Eigenvalues of psumming and lptype 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 (λn(T)) of a pabsolutely summing operator, p ≥ 2, satisfy ∑ n∈Nλn(T)p 1 p≤πp(T). This solves a problem of A. Pietsch. We give applications of this to integral operators in Lpspaces, 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∈Nλi(T) 2 n n 2≤πn2(T). More generally, we show that for T ∈ Πp(n), P = (p1, ..., pn), pi ≥ 2, the eigenvalues are absolutely psummable, 1 p= ∑ i=1 n 1 pi and ∑ n∈Nλn(T)p 1 p≤CpπnP(T). We also consider the distribution of eigenvalues of pnuclear operators on Lrspaces. In Section 3 we prove the Banach space analog of the classical Weyl inequality, namely ∑ n∈Nλn(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 MarkusMacaev. 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.
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Johnson, W. B., König, H., Maurey, B., & Retherford, J. R.
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Johnson, WBKönig, HMaurey, BRetherford, JR
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