Regular operator on a Hilbert C*-module - Imported Knowlpedia Content

Definition

Let $E$ be a Hilbert $C^*$ -module over a $C^*$ -algebra $A$ . A densely defined $A$ -linear operator $T:\operatorname{Dom}(T)\to E$ , whose domain is a dense right $A$ -submodule of $E$ , is regular if $T$ is closed, its adjoint $T^*$ is densely defined, and

\operatorname{Ran}(1+T^*T)

is dense in $E$ . The adjoint is defined by requiring $\langle Tx,y\rangle=\langle x,T^*y\rangle$ for every $x\in\operatorname{Dom}(T)$ . A regular operator is self-adjoint when $T=T^*$ , including equality of domains. Regularity is additional structure in Hilbert-module theory; closed densely defined operators need not be regular.

Graph and bounded transform

Regularity is equivalent to orthogonal complementability of the graph of $T$ in $E\oplus E$ , with the densely defined adjoint condition understood. It also makes

F_T=T(1+T^*T)^{-1/2}

a bounded adjointable operator with $\|F_T\|\leq1$ . Conversely, suitable contractions arise as bounded transforms of regular operators. This correspondence supplies the continuous functional calculus used in unbounded $KK$ -theory Lance, Chapters 9–10.

The self-adjoint case

For a self-adjoint regular $T$ , the operators $T\pm i$ have bounded adjointable inverses $E\to\operatorname{Dom}(T)\subset E$ , and continuous functional calculus is available for functions on $\mathbb R$ . Conversely, a closed symmetric operator whose $T+i$ and $T-i$ have dense range is self-adjoint and regular. These range conditions replace Hilbert-space arguments that would otherwise rely on automatic orthogonal complements.

Comparison with Hilbert spaces

When the coefficient algebra is $\mathbb C$ , every closed densely defined operator with densely defined adjoint is regular. This fails for general Hilbert $C^*$ -modules because closed submodules need not be orthogonally complemented. Hence “regular” here is unrelated to regularity of a measure, elliptic regularity, or a regular point of an operator pencil.

References

E. Christopher Lance, Hilbert C-Modules: A Toolkit for Operator Algebraists*, Cambridge University Press, 1995. Publisher DOI record. Relevant: Chapter 9 on regular operators and Chapter 10 on the bounded transform.
Bruce Blackadar, K-Theory for Operator Algebras, 2nd ed., Cambridge University Press, 1998. Publisher DOI record. Relevant: §13.3 on regular operators on Hilbert modules.