## A remark about twisting Schatten classes.(English)Zbl 1328.46015

N. J. Kalton [Trans. Am. Math. Soc. 333, No. 2, 479–529 (1992; Zbl 0776.46033)] proved that the space of Schatten class operators $$\mathcal{S}_p$$ can be twisted, i.e., there exists a $$B(H)$$-module, $$\Theta_p$$, containing a non-complemented copy of the Schatten class $$\mathcal{S}_p$$, such that the quotient is another copy of $$\mathcal{S}_p$$. Here it is shown that this quotient mapping is strictly singular, and that the dual of $$\Theta_p$$ is $$\Theta_q$$, where $$p$$ and $$q$$ have the usual relationship. While the latter result is to be expected, the author provides a service to the community by presenting a direct proof.

### MSC:

 46B28 Spaces of operators; tensor products; approximation properties 47B10 Linear operators belonging to operator ideals (nuclear, $$p$$-summing, in the Schatten-von Neumann classes, etc.) 46M18 Homological methods in functional analysis (exact sequences, right inverses, lifting, etc.)

### Keywords:

Schatten class; twisted sum; singular operator; dual space

Zbl 0776.46033
Full Text:

### References:

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