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.
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Acknowledgements
Part of this work was carried out at the University of Ilmenau, Germany. T.Ya. Azizov and A. Dijksma thank the University of Ilmenau and Prof. C. Trunk for their hospitality. We thank Prof. J. Behrndt (Technical University of Graz, Austria), Prof. B. Ćurgus (Western Washingtion University, Bellingham, WA, USA) and Prof. S.G. Pyatkov (Yugra State University, Hanty-Mansiisk, Russia) for helpful comments and references. Finally, we thank the referee for useful suggestions.
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Azizov, T.Y., Dijksma, A. & Gridneva, I.V. Conditional Reducibility of Certain Unbounded Nonnegative Hamiltonian Operator Functions. Integr. Equ. Oper. Theory 73, 273–303 (2012). https://doi.org/10.1007/s00020-012-1964-x
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DOI: https://doi.org/10.1007/s00020-012-1964-x