Integral triangular operators and Friedrichs model

TL;DR

This paper introduces semi-groups of triangular integral operators as analogues of fractional integrals, constructs Friedrichs models based on these operators, and proves their similarity to self-adjoint operators with absolutely continuous spectra.

Contribution

It develops new semi-groups of triangular integral operators and applies them to construct Friedrichs models, establishing their spectral properties.

Findings

Triangular integral operator semi-groups are analogous to fractional integrals.

Constructed Friedrichs models are linearly similar to self-adjoint operators.

Models have absolutely continuous spectra.

Abstract

In the present paper we investigate a semi-group of triangular integral operators $V_{\beta}$, which is an analogue of the semi-group of the fractional integral operators $J^{\beta}$. With the help of these semi-groups, we construct and study two classes of triangular Friedrichs models $A_{\beta}$ and $B_{\beta}$, respectively. Using generalized wave operators we prove that $A_{\beta}$ and $B_{\beta}$ are linearly similar to a self-adjoint operator with absolutely continuous spectrum.