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

Voir la notice de l'article provenant de la source Czech Digital Mathematics Library

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 
@article{ARM_2003__39_1_a1,
     author = {Ra'ed, Hatamleh},
     title = {A functional model for a family of operators induced by {Laguerre} operator},
     journal = {Archivum mathematicum},
     pages = {11--25},
     publisher = {mathdoc},
     volume = {39},
     number = {1},
     year = {2003},
     mrnumber = {1982208},
     zbl = {1109.47308},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ARM_2003__39_1_a1/}
}
TY  - JOUR
AU  - Ra'ed, Hatamleh
TI  - A functional model for a family of operators induced by Laguerre operator
JO  - Archivum mathematicum
PY  - 2003
SP  - 11
EP  - 25
VL  - 39
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/ARM_2003__39_1_a1/
LA  - en
ID  - ARM_2003__39_1_a1
ER  - 
%0 Journal Article
%A Ra'ed, Hatamleh
%T A functional model for a family of operators induced by Laguerre operator
%J Archivum mathematicum
%D 2003
%P 11-25
%V 39
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/ARM_2003__39_1_a1/
%G en
%F ARM_2003__39_1_a1
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/