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Laguerre operator; semigroup; Hilbert space; functional model
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 $e^{-itn}$.
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[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. MR 0225183 | Zbl 0202.13102
[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). MR 1070490 | Zbl 0738.47009
[4] Lax, P., Philips R. S.: Scattering theory. Academic Press, New York 1967. MR 0217440
[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
[6] Mc. Cully J.: The operational calculus of the Lagueree transform. Ph.D. University of Michigan (1957).
[7] Kamke, E.: Differentialgleichungen. Lösungsmethoden und Lösungen. Leipzig 1974. Zbl 0395.35001
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