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 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. 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, where AI=(A-A*)/2i.
- Inequality for eigenvalues
- Schatten-von Neumann operators
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