Chapter 22 Eigenvalues of operators and applications

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Classical results relate the order of decay of the eigenvalues of integral operators in Lp-spaces to the regularity and integrability properties of the deflning kernel: the smoother the kernel is, the faster the eigenvalues tend to zero. These operators are clearly compact. Riesz’ theory of compact operators admits an abstract quantitative analogue, as far as the decay of the eigenvalues is concerned, for operator classes like p-summing and nuclear operators or operators with summable s-numbers. These latter results will be presented here. Applications to integral operators will be given which extend the well-known classical theorems. The main ingredients of this theory are Weyl’s inequality relating eigenvalues and singular numbers, factorization methods for operator ideals - see also [7] - and interpolation theory. All classes of operators which we consider form operator ideals, i.e. they are stable under compositions with continuous operators. To apply the abstract results, an integral operator T in Lp is factored over a standard map, like a Sobolev imbedding map, which is known to belong to the operator ideal considered; then T itself belongs to the ideal. Remarks on the historical development of the theory can be found in Pietsch’s book [28]. All Banach spaces X;Y;::: will be complex. We let BX = fx 2 X fl flkxk • 1g and denote the linear continuous and compact operators, respectively, between X and Y by L(X;Y ) and K(X;Y ), respectively, with L(X) := L(X;X) and K(X) := K(X;X). A map T 2 L(X) is power-compact if there is n 2N such that T n 2 K(X). Let ae(T) denote the spectrum and ‰(T) the resolvent set of T. The classical Riesz’ spectral theory asserts for power-compact operators T 2 L(X):