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Finite representability of the Yang operator. (English) Zbl 1029.46008
To 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 $$T^{co}$$.

##### MSC:
 46B09 Probabilistic methods in Banach space theory 47L20 Operator ideals 47B99 Special classes of linear operators 46B07 Local theory of Banach spaces
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