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

Hatamleh Ra'ed

Address. Department of Mathematics, Irbid National University P.O.Box 2600, Irbid, JORDAN

E-mail: raedhat@yahoo.com

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.