Closely embedded Kreĭn spaces and applications to Dirac operators (Q624613)

The authors continue their study of Krein spaces induced by symmetric operators [\textit{P.\,Cojuhari} and \textit{A.\,Gheondea}, J.~Oper.\ Theory 61, No.\,2, 347--367 (2009; Zbl 1174.47016)]. Here the new concept of closely embedded Krein spaces is added and it is put into a correspondence with the notion of Krein spaces induced by symmetric operators. This new notion is then applied to Dirac operators and different results corresponding to a mass or a massless particle are obtained. Here a pair \((\mathcal K, \Pi)\) is called a \textit{Krein space induced by \(A\)} if \((\mathcal K, [\cdot , \cdot]_{\mathcal K})\) is a Krein space, \(A\) is a densely defined symmetric operator in a Hilbert space \((\mathcal H, (\cdot , \cdot)_{\mathcal H})\), and \(\Pi :\mathcal H \to \mathcal K\) is a linear mapping with dense range and a domain which contains the domain of \(A\) such that, for all \(x\) in the domain of \(A\) and all \(y\) in the domain of \(\Pi\), the following holds: \[ [\Pi x,\Pi y]_{\mathcal K}= (Ax,y)_{\mathcal H}. \] On the other hand, a Krein space \(\mathcal K\) is said to be \textit{closely embedded} in a Hilbert space \(\mathcal H\) if \(\mathcal D := \mathcal H \cap \mathcal K\) is dense in \(\mathcal K\), the embedding \(j: \mathcal D \to \mathcal H\) defines a closed operator, and the linear manifold \(\mathcal D\) admits a decomposition into a uniformly positive and a uniformly negative manifold in the Krein space \(\mathcal K\). The relation between induced Krein spaces and embedded Krein spaces is the following: If \(\mathcal K\) is closely embedded in \(\mathcal H\) with embedding \(j\), then \(A:=jj^+\) is a selfadjoint operator in \(\mathcal H\) and \((\mathcal K, j^+)\) is a Krein space induced by \(A\). If \((\mathcal K, \Pi)\) is a Krein space induced by \(A\), then the range of \(\Pi^+\) is a closely embedded Krein space in \(\mathcal H\).