Minimal extensions in tensor product spaces
Let \(X\) be any Banach space and let \(V\subseteq S\subseteq X\) be subspaces of \(X\) with \(\dim V<\infty\). For any operator \(M\in {\mathcal L}(S,V)\), the author defines the minimal extension constant by \[ \lambda_M(V, X)= \inf\{\| P\|: P\in{\mathcal L}(X, V), P|_S= M\}. \] For \(V= S\) and \(M= 1_V\) this generalizes the concept of a minimal projection constant \(\lambda(V,X)\). In the paper under review, the author proves as a main result the following nice formula: Let \(X\), \(Y\) be any separable Banach spaces and let \(V\subseteq S\subseteq X\) and \(W\subseteq Z\subseteq Y\) be any subspaces with \(\dim V\), \(\dim W<\infty\). Let \(\alpha\) be any reasonable and uniform crossnorm on \(X\otimes Y\). Then \[ \lambda_{M\otimes N}(V\otimes_\alpha W, X\otimes_\alpha Y)= \lambda_M(V, X)\lambda_N(W, Y) \] holds true for all \(M\in {\mathcal L}(S, V)\) and \(N\in{\mathcal L}(Z, W)\), respectively. The proof is essentially based on a quite recent characterization of minimal extensions of operators by certain integral operators given by \textit{B. L. Chalmers} and \textit{F. T. Metcalf} [J. Oper. Theory 32, 31-46 (1994; Zbl 0827.41016)]. In the final section, the main result is applied to function spaces and subspaces of polynomials of fixed degree.
subspaces of polynomials of fixed degree
scientific article; zbMATH DE number 1278849
scientific article