Sampling Theory Associated with a Symmetric Operator with Compact Resolvent and De Branges Spaces

Abstract.

Let $\mathcal{A}$ be a symmetric operator with compact resolvent defined in a Hilbert space $\mathbb{H}.$ For any fixed $a \in \mathbb{H}$ we consider an entire $\mathbb{H}{\text{ - valued}}$ function K _a which involves the resolvent of $\mathcal{A}.$ Associated with K _a we obtain, by duality in $\mathbb{H},$ a Hilbert space $\mathcal{H}_{a}$ of entire functions which becomes a De Branges space of entire functions. This property provides a characterization of $\mathcal{H}_{a}$ regardless of the anti-linear mapping which has $\mathcal{H}_{a}$ as its range space. There exists also a sampling formula allowing to recover any function in $\mathcal{H}_{a}$ from its samples at the sequence of eigenvalues of $\mathcal{A}.$