One property of functors of the real interpolation method
A Banach space X is said to be intermediate between a Banach pair \(\vec X\equiv (X_ 0,X_ 1)\) if \(X_ 0\cap X_ 1\subseteq X\subseteq X_ 0+X_ 1\) (where embeddings are continuous). For such Banach triples, \((X_ 0,X_ 1,X)\) is said to be interpolative relative to \((Y_ 0,Y_ 1,Y)\) iff for each linear operator \(T:X_ 1\to Y_ 1\), \(i=0,1\), T(X)\(\subseteq Y\). If \(X=Y\), X is said to be an interpolative space between the pairs \((X_ 0,X_ 1)\) and \((Y_ 0,Y_ 1)\). If moreover \(X_ i=Y_ i\), \(i=0,1\) then X is said to be an interpolative space between the spaces \(X_ 0\) and \(X_ 1\). An interpolative functor (i.f.) F is said to be exact if for each Banach pairs \(\vec X\equiv (X_ 1,X_ 1)\), \(\vec Y\equiv (Y_ 0,Y_ 1)\) and for each linear operator \(T:X_ i\to Y_ i,i=0,1\), \(\| T\|_{F(\vec X)\to F(\vec Y)}\leq \max_{i=0,1}(\| T\|_{X_ i\to Y_ i})\). For a Banach ideal space (B.I.S) G of two sided number sequences and real function f, denote by G(f) the space with the norm \(\| (a_ k)\|_{G(f)}=\| (a_ k| f(2^ k)|)\|_ G\). For a Banach space A intermediate between a Banach pair \(\vec A\equiv (A_ 0,A_ 1)\), define the exact i.f.'s \({\mathcal O}_ A^{\vec A}\) and \({\mathcal P}_ A^{\vec A}\) by \({\mathcal O}_ A^{\vec A}(\vec X)=\{\sum^{\alpha}_{k=1}T_ ka_ k:\sum^{\alpha}_{k=1}\max_{i=0,1}\| T_ k\|_{A_ i\to X_ i}\cdot \| a_ k\|_ A<\alpha \}\) with the norm inf\(\{\sum^{\alpha}_{k=1}\max_{i=0,1}\| T_ k\|_{A_ i\to X_ i}\| a_ k\|_ A\); \(x=\sum^{\alpha}_{k-1}T_ ka_ k\}\) and \({\mathcal P}_ A^{\vec A}(\vec X)=\{x:x\in X_ o+X_ 1\), such that for each linear operator \(T:X_ i\to A_ i\), \(i=0,1\), Tx\(\in A\}\). With the norm \(\sup_{T}\{\| Tx\|_ A;\max_{i=0,1}\| T\|_{X_ i\to A_ i}\leq 1\}\). For i.f.'s F, G, define \(F\leq G\) iff for each Banach pair \(\vec X=(X_ 0,X_ 1),F(\vec X)\subseteq G(\vec X)\). If \(F\leq G\), and \(G\leq F_ 0\), then we write \(F=G\). Then the author proves the following: Theorem: Let \(f_ 0,f_ 1\) be quasi-concave functions on (0,\(\infty)\) such that \(f_ 1,f_ 0^{-1}\) is increasing. If E is a BIS interpolative between \(\ell_ 1(f^{-1})\equiv (\ell_ 1(f_ 0^{- 1}),\ell_ 1(f_ 1^{-1}))\) and \(\ell^{\infty}(f^{-1})\equiv (\ell^{\alpha}(f_ 0^{-1})\), \(\ell^{\infty}(f_ 1^{-1}))\) then the only i.f. (upto equality defined above) F satisfying the relation \(F(\ell_ 1(f^{-1}))=F(\ell^{\alpha}(f^{-1}))=E\) is the exact i.f. \(F_{f_ 0,f_ 1,E}\) given by \(F_{f_ 0,f_ 1,E}(X_ 0,X_ 1)={\mathcal O}_ E^{\ell_ 1(f_ 1^{-1})}(X_ 0,X_ 1)={\mathcal P}_ E^{\ell^{\alpha}(f^{-1})}(X_ 0,X_ 1).\) There are other interesting results relating to a concept of sufficiency for i.f. (introduced by the author) and a new interpolation relation.
scientific article; zbMATH DE number 4060129