Bounds for eigenvalues of Schatten-von Neumann operators via self-commutators

Let H be a separable Hilbert space, A be a Schatten-von Neumann operator in H with the finite norm N2p(A)=[Trace(AA⁎)p]1/2p for an integer p≥1 and N1(T)=Trace(TT⁎)1/2 be the trace norm of a trace operator T. It is proved that∑k=1∞|λk(A)|2p≤[N2p4p(A)−14N12([A,A⁎]p)]1/2, where λk(A) (k=1,2,...) are the eigenvalues of A, [A,A⁎]p=Ap(A⁎)p−(A⁎)pAp; A⁎ is the adjoint to A. This results refines the classical inequality ∑k=1∞|λk(A)|2p≤N2p2p(A). Lower bounds for N1([A,A⁎]p) are also suggested. In addition, if A is a Hilbert-Schmidt operator, we improve the well-known inequality∑k=1∞|Imλk(A)|2≤N22(AI), where AI=(A−A⁎)/2i.