Grothendieck–Lidskiĭ theorem for subspaces of quotients of Lp-spaces

Abstract

Generalizing A. Grothendieck's (1955) and V.B. Lidski 's (1959) trace formulas, we have shown in a recent paper that for p ∈ [1,∞] and s ∈ (0, 1] with 1/s = 1+ |1/2− 1/p| and for every s-nuclear operator T in every subspace of any Lp(ν)-space the trace of T is well de ned and equals the sum of all eigenvalues of T. Now, we obtain the analogues results for subspaces of quotients (equivalently: for quotients of subspaces) of Lp-spaces. In the note [13], we have proved that if p ∈ [1,∞] and 1/s = 1+ |1/2−1/p|, then for any subspace (or quotient) of an Lp-space and for every s-nuclear operator T in the space the nuclear trace of T is well-de ned and equals the sum of all eigenvalues of T. The main fact, we are going to obtain here, is Theorem. Let Y be a subspace of a quotient (or a quotient of a subspace) of an Lp-space, 1 ≤ p ≤ ∞. If T ∈ Ns(Y, Y ) (s-nuclear), where 1/s = 1 + |1/2− 1/p|, then 1. the (nuclear) trace of T is well de ned, 2. ∑∞ n=1 |λn(T )| < ∞, where {λn(T )} is the system of all eigenvalues of the operator T (written in according to their algebraic multiplicities) and