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

Kolchinskij, V. I.

Teor. Veroyatn. Mat. Stat., Kiev 38, 69-76 (1988).

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

Keywords:

Banach spaces; bounded linear operator; dual spaces; dual operator; \(\epsilon \) -entropy; operator of type p