bar alpha vertical bar-1)/(d-1)vertical bar alpha vertical bar, alpha is an element of N-d) with E being the joint kernel of S*. to ensure an analytic model for S under some natural assumptions. In particular, the strictly higher dimensional obstruction to the intertwining of U-a,s(t) with St (resp. the intertwining of U-a,U-s. with S-s) and the multiplication tuple.M-z is characterized in terms of a kernel condition. These results can be considered as toral and spherical analogs of Shimorin's Theorem (the case of d = 1) stating that any left-invertible analytic operator admits an analytic model. (C) 2018 Elsevier Masson SAS. All rights reserved.