## Operators of type p and the metric entropy.(English)Zbl 0664.47013

Let E, F be Banach spaces, T:E$$\to F$$ a bounded linear operator, $$E^*$$, $$F^*$$ the dual spaces of E, F, $$T^*:E^*\to E^*$$ the dual operator of T. Let Q be a subset of the unit sphere in $$F^*$$, $$\| x\| =\sup \{| f(x)|:f\in Q\}$$, $$x\in F$$, $$R=T^*Q$$. Let $$M_ p$$ be a set of all discrete probabilities $$\mu$$ on E, such that $$\int_{E}\| x\|^ p\mu (dx)\leq 1$$, $$p\in [1,+\infty)$$. For $$\mu \in M_ p$$ denote $d_{\mu,p}(f,g)=(\int_{E}| f(x)-g(x)|^ p\mu (dx))^{1/p},\quad f,g\in E^*.$ Let $$Hd_{\mu,p}(R;\epsilon)$$ denote the $$\epsilon$$-entropy of R with respect to $$d_{\mu,p}$$. Suppose $$p\in (1,2]$$, $$q=p/(p-1).$$
Theorem. If $$\sup_{\mu \in M_ p}\int^{\infty}_{0}Hd^{1/q}_{\mu,p}(R;\epsilon)d\epsilon <+\infty$$, then T is an operator of type p. If T is an operator of type p, then $$\sup_{\mu \in M_ p}\sup_{\epsilon >0}\epsilon Hd^{1/q}_{\mu,p}(R;\epsilon)<+\infty.$$
As a corollary some recent results of G. Pisier are obtained.
Reviewer: V.I.Kolchinskij

### MSC:

 47B10 Linear operators belonging to operator ideals (nuclear, $$p$$-summing, in the Schatten-von Neumann classes, etc.) 46B20 Geometry and structure of normed linear spaces