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Type and cotype numbers of operators on Banach spaces. (English) Zbl 0726.47013

The author introduces cotype numbers of an operator T: \(X\to Y\) between normed spaces, \[ x_ n(T| {\mathfrak P}_ G)=\sup \{a_ n(TS)| \quad S: \ell^ m_ 2\to X,\quad \ell (S)\leq 1,\quad m\in {\mathbb{N}}\}, \] where \(a_ n\) denote the approximation numbers and \(\ell (S)\) the \(\ell\)-norm of S. He studies the relation of these numbers to other s- numbers and operator ideals, in particular the relation with the notion of weak cotype. It is shown using this notion that if the nuclear operators on X have eigenvalues of order \(O(n^{-1/r})\), the space is of weak type p for \(1/p+1/r=3/2\) and weak cotype q for \(1/q=1/r-1/2\). Conversely, if X is of weak type p and weak cotype q, the eigenvalues of nuclear operators on X are of order \((n^{-1/r})\) with \(1/r=1+1/q-1/p\). The limit case is of the one f Pisier’s weak Hilbert spaces \((p=q=2)\). In various examples, the cotype numbers are estimated.
Reviewer: H.König (Kiel)

MSC:

47B06 Riesz operators; eigenvalue distributions; approximation numbers, \(s\)-numbers, Kolmogorov numbers, entropy numbers, etc. of operators
47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
47L20 Operator ideals
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