Gowda, Huche; Lokesha, V. On spectral properties of operators on Banach spaces. (English) Zbl 1043.47005 Far East J. Math. Sci. (FJMS) 6, No. 3, 325-331 (2002). 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 \(\sigma(T) \subset W_p(T)^-\). Reviewer: Taduri S. Rao (Bangalore) MSC: 47A10 Spectrum, resolvent 46C50 Generalizations of inner products (semi-inner products, partial inner products, etc.) Keywords:reflexivity; generalized duality mapping; generalized semi-inner product; spectrum PDFBibTeX XMLCite \textit{H. Gowda} and \textit{V. Lokesha}, Far East J. Math. Sci. (FJMS) 6, No. 3, 325--331 (2002; Zbl 1043.47005)