On the Number of Components of the Essential Spectrum of One 2 × 2 Operator Matrix

Abstract

In this paper, a \(2 \times 2\) block operator matrix \(H\) is considered as a bounded and self-adjoint operator in a Hilbert space. The location of the essential spectrum \({{\sigma }_{{{\text{ess}}}}}(H)\) of operator matrix \(H\) is described via the spectrum of the generalized Friedrichs model, i.e., the two- and three-particle branches of the essential spectrum \({{\sigma }_{{{\text{ess}}}}}(H)\) are singled out. We prove that the essential spectrum \({{\sigma }_{{{\text{ess}}}}}(H)\) consists of no more than six segments (components).