Conditional Reducibility of Certain Unbounded Nonnegative Hamiltonian Operator Functions

Abstract

Let J and \({{\mathfrak{J}}}\) be operators on a Hilbert space \({{\mathcal{H}}}\) which are both self-adjoint and unitary and satisfy \({J{\mathfrak{J}}=-{\mathfrak{J}}J}\) . We consider an operator function \({{\mathfrak{A}}}\) on [0, 1] of the form \({{\mathfrak{A}}(t)={\mathfrak{S}}+{\mathfrak{B}}(t)}\) , \({t \in [0, 1]}\) , where \({\mathfrak{S}}\) is a closed densely defined Hamiltonian (\({={\mathfrak{J}}}\) -skew-self-adjoint) operator on \({{\mathcal{H}}}\) with \({i {\mathbb{R}} \subset \rho ({\mathfrak{S}})}\) and \({{\mathfrak{B}}}\) is a function on [0, 1] whose values are bounded operators on \({{\mathcal{H}}}\) and which is continuous in the uniform operator topology. We assume that for each \({t \in [0,1] \,{\mathfrak{A}}(t)}\) is a closed densely defined nonnegative (=J-accretive) Hamiltonian operator with \({i {\mathbb{R}} \subset \rho({\mathfrak{A}}(t))}\) . In this paper we give sufficient conditions on \({{\mathfrak{S}}}\) under which \({{\mathfrak{A}}}\) is conditionally reducible, which means that, with respect to a natural decomposition of \({{\mathcal{H}}}\) , \({{\mathfrak{A}}}\) is diagonalizable in a 2×2 block operator matrix function such that the spectra of the two operator functions on the diagonal are contained in the right and left open half planes of the complex plane. The sufficient conditions involve bounds on the resolvent of \({{\mathfrak{S}}}\) and interpolation of Hilbert spaces.