Geodesic
A functional model for a family of operators induced by Laguerre operator
Archivum mathematicum, Tome 39 (2003) no. 1, pp. 11-25.

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The paper generalizes the instruction, suggested by B. Sz.-Nagy and C. Foias, for operatorfunction induced by the Cauchy problem \[ T_t : \left\lbrace \begin{array}{ll}th^{\prime \prime }(t) + (1-t)h^\prime (t) + Ah(t)=0\\ h(0) = h_0 (th^\prime )(0)=h_1 \end{array}\right.\] 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 Zolotarev, V., [3]. Finally, we derive a discrete functional model of family $T_t$ and operator $A$ applying the Laguerre transform \[ f(x)\rightarrow \int _0^\infty f(x) \,P_n(x)\,e^{-x} dx \] where $P_n(x)$ are Laguerre polynomials [6, 7]. We show that the Laguerre transform is a straightening transform which transfers the family $T_t$ (which is not semigroup) into discrete semigroup $e^{-itn}$.
Classification : 34G99, 47A40, 47A48, 47A50, 47D06, 47E05
Keywords: Laguerre operator; semigroup; Hilbert space; functional model 
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     title = {A functional model for a family of operators induced by {Laguerre} operator},
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Ra'ed, Hatamleh. A functional model for a family of operators induced by Laguerre operator. Archivum mathematicum, Tome 39 (2003) no. 1, pp. 11-25. http://geodesic.mathdoc.fr/item/ARM_2003__39_1_a1/