Eigenvalues of Operators on Banach Spaces

Abstract

The q-nuclear operators N_q, q- and (q,2)-summing operators π_q and π_q,2 as well as the s-number ideals S ^a_q and S ^x_q coincide with Schatten classes S_p (H) on Hilbert spaces H, for appropriate values of p = p(q) (1.d.12). Thus they form extensions of the Schatten classes to operator ideals on (all) Banach spaces. On a fixed Banach space, all maps in these ideals are power-compact and hence Riesz operators (1.a.5). By Weyl’s inequality (1.b.9), S_p(H)-operators have p-th power summable eigenvalues. It is thus natural to ask: What is the optimal order of summability of the eigenvalues of the above classes of operators on Banach spaces? This is the main topic of this chapter: we extend Weyl’s inequality to the above operator ideals on Banach spaces.