Mathematics – Operator Algebras
Scientific paper
2007-05-17
Mathematics
Operator Algebras
13 pages
Scientific paper
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.
Frank Michael
Sharifi Kamran
No associations
LandOfFree
Adjointability of densely defined closed operators and the Magajna-Schweizer Theorem does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Adjointability of densely defined closed operators and the Magajna-Schweizer Theorem, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Adjointability of densely defined closed operators and the Magajna-Schweizer Theorem will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-142684