# On simultaneous approximation of several eigenvalues of a semi-definite self-adjoint linear operator in a Hilbert space

## TL;DR

This paper addresses the problem of approximating multiple eigenvalues of a semi-definite self-adjoint operator in a Hilbert space using finite-dimensional operators, with accuracy improving as the dimension increases.

## Contribution

It provides a method for simultaneous approximation of several eigenvalues of semi-definite self-adjoint operators with increasing accuracy as the finite-dimensional space expands.

## Key findings

- Approximation accuracy improves as the dimension s increases.
- Method applies to operators with non-empty discrete spectrum.
- Convergence of eigenvalue approximations is established.

## Abstract

A lower semi-definite self-adjoint linear operator in a Hilbert space is taken whose discrete spectrum is not empty and comprises at least several eigenvalues $\lambda_{min}=\lambda_1\leqslant\ldots\leqslant\lambda_m<\sigma_{ess}$. The problem of approximation of these eigenvalues by eigenvalues of some linear operator in a finite-dimensional space of the dimension $s$ is considered and solved. The accuracy of the approximation obtained becomes unlimitedly high as $s\to\infty$.