Complex analysis of symmetric operators, I

TL;DR

This paper develops a geometric framework for symmetric operators using Weyl curves, establishing a correspondence with Nevanlinna curves and introducing concepts like entire operators and their eigenvalue distributions.

Contribution

It introduces the notion of Weyl curves for symmetric operators, linking operator theory with complex geometry and value distribution theory, and develops a canonical functional model based on characteristic vector bundles.

Findings

One-to-one correspondence between symmetric operators and Nevanlinna curves.

Introduction of entire operators with Weyl curves in Grassmannians.

Eigenvalue distribution matches the value distribution of Weyl curves.

Abstract

Based on the relationship of symmetric operators with Hermitian symmetric spaces, we introduce the notion of \emph{Weyl curve} for a symmetric operator $T$, which is the geometric abstraction and generalization of the well-known Weyl functions. We prove that there is a one-to-one correspondence between unitary equivalence classes of simple symmetric operators and congruence classes of Nevanlinna curves (the geometric analogue of operator-valued Nevanlinna functions). To prove this result, we introduce a \emph{canonical} functional model for $T$ in terms of its \emph{characteristic vector bundles}. In this geometric formalism, when the deficiency indices are $(n,n)$ ($n\leq +\infty$) we also introduce the notion of \emph{entire operators}, whose Weyl curves are entire in the Grassmannian $Gr(n,2n)$. If $n$ is finite, this makes it possible to introduce modern value distribution theory of entire curves into the picture and to demonstrate that the distribution of eigenvalues of a \emph{generic} abstract boundary value problem is the same thing as the value distribution of the Weyl curve with respect to the Cartier divisor induced by the corresponding boundary condition. Many other new concepts are introduced and many new results are obtained as well.