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Samuel multiplicity and the structure of semi-Fredholm operators. (English) Zbl 1070.47007

In this paper, the shift Samual multiplicity, denoted s mul (T), and the backward shift Samual multiplicity, denoted bs mul (T), are introduced for any semi-Fredholm operator T on a separable complex Hilbert space H:

s mul (T)=lim t (dim(H/T k (H))/k),
bs mul (T)=lim t (dim(ker(T k )))/k)·

These numerical invariants refining the Fredholm index can be regarded as stabilized dimensions of the cokernel H/T(H) and kernel ker(T), respectively. A geometric interpretation of s mul (T) and bs mul (T) leads to a local version of Apostol’s triangular representation theorem and to Gohberg’s punctured neighborhood theorem. Banach space operators are also considered.


MSC:
47A53(Semi-)Fredholm operators; index theories
47A13Several-variable spectral theory
46H25Normed modules and Banach modules, topological modules