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Cubo (Temuco)

versión On-line ISSN 0719-0646

Cubo vol.12 no.3 Temuco  2010

http://dx.doi.org/10.4067/S0719-06462010000300015 

CUBO A Mathematical Journal Vol.12, N° 03, (241–253). October 2010

 

On the Weyl Transform with Symbol in the Gel’fand-Shilov Space and its Dual Space

 

YASUYUKI OKA

Department of Mathematics, Sophia University 7-1 Kioicho, Chiyoda-ku, Tokyo 102-8554, Japan email: yasuyu-o@hoffman.cc.sophia.ac.jp


ABSTRACT

In this paper, we claim two subjects. One is that the Weyl transform with symbol in the Gel’fand-Shilov space l rr , r ≥ 1/2 , is a trace class operator. The other one is that the Weyl transform with symbol in the generalized function (l rr )1, r ≥ 1/2 , is a continuous linear transformation from the Gel’fand-Shilov space l rr to (l rr )1. As r > 1, Z. Lozanov- Crvenkovic and D. Perišic have proved in [6] this result. Our second claim includes their result.

Key words and phrases: Weyl transform, Gel’fand-Shilov space, Fourier-Wigner transform, trace class operator, Schwartz’s kernel theorem.


RESUMEN

En este artículo afirmamos dos asuntos. El primero es que la transformada de Weyl con símbolo en el espacio de Gel’fand-Shilov l rr , r ≥ 1/2 , es un operador de clase trazo. El segundo asunto es que la transformación de Weyl con símbolo en las funciones generalizadas (l rr )1, r ≥ 1/2 , es una transformación lineal continua del espacio Gel’fand-Shilov l rr to (l rr )1 . Como r > 1, Z. Lozanov-Crvenkovic y D. Perišic probaron en [6] este resultado. Nuestro resultado incluye su resultado.

Math. Subj. Class.: 46F05; 46F15; 81R15; 81S40.


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