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

Abstract

In this notes unbounded regular operators on Hilbert $C^{\ast }$-modules over arbitrary $C^{\ast }$-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 $P_F{P}_{G(t{)}^{\perp }}$ is dense in its biorthogonal complement if and only if $t$ is regular. For a given $C^{\ast }$-algebra $\mathcal{A}$ any densely defined $\mathcal{A}$-linear closed operator $t$ between Hilbert $C^{\ast }$-modules is regular, if and only if any densely defined $\mathcal{A}$-linear closed operator $t$ between Hilbert $C^{\ast }$-modules admits a densely defined adjoint operator, if and only if $\mathcal{A}$ is a $C^{\ast }$-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.

13 pages

References in corpus (2)

• Characterizing C*-algebras of compact operators by generic categorical properties of Hilbert C*-modules
• Regular Operators on Hilbert C^*-modules

Cited by in corpus (10)

• Characterizing C*-algebras of compact operators by generic categorical properties of Hilbert C*-modules
• Generalized inverses and polar decomposition of unbounded regular operators on Hilbert $C^{\ast }$-modules
• The product of operators with closed range in Hilbert C*-modules
• EP modular operators and their products
• Moore-Penrose inverse of Gram operator on Hilbert C*-modules
• The gap between unbounded regular operators
• Generic properties of module maps and characterizing inverse limits of C*-algebras of compact operators
• Orthogonality-preserving, C*-conformal and conformal module mappings on Hilbert C*-modules
• Normality of adjointable module maps
• Structure theory of homologically trivial and annihilator locally C*-algebras