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A functional model for a family of operators induced by Laguerre operator

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*Hatamleh Ra'ed*

**Address.** Department of Mathematics, Irbid National University P.O.Box
2600, Irbid, JORDAN
**E-mail:** raedhat@yahoo.com

**Abstract.**

The paper generalizes the instruction, suggested by B.\,Sz.-Nagy and
C. Foias, for operatorfunction

induced by the Cauchy problem $$ T_t : \cases th^{\prime\prime}(t)
+ (1-t)h^\prime (t) + Ah(t)=0\\

h(0) = h_0 (th^\prime)(0)=h_1 \endcases $$ 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., \cite{3}. Finally, we derive a discrete
functional model of family $T_t$ and

operator $A$ applying the Laguerre transform $$ f(x)\to \int_0^\infty
f(x) \,P_n(x)\,e^{-x} dx $$ 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 $e^{-itn}$.

**AMSclassification.** Primary: 47D06, 47A40; Secondary: 47A50, 47A48,
42A50.

**Keywords.** Laguerre operator, semigroup, Hilbert space, functional
model.