Unbounded Operators on Hilbert $C^*$-Modules

TL;DR

This paper introduces and studies new classes of unbounded operators on Hilbert $C^*$-modules, focusing on their adjoints, graph regularity, and orthogonal properties, with applications to various algebraic structures.

Contribution

It develops a theory for essentially defined operators on Hilbert $C^*$-modules, including graph regularity and orthogonal closure, with new characterizations and examples.

Findings

Operators have well-defined adjoints under weaker domain assumptions

Characterization of graph regular operators in terms of orthogonal complements

Examples include operators on $C_0(X)$, Weyl algebra, Toeplitz algebra, and Heisenberg group

Abstract

Let $E$ and $F$ be Hilbert $C^*$-modules over a $C^*$-algebra $\CAlg{A}$. New classes of (possibly unbounded) operators $t:E\to F$ are introduced and investigated. Instead of the density of the domain $\Def(t)$ we only assume that $t$ is essentially defined, that is, $\Def(t)^\bot=\{0\}$. Then $t$ has a well-defined adjoint. We call an essentially defined operator $t$ graph regular if its graph $\Graph(t)$ is orthogonally complemented in $E\oplus F$ and orthogonally closed if $\Graph(t)^{\bot\bot}=\Graph(t)$. A theory of these operators is developed. Various characterizations of graph regular operators are given. A number of examples of graph regular operators are presented ($E=C_0(X)$, a fraction algebra related to the Weyl algebra, Toeplitz algebra, Heisenberg group). A new characterization of affiliated operators with a $C^*$-algebra in terms of resolvents is given.