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 Ka,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 Ha,T denotes the reproducing kernel Hilbert space associated with Ka,T, then there exists an isometry Ua,T from Ha,T into H. It turns out that Ua,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 St and the constant multi-sequence at with value 1 (resp. commuting spherical Cauchy dual d-tuple Ss and the multi-sequence as,α:=(d+|α|−1)!(d−1)!α!,α∈Nd) 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 Ua,St with St (resp. the intertwining of Ua,Ss with Ss) and the multiplication tuple Mz 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.