On the existence of maximal semidefinite invariant subspaces for \(J\)-dissipative operators (Q2884665)

The main problem considered in the paper concerns necessary and sufficient conditions for the existence of maximal nonnegative/nonpositive invariant subspaces for \(J\)-dissipative operators. Here, \(J\)-dissipative is understood in the sense of Krein spaces, i.e., an operator \(A\) in a Krein space \((H,[\cdot,\cdot])\) is \(J\)-dissipative if the real part of \([Ax,x]\) for \(x\in\,\)dom\(A\) is nonpositive.NEWLINENEWLINEThe study of semi-definite invariant subspaces of \(J\)-dissipative operators is a classical topic in operator theory and was initiated by \textit{L. S.\ Pontryagin} [Izv. Akad.\ Nauk USSR, Ser.\ Mat.\ 8, 243--280 (1944; Zbl 0061.26004)] and \textit{S. L.\ Sobolev} [Zh.\ Prikl. Mekh. Tekh. Fiz. 3, 20--55 (1960; Zbl 0105.18103)].NEWLINENEWLINEThe results of L. S.\ Pontryagin and S. L.\ Sobolev were generalized to various classes of operators by many authors, among them T. Ya.\ Azizov, M. G.\ Krein, H.\ Langer and A. A.\ Shkalikov. In these contributions, the operator under consideration is represented as a \(2\times 2\) operator matrix with respect to a fundamental decomposition of the Krein space \(H=H_+\times H_-\). Under some compactness assumption, which is usually connected with the off-diagonal entries of the \(2\times 2\) operator matrix, the existence of a maximal nonnegative/nonpositive invariant subspace follows.NEWLINENEWLINEIn contrast to the above mentioned approach, the present paper uses results from the interpolation theory in Banach spaces. There is no compactness condition needed, instead, conditions for some kind of subordination of the off-diagonal operators to the diagonal operators is used to show the existence of a maximal nonnegative/nonpositive invariant subspace. More precisely, let \(L\) be a maximal \(J\)-dissipative operator with \(i\mathbb R \subset \rho(L)\), let \(F_1\) be the completion of its domain with respect to the norm \(x\mapsto -\)Re\(\,[Lx,x]+\|x\|^2\) and let \(F_{-1}\) be the completion of \(H\) with respect to the negative norm which is defined with the help of \(F_1\) as usual. Then it is shown that \(L\) has maximal nonnegative/nonpositive invariant subspace, provided that the inner product \(|[L\cdot,\cdot]|\) is continuous with respect to the norm of \(F_1\) and that NEWLINE\[NEWLINE \left(F_1,F_{-1}\right)_{\frac{1}{2},2} =H, NEWLINE\]NEWLINE where the above notion is understood in the sense of interpolation.