| 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 |
| . |
