Geodesic
Investigations of the Numerical Range of a Operator Matrix
Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, no. 2 (2014), pp. 50-63.

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We consider a $2\times2$ operator matrix $A$ (so-called generalized Friedrichs model) associated with a system of at most two quantum particles on ${\mathrm d}-$ dimensional lattice. This operator matrix acts in the direct sum of zero- and one-particle subspaces of a Fock space. We investigate the structure of the closure of the numerical range $W(A)$ of this operator in detail by terms of its matrix entries for all dimensions of the torus ${\mathbf T}^{\mathrm d}$. Moreover, we study the cases when the set $W(A)$ is closed and give necessary and sufficient conditions under which the spectrum of $A$ coincides with its numerical range.
Keywords: operator matrix, generalized Friedrichs model, Fock space, numerical range, point and approximate point spectra, annihilation and creation operators, first Schur compliment. 
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T. H. Rasulov; E. B. Dilmurodov. Investigations of the Numerical Range of a Operator Matrix. Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, no. 2 (2014), pp. 50-63. http://geodesic.mathdoc.fr/item/VSGTU_2014_2_a4/

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