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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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