On equivalent strictly \(G\)-convex renormings of Banach spaces (Q539177)

This paper deals with a natural generalization of strict convexity for Banach spaces called strict \(G\)-convexity. Given a Banach space \(X\), the choice of signs in the definition of strict convexity is replaced by the choice of an operator in a fixed subset \(G\) of the space of bounded linear operators on \(X\). Some results concerning strictly \(G\)-convex renormings and extensions of strictly \(G\)-convex norms are obtained. More precisely, it is proved that a Banach space which is strictly \(G\)-convex for a finite group \(G\) can be renormed such that it is strictly \(G\)-convex and every operator in \(G\) becomes an isometry in the new norm. With regard to extension results, it is shown that in a strictly \(G\)-convex Banach space for a finite group symmetric about the origin, every equivalent strictly \(G\)-convex norm defined on a subspace invariant for \(G\) can be extended to the whole space. Besides, it is proved that, for uncountable \(\Gamma\), the subspace of \(\ell_\infty(\Gamma)\) formed by elements with countable supports cannot be equivalently renormed to be strictly \(G\)-convex for any separable bounded operator family \(G\). More information related to this topic can be found in two prior works of the author [Visn. Khark. Univ., Ser. Mat. Prykl. Mat. Mekh. 826, No.~58, 197--210 (2008; Zbl 1164.46306); the author and \textit{V. Kadets} [Serdica Math. J. 35, No.~1, 1--14 (2009; Zbl 1224.46020)].