Uniqueness of rotation invariant norms
For each \(N=1,2,\dots\), let \(S^N\) denote the \(N\)-dimensional Euclidean sphere endowed with the Lebesgue measure. Given a set \(J\) of continuous linear operators on a Banach space \(X\), we say that \(J\) determines the norm topology of \(X\) if any complete norm \(|\cdot|\) on \(X\) such that the operator \(T\) maps \((X,|\cdot|)\) continuously to \((X,|\cdot|)\) for each \(T\in J\), is equivalent to the given norm \(\|\cdot\|\) on \(X\). It was shown by \textit{K.\,Jarosz} [``Uniqueness of translation invariant norms'', J.~Funct.\ Anal.\ 174, No.\,2, 417--429 (2000; Zbl 0981.46006)] that the set of operators on \(L^p(S^1)\), with \(1<p<\infty\), corresponding to all rotations on \(S^1\), determines the norm topology of \(L^p(S^1)\). \textit{A.\,R.\thinspace Villena} established in [``Invariant functionals and the uniqueness of invariant norms'', J.~Funct.\ Anal.\ 215, No.\,2, 366--398 (2004; Zbl 1067.46041)] a similar result for \(L^p(S^N)\), \(N\geq 2\). In the present, interesting paper, the authors study the question: how many rotations are necessary to determine the topology of \(L^p(S^N)\) with \(1<p<\infty\) and \(N\geq 1\)? One of the main results shows that, if \(N\geq 2\), then there exist finitely many rotations of \(S^N\) such that the set consisting of the corresponding rotation operators on \(L^p(S^N)\) determine the norm topology of the space for \(1<p\leq \infty\). On the other hand, it is shown that the norm topology of \(L^2 (S^1)\) cannot be determined by the set of operators corresponding to rotations by elements of any thin set of rotations of \(S^1\).
scientific article; zbMATH DE number 5379952