A Bound for Imaginary Parts of Eigenvalues of Hilbert - Schmidt Operators

Abstract

Let     $A$ be a  Hilbert - Schmidt    operator in a separable  Hilbert space, $A^*$ is the adjoint to $A$, and $N_{2}(A)=$ $[Trace\; (AA^*)]^{1/2}$. It is proved that $$ \bs 1 \8 (Im\;\la_k )^2\le N_2^2(A_I)- \fr 18 \left(\ze(A)- \sqrt{\ze^2(A)+2\sqrt 2 N_2([A,A^*])}\right)^2, $$  where $A_I=(A-A^*)/2i$, $\la_k$ $(k=1, 2, ...)$ are the  eigenvalues of $A$, $\ze(A):=\sup_{j, k=1, 2, ...; \;j\neq k} |\la_j-\la_{k}|$ is the spread of the eigenvalues $A$ and $[A,A^*]=AA^*-A^*A$.  That result refines the classical  inequality $$ \bs  1 \8 (Im\;\la_k )^2\le  N_2^2(A_I). $$

{\bf Key words}: Hilbert-Schmidt operator,  inequality for eigenvalues

{\bf AMS (MOS) subject classification}:   47B10, 47B06