Eigenvalues of Operators on Banach Spaces
The q-nuclear operators Nq, q- and (q,2)-summing operators πq and πq,2 as well as the s-number ideals S q a and S q x coincide with Schatten classes Sp (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), Sp(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.
KeywordsHilbert Space Banach Space Banach Lattice Diagonal Operator Entropy Number
Unable to display preview. Download preview PDF.