Operator theoretical study of inverse problems in linear systems

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Nakamura investigated an inverse problem in the time-varying linear system according to the models of contractions and J-contractions on Hilbert spaces and indefinite inner product spaces, respectively, which are developed by de Branges and Rovnyak. Also he studied a method of constructing the lossless linear system by a given scattering matrix.The backward shift, which is the adjoint operator of the unilateral shift operator, is a most basic linear operator in the operator models of linear systems. When the backward shift acts on a de Branges-Rovnyak space, it is a expansive operator and unitarily equivalent to a one-dimensional perturbation of the shift operator. According to this fact he investigated expansive operators which are one-dimensional perturbation of the shift operator in full detail and obtained conditions for those operators to be similar or quasisimilar to the shift operator. Further spectra and invariant subspaces of those operators were described explicitly.Nakamura also investigated the inequality of Popoviciu from a view point of a variation of the Cauchy-Schwarz inequality in an indefinite inner product form, and gave a new transparent proof of the inequality. As it's application the inequality of Bellman was proved clearly.Toda and Nakai advanced the research from a view point of Complex Analysis, and obtained results on subsets of C^<n+1> in general position and Brelot spaces of Schrodinger equations.Yosimura and Ohyama advanced the research from a view point of Topology, and obtained results on the K_<**>-local types of the smash product of the real projective spaces and Vassiliev invariants in Knot Theory.Adachi advanced the research from a view point of Differential Topology, and obtained a results on circles on a quaternionic space form.