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.


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
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