Eigenvalues of p-summing and lp-type operators in Banach spaces

AbstractThis paper is a study of the distribution of eigenvalues of various classes of operators. In Section 1 we prove that the eigenvalues (λn(T)) of a p-absolutely summing operator, p ⩾ 2, satisfy ∑n∈N|λn(T)|p1p⩽πp(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 Π2(n), P = (p1, …, pn) and show that for T ∈ Π2(n), 2 = (2, …, 2) ∈ Nn, the eigenvalues of T satisfy ∑i∈N|λi(T)|2nn2⩽πn2(T). More generally, we show that for T ∈ Πp(n), P = (p1, …, pn), pi ⩾ 2, the eigenvalues are absolutely p-summable, 1p=∑i=1n1pi and ∑n∈N|λn(T)|p1p⩽CpπnP(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 ∑n∈N|λn(T)|p ⩽ Cp∑n∈N αn(T)p, 0 < p < ∞, where αn 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