Vol. 31, No. 1, 1969

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Existence of a spectrum for nonlinear transformations

John William Neuberger

Vol. 31 (1969), No. 1, 157–159

Denote by S a complex (nondegenerate) Banach space. Suppose that T is a transformation from a subset of S to S. A complex number λ is said to be in the resolvent of T if (λI T)1 exists, has domain S and is Fréchet differentiable (i.e., if p is in ∕S there is a unique continuous linear transformation F = [(λI T)1](p) from S to S so that

lim ∥q − p∥− 1∥(λI − T)−1q− (λI − T )− 1p − F(q− p)∥ = 0

and locally Lipschitzean everywhere on S. A complex number is said to be in the spectrum of T if it is not in the resolvent of T.

Suppose in addition that the domain of T contains an open subset of S on which T is Lipschitzean.

Theorem. T has a (nonempty) spectrum.

Mathematical Subject Classification
Primary: 47.80
Published: 1 October 1969
John William Neuberger