Bounds for eigenvalues of Schatten-von Neumann operators via self-commutators
The author gives upper bounds for the Schatten \(2p\)-norm, with integer \(p\geq 1\), of operators in terms of the trace of certain commutators. Suppose that \(\mathcal H\) is a separable Hilbert space and \(A\) is a Schatten-von Neumann operator on \(\mathcal H\) with finite norm \(N_{2p}(A)=\) Trace\((AA^*)^{\frac{1}{2p}}\), where \(p\in\mathbb N\). He establishes the following two inequalities: NEWLINE\[NEWLINE\sum_{k=1}^\infty|\lambda_k(A)|^{2p}\leq\Bigg[N_{2p}^{4p}(A)-\frac{1}{4}N_1^2([A,A^*]_p)\Bigg]^{1/2},NEWLINE\]NEWLINE where \(A^*\) is the adjoint of \(A\), \(\lambda_k(A)\), \(k=1,2,\dots,\) are the eigenvalues of \(A\) and \([A,A^*]_p=A^p(A^*)^p-(A^*)^pA^p\). Moreover, if \(A\) is a Hilbert-Schmidt operator, then NEWLINE\[NEWLINE\sum_{k=1}^\infty|\mathrm{Im }\lambda_k(A)|^2\leq N_2^2(A_I),NEWLINE\]NEWLINE where \(A_I=\frac{A-A^*}{2i}.\)
scientific article; zbMATH DE number 6349106