# Self-Adjoint Operators in Extended Hilbert Spaces $H\oplus W$: An Application of the General GKN-EM Theorem

## TL;DR

This paper develops a framework for constructing self-adjoint operators in extended Hilbert spaces, generalizing classical theorems to include finite-dimensional extensions, motivated by differential operators with polynomial eigenfunctions.

## Contribution

It introduces a generalized GKN-EM theorem for characterizing self-adjoint extensions in extended Hilbert spaces, expanding the classical operator theory to include finite-dimensional extensions.

## Key findings

- Established a characterization of self-adjoint extensions in $H  W$.
- Generalized the classical GKN theorem to the GKN-EM theorem.
- Provided examples illustrating the theoretical results.

## Abstract

We construct self-adjoint operators in the direct sum of a complex Hilbert space $H$ and a finite dimensional complex inner product space $W$. The operator theory developed in this paper for the Hilbert space $H\oplus W$ is originally motivated by some fourth-order differential operators, studied by Everitt and others, having orthogonal polynomial eigenfunctions. Generated by a closed symmetric operator $T_{0}$ in $H$ with equal and finite deficiency indices and its adjoint $T_{1}$, we define \textit{families} of minimal operators $\{\widehat{T}_{0}\}$ and maximal operators $\{\widehat{T}_{1}\}$ in the extended space $H\oplus W$ and establish, using a recent theory of complex symplectic geometry, developed by Everitt and Markus, a characterization of self-adjoint extensions of $\{\widehat{T}_{0}\}$ when the dimension of the extension space $W$ is not greater than the deficiency index of $T_{0}$. A generalization of the classical Glazman-Krein-Naimark (GKN) Theorem - called the GKN-EM Theorem to acknowledge the work of Everitt and Markus - is key to finding these self-adjoint extensions in $H\oplus W.$ We consider several examples to illustrate our results.