Spectrum of infinite-dimensional Banach spaces
Let X be an infinite dimensional Banach space. Following \textit{L. Schwartz} [Bull. Am. Math. Soc., New Ser. No.4, 135-141 (1982; Zbl 0555.46007)] the author defines the \(\ell_ p\)-spectrum of X as the set S(X) of all those \(p\in [1,\infty]\) for each of which \(\ell_ p\) is finitely represented in X. A subset \(A\subset [1,\infty)\) is called spectral provided \(A=S(X)\) for some X. The author solves a problem posed by L. Schwartz of describing the spectral subsets of [1,\(\infty)\). Namely, a subset A of [1,\(\infty)\) is spectral if and only if it has a regular structure. The latter means that \(A=[p,2]\cup B\) for some compact subset \(B\subset [2,\infty)\) and for some \(p\in [1,2]\).
scientific article; zbMATH DE number 4105606