## 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.)

### Citations:

Zbl 0303.46018; Zbl 0341.46012
Full Text:

### References:

 [1] Bessaga, Studia Math. 17 pp 151– (1958) [2] Bourgain, Amer. J. Math. 100 pp 845– (1978) [3] James, Ann. of Math. 52 pp 518– (1950) [4] James, Transactions American Math. Soc. 113 pp 129– (1964) [5] Josefson, Arkiv for Matematik 13 pp 79– (1975) [6] Lindenstrauss, Bull. Amer. Math. Soc. 72 pp 967– (1966) [7] Nissenzweig, Isreal J. Math. 22 pp 266– (1975) [8] Pelczynski, Bull. Acad. Polon Sci. 13 pp 31– (1965) [9] Bull. Acad. Polon Sci. 13 pp 37– (1965) [10] Pietsch, Studia Math. 28 pp 333– (1967) [11] Rosenthal, Proc. Nat. Acad. Sci. (USA) 71 pp 2411– (1974)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.