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**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.

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 |