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

Hatamleh Ra'ed

Archivum Mathematicum (2003)

  • Volume: 039, Issue: 1, page 11-25
  • ISSN: 0044-8753

Abstract

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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 : t h ' ' ( t ) + ( 1 - t ) h ' ( t ) + A h ( t ) = 0 h ( 0 ) = h 0 ( t h ' ) ( 0 ) = h 1 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 ) → ∫ 0 ∞ f ( x ) P n ( x ) e - x d x 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 - i t n .

How to cite

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Ra'ed, Hatamleh. "A functional model for a family of operators induced by Laguerre operator." Archivum Mathematicum 039.1 (2003): 11-25. <http://eudml.org/doc/249144>.

@article{Raed2003,
abstract = {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 $e^\{-itn\}$.},
author = {Ra'ed, Hatamleh},
journal = {Archivum Mathematicum},
keywords = {Laguerre operator; semigroup; Hilbert space; functional model; Laguerre operator; semigroup; Hilbert space},
language = {eng},
number = {1},
pages = {11-25},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {A functional model for a family of operators induced by Laguerre operator},
url = {http://eudml.org/doc/249144},
volume = {039},
year = {2003},
}

TY - JOUR
AU - Ra'ed, Hatamleh
TI - A functional model for a family of operators induced by Laguerre operator
JO - Archivum Mathematicum
PY - 2003
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 039
IS - 1
SP - 11
EP - 25
AB - 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 $e^{-itn}$.
LA - eng
KW - Laguerre operator; semigroup; Hilbert space; functional model; Laguerre operator; semigroup; Hilbert space
UR - http://eudml.org/doc/249144
ER -

References

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  1. Theory of operator colligation in Hilbert space, J. Wiley, N. Y. 1979, Eng. translation. MR0634097
  2. Analyse harmonique des operateurs de l’espace de Hilbert, Mason, Paris and Akad. Kiado, Budapest 1967; Eng. translation North-Holland, Amsterdam and Akad. Kiado, Budapest 1970. Zbl0202.13102MR0225183
  3. Time cones and a functional model on a Riemann surface, Mat. Sb. 181 (1990), 965–995; Eng. translation in Math. USSR sb. 70 (1991). Zbl0738.47009MR1070490
  4. Scattering theory, Academic Press, New York 1967. MR0217440
  5. Dilatation theory and spectral analysis of nonsefadjoint operators, Math. programming and Related Questions (Proc. Sevent Winter School, Drogolych, 1994); Theory of Operators in Linear Spaces, Tsentral. Ekonom.-Math. Inst. Akad. Nauk SSSR, Moscow 1976, 3–69; Eng. translation in Amer. Math. Soc. Transl. (2) 115 (1980). MR0634807
  6. The operational calculus of the Lagueree transform, Ph.D. University of Michigan (1957). 
  7. Differentialgleichungen. Lösungsmethoden und Lösungen, Leipzig 1974. Zbl0395.35001

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