Estimates for the spectrum of a third-order operator matrix with a parameter

In this talk, we propose an alternative formula for computing the cubic numerical range of bounded self-adjoint $3\times 3$ operator matrices. We describe the essential and discrete spectra of a third-order operator matrix in Fock space. Furthermore, we establish a spectral inclusion for the spectra of the block elements. Finally, using the cubic numerical range, Gershgorin’s theorem, and classical perturbation theory, we obtain spectral estimates for this operator matrix.