Spectral inclusion for unbounded block operator matrices
The quadratic numerical \(W^2(\mathcal A)\) range of the block operator matrix \[ \mathcal A = \begin{pmatrix} A & B \cr C & D\cr \end{pmatrix}, \qquad \mathcal D(\mathcal A) = \mathcal D(A)\cap\mathcal D(C) \oplus \mathcal D(B)\cap\mathcal D(D) \subseteq \mathcal H_1\oplus \mathcal H_2, \] is the set of all eigenvalues of all \(2\times 2\) matrices \[ \mathcal A_{f,g} = \begin{pmatrix} (Af,f) & (Bg,f) \cr (Cf,g) & (Dg,g)\cr \end{pmatrix} \in M_2(\mathbb C), \] where \(f\in \mathcal D(A)\cap\mathcal D(C)\), \(g\in \mathcal D(B)\cap\mathcal D(D)\) with \(\|f\| = \|g\| = 1\) and \(\mathcal H_1,\mathcal H_2\) are Hilbert spaces. It is assumed that \(A, B, C\) and \(D\) are closed and densely defined linear operators. In contrast to the numerical range \(W(\mathcal A)\), the quadratic numerical range is not necessarily convex. In the main part of the article, the author proves the spectral inclusions \[ \sigma_p(\mathcal A) \subseteq W^2(\mathcal A) \subseteq W(\mathcal A), \quad \sigma_{app}(\mathcal A) \subseteq \overline{ W^2(\mathcal A) } \subseteq \overline{ W(\mathcal A)} \] if \(\mathcal A\) is diagonally dominant or off-diagonally dominant of order \(0\). Conditions that guarantee that \(\mathcal A\) is diagonally dominant or off-diagonally dominant are given in the first part of the paper. The paper concludes with an application to off-diagonally dominant selfadjoint block operator matrices with separated diagonal elements (a typical example is the Dirac operator in \(\mathbb R^3\)) and block operator matrices where \(A\) and \(D\) are selfadjoint and \(C=-B^\ast\) or \(C=B^\ast\) (operators of the latter type appear as models for a two channel Hamiltonian). The paper generalizes results of \textit{H.\,Langer, A.\,Markus, V.\,Matsaev} and \textit{C.\,Tretter} [Linear Algebra Appl.\ 330, No.\,1--3, 89--112 (2001; Zbl 0998.15035)] and \textit{H.\,Langer} and \textit{C.\,Tretter} [J.~Oper.\ Theory 39, No.\,2, 339--359 (1998; Zbl 0996.47006)].
scientific article; zbMATH DE number 5565776
scientific article