Generalized inverses and polar decomposition of unbounded regular operators on Hilbert $C^{\ast }$-modules

Abstract

In this note we show that an unbounded regular operator $t$ on Hilbert $C^{\ast }$-modules over an arbitrary $C^{\ast }$ algebra $\mathcal{A}$ has polar decomposition if and only if the closures of the ranges of $t$ and $|t|$ are orthogonally complemented, if and only if the operators $t$ and $t^{\ast }$ have unbounded regular generalized inverses. For a given $C^{\ast }$-algebra $\mathcal{A}$ any densely defined $\mathcal{A}$-linear closed operator $t$ between Hilbert $C^{\ast }$-modules has polar decomposition, if and only if any densely defined $\mathcal{A}$-linear closed operator $t$ between Hilbert $C^{\ast }$-modules has generalized inverse, if and only if $\mathcal{A}$ is a $C^{\ast }$-algebra of compact operators.

11 pages / corrected typos and cross-references

References in corpus (3)

• Characterizing C*-algebras of compact operators by generic categorical properties of Hilbert C*-modules
• Adjointability of densely defined closed operators and the Magajna-Schweizer Theorem
• Regular Operators on Hilbert C^*-modules

Cited by in corpus (11)

• Characterizing C*-algebras of compact operators by generic categorical properties of Hilbert C*-modules
• The product of operators with closed range in Hilbert C*-modules
• EP modular operators and their products
• The gap between unbounded regular operators
• Moore-Penrose inverse of Gram operator on Hilbert C*-modules
• 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
• The reverse order law for Moore-Penrose inverses of operators on Hilbert C*-modules
• Power-norms based on Hilbert $C^{\ast }$-modules
• Normality of adjointable module maps
• The polar decomposition for adjointable operators on Hilbert $C^{\ast }$-modules and $n$-centered operators