Adjointability of densely defined closed operators and the Magajna-Schweizer Theorem
Michael Frank, Kamran Sharifi
TL;DR
The work analyzes adjointability and regularity of densely defined operators on Hilbert $\mathcal{A}$-modules, showing that if the graph $G(t)$ is orthogonally complemented and a projected range condition holds, then $t$ possesses a densely defined adjoint. This leads to a broader criterion: an operator is regular precisely when it is adjointable, closed, and $c\,\mathrm{1}+t^*t$ is bijective for all $c>0$, with geometric refinements due to Kucerovsky. A central consequence is that over $C^*$-algebras of compact operators, every densely defined closed operator is regular, aligning with the Magajna-Schweizer characterization and Pal’s results on semiregular operators. The paper also ties these operator-theoretic properties to structural features of the underlying algebra, offering multiple equivalent conditions for compactness and adjointability in this setting.
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.