Spectral inclusion properties of the numerical range in a space with an indefinite metric
A classical result in operator theory is the spectral inclusion property for bounded operators in Hilbert spaces: all eigenvalues of a bounded operator are included in the numerical range and the spectrum is included in the closure of the numerical range. In a Krein space, there exists an adopted notion of a numerical range \(W_{\mathcal J}(A)\) with respect to the indefinite inner product \([\cdot, \cdot]\) for a (possibly unbounded) operator \(A\), i.e., \[ W_{\mathcal J}(A) := \{[Ax,x][x,x]^{-1} : x \in \text{dom}\,A,\quad [x,x]\neq 0\}. \] For non-negative operators (i.e., \([Ax,x]\geq 0\) for \(x\in \text{dom}\, A\) and \(\rho(A)\neq \emptyset\)), the spectral inclusion property for \(W_{\mathcal J}(A)\) is a consequence of the spectral function, see, e.g., \textit{H.\ Langer} [Lect.\ Notes Math.\ 948, 1--46 (1982; Zbl 0511.47023)]. Here, the authors give an elementary proof of this fact which is direct and does not make use of the spectral function.
uniformly \(\mathcal J\)-positive operator
scientific article; zbMATH DE number 5924524