The Atkinson Theorem in Hilbert $C^*$-Modules over $C^*$-Algebras of Compact Operators

Abstract

The concept of unbounded Fredholm operators on Hilbert $C^*$-modules over an arbitrary $C^*$-algebra is discussed and the Atkinson theorem is generalized for bounded and unbounded Feredholm operators on Hilbert $C^*$-modules over $C^*$-algebras of compact operators. In the framework of Hilbert $C^*$-modules over $C^*$-algebras of compact operators, the index of an unbounded Fredholm operator and the index of its bounded transform are the same.