To the theory of operators that are bounded on cones in weighted spaces of numerical sequences
In this paper, the authors study an interesting result. Namely, they are interested in the interpolation of operators bounded on a pair of cones in Banach spaces. In particular, they consider the case of operators bounded on cones in weighted spaces of numerical sequences. The authors announce applications of their results which will be published in a subsequent paper. The basic definitions and results are similar to those of the classical theory of interpolation and we omit them, but the corresponding proofs are more involved. Alas, the statements are rather complicated; for instance, the statement of Theorem 3.2 runs as follows: Theorem 3.2. let \(E_i=c_0(a_i)\) and \(F_i=c_0(b_i)\), \(i=0,1\). Let \(E=c_0(a)\) and let \(F=c_0(b)\). Assume that \(E_1\subset E\subset E_0\), \(F_1\subset F\subset F_0\), and the Banach triple \((E_0,E_1,E)\) has the interpolation property with respect to the Banach triple \((F_0,F_1,F)\). Let \(\mathcal A\) be a set of cones in \(\omega^+\) such that the following conditions hold for any cone \(Q\in\mathcal A\):{\parindent=0.6cm \begin{itemize}\item[(1)] \(Q\) is a lower semi-lattice in \(\omega\); \item[(2)] \(Q\cap E_1^+\) is a total cone in the space \(E_1\); \item[(3)] \(Q\cap E_1^{++}\neq\emptyset\). \end{itemize}} Then the family of triples of cones \[ M=\{(E_0^+. Q\cap E_1^+,Q\cap E^+):Q\in\mathcal A\} \] has the uniform interpolation property with respect to the Banach triple \((F_0,F_1,F)\). In conclusion, this paper deserves more attention under the condition that the promised applications paper will be available.
scientific article; zbMATH DE number 6537186