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Limited operators and strict cosingularity. (English) Zbl 0601.47019

Recall that a bounded linear operator T:X\(\to Y\) between Banach spaces is called strictly cosingular if the only Banach spaces E for which one can find surjective bounded linear operators \(q_ X:X\to E\) and \(q_ Y:Y\to E\) for which \(q_ X=q_ YT\) are finite dimensional. The main result of this note is the following theorem:
A bounded linear operator T:X\(\to Y\) is strictly cosingular if its adjoint \(T^*:Y^*\to X^*\) takes sequences tending to zero for \(\sigma (Y^*,Y)\) into sequences tending to zero in norm.
This result may be viewed as an operator theoretic generalization of the B. Josefson and A. Nissenzweig theorem [cf. Ark. Math. 13, 79-89 (1975; Zbl 0303.46018) and Isr. J. Math. 22 (1975), 266-272 (1976; Zbl 0341.46012)].
Reviewer: G.Pisier

MSC:

47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.)
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