Samuel Multiplicity in Accordance with Structure and index of Fredholm Operators

Abstract

We study the curvature invariant of a Hilbert module over the complex plain and for modules over free semigroup Algebras . The Berger-show theorem in the Hardy module over the Bidisk and Hilbert modules are considered . We determine the Samuel multiplicity and give the structure of semi-Fredholm operators . We also establish the Hilbeert-Samuel multiplicity of Fredholm tuples and Samuel multiplicity for several commuting operators . We show the Hilbert polynomials ,Arveson's curvature invariant and Additive invariants on the Hardy space over the polydisk . We discuss the Dirac of a commuting d-tuples and give the structure of Inner multipliers on spaces with complex Nevanlinna-Pick kernels . We find estimates for HilbertianKoszul homology . We obtain the stability of index for semi-Fredholm Chains and consider the Fredholm index of pair of commuting operators .