Spear Operators Between Banach Spaces by Vladimir Kadets Miguel Martín Javier Merí & Antonio Pérez

Author:Vladimir Kadets, Miguel Martín, Javier Merí & Antonio Pérez
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

Proof

The equivalences are consequence of Proposition 3.10, Theorems 3.11 and 3.14, together with Proposition 3.25 to pass from to . □

Next, we get from the previous sections some conditions for an operator having the aDP to be lush. The main result in this line is the next one, which follows from Theorem 3.17 applied to T =Id X .

Theorem 3.29

Let X, Y be Banach spaces and let be a norm-one operator. Suppose that B X is SCD. Then, G has the aDP if and only if G is lush.

As all the properties involved in the above result are separably determined, we have the following generalization.

Corollary 3.30

Let X, Y be Banach spaces and let be a norm-one operator. Suppose that is SCD for every separable subspace X 0 ⊂ X. Then, G has the aDP if and only if G is lush.

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