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On spectral properties of operators on Banach spaces. (English) Zbl 1043.47005

Let \(X\) be a Banach space. A generalized duality mapping \(\phi : X \rightarrow 2^{X^\ast}\) is defined by \(\phi(x) = \{x^\ast \in X^\ast : x^\ast(x) = \| x\| ^p\) and \(\| x^\ast\| = \| x\| ^{p-1}\), \(1\leq p<\infty\}\). When \(X\) is smooth, such a \(\phi\) induces a generalized semi-inner product (g.s.i.p) by \(\phi(x)(y) = [y,x]\). For an operator \(T\) on \(X\), let \(W_p(T) = \{[Tx,x] : \| x\| = 1\), \(1 \leq p < \infty \}\). For a reflexive g.s.i.p space \(X\), the authors relate the spectrum of \(T \in B(X)\) to the above set by showing that \(\sigma(T) \subset W_p(T)^-\).

MSC:

47A10 Spectrum, resolvent
46C50 Generalizations of inner products (semi-inner products, partial inner products, etc.)
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