Nuclearity Of Operators Research Articles

Grothendieckʼs nuclear operator theorem revisited with an application to p-null sequences

Let X and Y be Banach spaces and let α be a tensor norm. The principal result is the following theorem. If either X⁎⁎⁎ or Y has the approximation property, then each α-nuclear operator T:X⁎→Y such that T⁎(Y⁎)⊂X can be approximated in the α-nuclear norm by finite-rank operators of type X⊗Y. In the special case of (Grothendieck) nuclear operators, the theorem provides a strengthening for the classical theorem on the nuclearity of operators with a nuclear adjoint. The hypotheses about the approximation property are essential. The main application yields an affirmative answer to [C. Piñeiro, J.M. Delgado, p-Convergent sequences and Banach spaces in which p-compact sets are q-compact, Proc. Amer. Math. Soc. 139 (2011) 957–967]: for p⩾1, a sequence (xn)⊂X is p-null if and only if limxn=0 and (xn) is relatively p-compact in X.

The nuclearity of operators generated by Hölder continuous kernels

Introduction. The space Cp is the class of operators on a Hilbert space for which the norm ∥K∥p = [trace (KK*)P/2]/p is finite. Equivalently, a compact operator is in Cp ifwhere the μn are the so-called ‘singular values’ of K (characteristic values of the non-negative compact operator [K] ≡ (KK*)½). The case p = 2 gives the familiar class of Hilbert–Schmidt operators, while C1 is the collection of trace-class or nuclear operators considered by Schatten(12), Lidskii(11), and Gohberg and Krein(7), among others.