Cyclic Operators, Linear Functionals and RKHS

Given a commuting n-tuple of bounded linear operators on a Hilbert space, together with a distinguished cyclic vector, Jim Agler defined a linear functional Λₓ, ₇ on the polynomial ring Cz, z. ``Near subnormality properties'' of an operator T are translated into positivity properties of Λₓ, ₇. In this paper, we approach ``near subnormality properties'' in a different way by answering the following question: when is Λₓ, ₇ given by a compactly supported distribution? The answer is in terms of the off-diagonal growth condition of a two-variable kernel function Fₓ, ₇ on Cⁿ. Using the reproducing kernel Hilbert spaces (RKHS) defined by the kernel function Fₓ, ₇, we give a function model for all cyclic commuting n-tuples. This potentially gives a different approach to operator models. The reproducing kernels of the Fock space are used in the construction of Fₓ, ₇, but one may also replace the Fock space by other RKHS. We give many examples in the last section.