Symmetric Operators and Reproducing Kernel Hilbert Spaces

Abstract

We establish the following sufficient operator-theoretic condition for a subspace $${S \subset L^2 (\mathbb{R}, d\nu)}$$ to be a reproducing kernel Hilbert space with the Kramer sampling property. If the compression of the unitary group U(t) := e itM generated by the self-adjoint operator M, of multiplication by the independent variable, to S is a semigroup for t ≥ 0, if M has a densely defined, symmetric, simple and regular restriction to S, with deficiency indices (1, 1), and if ν belongs to a suitable large class of Borel measures, then S must be a reproducing kernel Hilbert space with the Kramer sampling property. Furthermore, there is an isometry which acts as multiplication by a measurable function which takes S onto a reproducing kernel Hilbert space of functions which are analytic in a region containing $${\mathbb{R}}$$ , and are meromorphic in $${\mathbb{C}}$$ . In the process of establishing this result, several new results on the spectra and spectral representations of symmetric operators are proven. It is further observed that there is a large class of de Branges functions E, for which the de Branges spaces $${\mathcal{H}(E) \subset L^{2}(\mathbb{R}, |E(x)|^{-2}dx)}$$ are examples of subspaces satisfying the conditions of this result.