## Dimensional gradations and cogradations of operator ideals. The weak distance between Banach spaces.(English)Zbl 0538.47029

Semin. Anal. Fonct. 1980-1981, Exp. No.IX, 31 p. (1981).
The author introduces and studies gradations of a Banach ideal of operators ($${\mathfrak A},\alpha)$$, i.e. an increasing sequence of ideal norms $$\alpha_ k$$ with $$\alpha(u)=\lim_ k\alpha_ k(u)$$ for all $$u\in {\mathfrak A}(E,F)$$ such that $$\alpha_ k(u)=\sup \alpha_ k(u|_{E_ k})$$ where $$E_ k\subseteq E$$ with $$\dim E_ k\leq k$$. Among the basic examples are the (q,r)-summing norms $$\pi_{q,r}$$ with their gradations $$\pi^{(k)}_{q,r}$$ defined by k vectors. They are submultiplicative. For any 2$$\leq q\leq \infty$$, one has $$(*)\quad \pi_{q,2}\leq c_ q\pi^{(k)}_{q,2}(u)$$ for operators u with rank $$u\leq k$$. A corresponding statement is false for $$\pi_ 1$$ (Theorem 5).
Further examples are given by the Gaussian type p and cotype q ideal norms $$\alpha_ p,\beta_ q$$ and their gradations $$\alpha_{p,k}$$ and $$\beta_{q,k}$$. Similar relations as (*) can be shown. They imply a local Maurey-type extension theorem in terms of the Gaussian gradation norms. For the p-factorable operators $$(\Gamma_ p,\gamma_ q)$$ the gradations $$\gamma_{p,k}$$ are introduced. It is shown that $$\gamma_{2,km}(u)\leq 2\sqrt{m}\gamma_{2,k}(u).$$ The problem of submultiplicativity of these gradations is related to the distance problem in spaces of type $$p>1$$. The dual notion of cogradation is illustrated by the (t,s)-nuclear norm $$\nu_{t,s}$$, the quasinorms $$\nu^{(k)}_{t,s}$$ defined by representations of length k and its ”normification” $${\hat \nu}^{(k)}_{t,s}$$, being the adjoint of $$\pi^{(k)}_{t',s'}$$. Another example is the cogradation $${\hat \gamma}_ p^{(k)}$$ of $$\gamma_ p$$ or, more general, $${\hat \gamma}_ E^{(k)}$$ of $${\hat \gamma}_ E$$, the E-factorable norm. This leads to the weak Banach-Mazur distance which is studied at the end of the paper.

### MSC:

 47L10 Algebras of operators on Banach spaces and other topological linear spaces 46B20 Geometry and structure of normed linear spaces 47B06 Riesz operators; eigenvalue distributions; approximation numbers, $$s$$-numbers, Kolmogorov numbers, entropy numbers, etc. of operators
Full Text: