Symmetric pairs of unbounded operators in Hilbert space, and their applications in mathematical physics

TL;DR

This paper explores symmetric pairs of unbounded operators in Hilbert spaces, demonstrating their utility in establishing closability, computing adjoints, and applying these concepts to stochastic calculus, Tomita-Takesaki theory, and infinite graph functions.

Contribution

It introduces the use of symmetric pairs to analyze operator properties, providing new methods for closability and adjoint computation in various mathematical physics contexts.

Findings

Malliavin derivative and Skorokhod integral are closable with mutually adjoint closures.

Basic involutions in Tomita-Takesaki theory are closable with mutually adjoint closures.

Symmetric pairs apply to functions of finite energy on infinite graphs, involving Laplace and inclusion operators.

Abstract

In a previous paper, the authors introduced the idea of a symmetric pair of operators as a way to compute self-adjoint extensions of symmetric operators. In brief, a symmetric pair consists of two densely defined linear operators $A$ and $B$, with $A \subseteq B^\star$ and $B \subseteq A^\star$. In this paper, we will show by example that symmetric pairs may be used to deduce closability of operators and sometimes even compute adjoints. In particular, we prove that the Malliavin derivative and Skorokhod integral of stochastic calculus are closable, and the closures are mutually adjoint. We also prove that the basic involutions of Tomita-Takesaki theory are closable and that their closures are mutually adjoint. Applications to functions of finite energy on infinite graphs are also discussed, wherein the Laplace operator and inclusion operator form a symmetric pair.