Finite representability of the Yang operator.

*(English)*Zbl 1029.46008To every operator \(T:X\to Y\) acting between two Banach spaces, one can associate the operator \(T^{co}:X^{**}/X\to Y^{**}/Y\), called the Yang operator, by \(T^{co}(x^{**}+X)=T^{**}(x^{**})+Y\). The authors wish to establish two sorts of finite representability of the Yang operator. First, given two operators \(T\in {\mathcal L}(X,Y)\) and \(S\in {\mathcal L}(W,Z) \) and a number \(d\geq 1\), say that \(T\) is locally \(d\)-supportable in \(S\) provided that, for every \(\epsilon>0\) and every finite- dimensional subspace \(E\) of \(X\), there is a \((d+\epsilon)\)-injection \(U\) in \({\mathcal L}(E,W)\) and an operator \(V\) in \( {\mathcal L}(T(E),Z)\) satisfying \(\|V\|<d+\epsilon\) and \(\|SU-VT|_E \|\leq \epsilon\). (The condition that \(U\) is a \((d+\epsilon)\)-injection means that \((d+\epsilon)^{-1} \leq \|Ux\|\leq (d+\epsilon)\) for every unit vector \(x\) in \(E\).)

On the other hand, given \(T\) and \(S\) as above and a number \(c>0\), say that \(T\) is locally \(c\)-representable in \(S\) if, for every \(\epsilon>0\) and every pair of operators \(A\in {\mathcal L} (E,X)\) and \(B\in {\mathcal L}(Y,F)\) with \(E\) and \(F\) finite-dimensional, there exist operators \(A_1\in {\mathcal L}(E,W)\) and \(B_1\in {\mathcal L}(Z,F) \) satisfying \(\|A_1\|\cdot \|B_1\|\leq (c+\epsilon) \|A\|\cdot \|B\|\) and \(BTA = B_1SA_1\). These are generalizations of the notions of finite representability due to Bellenot and to Heinrich, respectively.

A series of lemmas leads to new proofs (Theorems 3.4 and 3.5) that \(T^ {**}\) is finitely representable in \(T\) (in either the Bellenot or Heinrich sense) but also provides additional information related to \(T^ {co}\). These properties are then put together to show (Theorems 3.7 and 3.8) that the Yang operator is both locally 6-supportable and locally 6-representable in \(T\). A final result, in the context of ultrafilters, shows that any regular, ultrapower-stable ideal of operators that contains \(T\) also contains \(T^{co}\).

On the other hand, given \(T\) and \(S\) as above and a number \(c>0\), say that \(T\) is locally \(c\)-representable in \(S\) if, for every \(\epsilon>0\) and every pair of operators \(A\in {\mathcal L} (E,X)\) and \(B\in {\mathcal L}(Y,F)\) with \(E\) and \(F\) finite-dimensional, there exist operators \(A_1\in {\mathcal L}(E,W)\) and \(B_1\in {\mathcal L}(Z,F) \) satisfying \(\|A_1\|\cdot \|B_1\|\leq (c+\epsilon) \|A\|\cdot \|B\|\) and \(BTA = B_1SA_1\). These are generalizations of the notions of finite representability due to Bellenot and to Heinrich, respectively.

A series of lemmas leads to new proofs (Theorems 3.4 and 3.5) that \(T^ {**}\) is finitely representable in \(T\) (in either the Bellenot or Heinrich sense) but also provides additional information related to \(T^ {co}\). These properties are then put together to show (Theorems 3.7 and 3.8) that the Yang operator is both locally 6-supportable and locally 6-representable in \(T\). A final result, in the context of ultrafilters, shows that any regular, ultrapower-stable ideal of operators that contains \(T\) also contains \(T^{co}\).

Reviewer: Timothy Feeman (Villanova)