Browsing by Subject "linear operators"

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Access status: Open Access ,
An inequality for imaginary parts of eigenvalues of non-compact operators with Hilbert-Schmidt Hermitian components
(Wydawnictwa AGH, 2024) Gil', Michael
Let $A$ be a bounded linear operator in a complex separable Hilbert space, $A^{*}$ be its adjoint one and $A_I:=(A-A^*)/(2i)$. Assuming that $A_I$ is a Hilbert-Schmidt operator, we investigate perturbations of the imaginary parts of the eigenvalues of $A$. Our results are formulated in terms of the »extended« eigenvalue sets in the sense introduced by T. Kato. Besides, we refine the classical Weyl inequality $\sum_{k=1}^\infty (\operatorname{Im} \lambda_k(A))^2 \leq N_2^2(A_I)$, where $\lambda_{k}(A)$ $(k=1,2, \ldots )$ are the eigenvalues of $A$ and $N_2(\cdot)$ is the Hilbert-Schmidt norm. In addition, we discuss applications of our results to the Jacobi operators.
Access status: Open Access ,
Simple eigenvectors of unbounded operators of the type »normal plus compact«
(2015) Gil', Michael
The paper deals with operators of the form $A=S+B$, where $B$ is a compact operator in a Hilbert space $H$ and $S$ is an unbounded normal one in $H$, having a compact resolvent. We consider approximations of the eigenvectors of $A$, corresponding to simple eigenvalues by the eigenvectors of the operators $A_{n}=S+B_{n}$ ($n=1,2, \ldots$), where $B_n$ is an $n$-dimensional operator. In addition, we obtain the error estimate of the approximation.