Adjointability of densely defined closed operators and the Magajna-Schweizer Theorem

TL;DR

This paper investigates the conditions under which densely defined operators on Hilbert C*-modules are adjointable and regular, establishing a connection with the structure of the underlying C*-algebra and providing new characterizations.

Contribution

It provides new characterizations of regular operators and adjointability on Hilbert C*-modules, linking these properties to the algebra being of compact operators, and refines existing results with improved proofs.

Findings

Densely defined operators have adjoints iff their graphs are orthogonal summands.

Regularity of operators is equivalent to the graph being orthogonally complemented.

Characterization of C*-algebras of compact operators via regularity of all densely defined operators.

Abstract

In this notes unbounded regular operators on Hilbert $C^*$-modules over arbitrary $C^*$-algebras are discussed. A densely defined operator $t$ possesses an adjoint operator if the graph of $t$ is an orthogonal summand. Moreover, for a densely defined operator $t$ the graph of $t$ is orthogonally complemented and the range of $P_FP_{G(t)^\bot}$ is dense in its biorthogonal complement if and only if $t$ is regular. For a given $C^*$-algebra $\mathcal A$ any densely defined $\mathcal A$-linear closed operator $t$ between Hilbert $C^*$-modules is regular, if and only if any densely defined $\mathcal A$-linear closed operator $t$ between Hilbert $C^*$-modules admits a densely defined adjoint operator, if and only if $\mathcal A$ is a $C^*$-algebra of compact operators. Some further characterizations of closed and regular modular operators are obtained. Changes 1: Improved results, corrected misprints, added references. Accepted by J. Operator Theory, August 2007 / Changes 2: Filled gap in the proof of Thm. 3.1, changes in the formulations of Cor. 3.2 and Thm. 3.4, updated references and address of the second author.