# Article

 Title: A functional model for a family of operators induced by Laguerre operator (English) Author: Ra'ed, Hatamleh Language: English Journal: Archivum Mathematicum ISSN: 0044-8753 (print) ISSN: 1212-5059 (online) Volume: 39 Issue: 1 Year: 2003 Pages: 11-25 Summary lang: English . Category: math . Summary: 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 $e^{-itn}$. (English) Keyword: Laguerre operator Keyword: semigroup Keyword: Hilbert space Keyword: functional model MSC: 34G99 MSC: 47A40 MSC: 47A48 MSC: 47A50 MSC: 47D06 MSC: 47E05 idZBL: Zbl 1109.47308 idMR: MR1982208 . Date available: 2008-06-06T22:41:05Z Last updated: 2012-05-10 Stable URL: http://hdl.handle.net/10338.dmlcz/107850 . Reference: [1] Livshits, M. S., Yantsevich, A. A.: Theory of operator colligation in Hilbert space.J. Wiley, N. Y. 1979, Eng. translation. MR 0634097 Reference: [2] Sz.-Nagy, B., Foias, C.: 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. Zbl 0202.13102, MR 0225183 Reference: [3] Zolotarev, V. A.: Time cones and a functional model on a Riemann surface.Mat. Sb. 181 (1990), 965–995; Eng. translation in Math. USSR sb. 70 (1991). Zbl 0738.47009, MR 1070490 Reference: [4] Lax, P., Philips R. S.: Scattering theory.Academic Press, New York 1967. MR 0217440 Reference: [5] Pavlov, B. S.: 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). MR 0634807 Reference: [6] Mc. Cully J.: The operational calculus of the Lagueree transform.Ph.D. University of Michigan (1957). Reference: [7] Kamke, E.: Differentialgleichungen. Lösungsmethoden und Lösungen.Leipzig 1974. Zbl 0395.35001 .

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