Interpolation of operator ideals with an application to eigenvalue distribution problems

We derive results on the interpolation of complete quasinormed operator ideals, mainly for the absolutelyp-summing and thes-number idealsS _p ^s defined by Pietsch. By estimating theK-Functional of Peetre, we get that the interpolation ideal (S _p1 ^s ,S _p2 ^s ),_,p is contained inS _p ^s and is even equal to it in the case of the approximation numbers. A similar fact is proved for absolutely (p, q)-summing operators, interpolating the first index. We show further that the absolutelyp-summing operators onc ₀ are contained in the complex interpolation space ( _p1 (c _o), _p2 (c _o))_].The previous results are then applied to prove summability properties for the eigenvalues of operators in Banach spaces, which are products ofS _p1 ^s -type and absolutelyp _j -summing operators. Roughly speaking, the summability order is the harmonic sum of thep _i - andp _j -indices, wherep _j 2. In the case of Hilbert spaces, this reduces to the well-known Weyl-inequality. The method uses an abstract interpolation estimate for ideal quasinorms which may be useful also for other operator ideals.