Spectral Mapping Theorems by Robin Harte

Author:Robin Harte
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

(4.7.7)

now it is Schechter’s theorem that

(4.7.8)

is equivalent to the Weyl condition (1.​9.​1) for each of the two Calkin homomorphisms. The logarithmic law of the index says that (1.​9.​8) holds: if and are both Fredholm then the index of the product is the sum of the index of the factors and . We remark that the closed range condition (4.7.3) is important here: if we define to be weakly Fredholm if it satisfies the conditions (4.7.1) and (4.7.2), but possibly not (4.7.3), and define the index by the formula (4.7.7), then the logarithmic law (1.​9.​8) may possibly fail.

When then the condition that is Fredholm with

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