On the Bochner theorem on orthogonal operators
HTML articles powered by AMS MathViewer
- by Zinoviy Grinshpun PDF
- Proc. Amer. Math. Soc. 131 (2003), 1591-1600 Request permission
Abstract:
We prove the following theorem. Any isometric operator $U$, that acts from the Hilbert space $H_1(\Omega )$ with nonnegative weight $p(x)$ to the Hilbert space $H_2(\Omega )$ with nonnegative weight $q(x)$, allows for the integral representation \[ Uf=\frac {1}{q(\xi )} \frac {\partial ^n}{\partial \xi _1\ldots \partial \xi _n}\int _{\Omega } \overline {L(\xi ,t)}f(t)p(t)dt, \] \[ U^{-1}f= \frac {1}{p(\xi )}\frac {\partial ^n}{\partial \xi _1\ldots \partial \xi _n} \int _{\Omega }\overline {K(\xi ,t)}f(t)q(t)dt, \] where the kernels $L(\xi ,t)$ and $K(\xi ,t)$ satisfy certain conditions that are necessary and sufficient for these kernels to generate the corresponding isometric operators.References
- N. I. Akhiezer and I. M. Glazman, Theory of linear operators in Hilbert space. Vol. I, Frederick Ungar Publishing Co., New York, 1961. Translated from the Russian by Merlynd Nestell. MR 0264420
- Z. S. Grinshpun, Analytic form of isometric operators in weighted Hilbert spaces, Izv. Akad. Nauk Kazakh. SSR Ser. Fiz.-Mat. 3 (1983), 17–20 (Russian, with Kazakh summary). MR 713324
- Saunders MacLane, Steinitz field towers for modular fields, Trans. Amer. Math. Soc. 46 (1939), 23–45. MR 17, DOI 10.1090/S0002-9947-1939-0000017-3
- Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger, and Francesco G. Tricomi, Higher transcendental functions. Vol. I, Robert E. Krieger Publishing Co., Inc., Melbourne, Fla., 1981. Based on notes left by Harry Bateman; With a preface by Mina Rees; With a foreword by E. C. Watson; Reprint of the 1953 original. MR 698779
Additional Information
- Zinoviy Grinshpun
- Affiliation: Department of Mathematics and Computer Science, Bar-Ilan University, Ramat-Gan 52900, Israel
- Email: miriam@macs.biu.ac.il
- Received by editor(s): April 3, 2001
- Received by editor(s) in revised form: January 11, 2002
- Published electronically: September 20, 2002
- Communicated by: Joseph A. Ball
- © Copyright 2002 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 131 (2003), 1591-1600
- MSC (2000): Primary 44A05, 44A15, 46F12
- DOI: https://doi.org/10.1090/S0002-9939-02-06707-2
- MathSciNet review: 1949890