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A functional model for a family of operators induced by Laguerre operator. (English) Zbl 1109.47308
Summary: The paper generalizes the construction, suggested by B. Sz.-Nagy and C. Foias, for an operator function induced by the Cauchy problem \[ T_t\: \begin{cases} th'' (t) + (1 - t) h' (t) + Ah (t) = 0,\\ h (0) = h_0 (th') (0) = h_1. \end{cases} \] A unitary dilatation for \(T_t\) is constructed in the present paper, then a translational model for the family \(T_t\) is presented using a model construction scheme, suggested by V. A. Zolotare. Finally, we derive a discrete functional model of the family \(T_t\) and the operator \(A\), applying the Laguerre transform \[ f (x) \mapsto \int ^{\infty }_0 f (x) P_n (x)\text e^{-x}\,dx, \] where \(P_n (x)\) are Laguerre polynomials. We show that the Laguerre transform is a straightening transform which transfers the family \(T_t\) (which is not a semigroup) into the discrete semigroup \(e^{-itn}\).
MSC:
47D06 One-parameter semigroups and linear evolution equations
47A40 Scattering theory of linear operators
47A50 Equations and inequalities involving linear operators, with vector unknowns
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