Eigenvalues of p-summing and lp-type operators in Banach spaces
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This paper is a study of the distribution of eigenvalues of various classes of operators. In Section 1 we prove that the eigenvalues (n(T)) of a p-absolutely summing operator, p 2, satisfy nN|n(T)|p 1 pp(T). This solves a problem of A. Pietsch. We give applications of this to integral operators in Lp-spaces, weakly singular operators, and matrix inequalities. In Section 2 we introduce the quasinormed ideal 2(n), P = (p1, ..., pn) and show that for T 2(n), 2 = (2, ..., 2) Nn, the eigenvalues of T satisfy iN|i(T)| 2 n n 2n2(T). More generally, we show that for T p(n), P = (p1, ..., pn), pi 2, the eigenvalues are absolutely p-summable, 1 p= i=1 n 1 pi and nN|n(T)|p 1 pCpnP(T). We also consider the distribution of eigenvalues of p-nuclear operators on Lr-spaces. In Section 3 we prove the Banach space analog of the classical Weyl inequality, namely nN|n(T)|p Cp nN n(T)p, 0 < p < , where n denotes the Kolmogoroff, Gelfand of approximation numbers of the operator T. This solves a problem of Markus-Macaev. Finally we prove that Hilbert space is (isomorphically) the only Banach space X with the property that nuclear operators on X have absolutely summable eigenvalues. Using this result we show that if the nuclear operators on X are of type l1 then X must be a Hilbert space. 1979.
