Functional analytic characterizations of the Gelfand-Shilov spaces $S\sp{\beta}\sb{\alpha}$

Let P denote the differentiation operator i d/dx and %plane1D;4AC; the operator of multiplication by x in L2(). With suitable domains the operators P and %plane1D;4AC; are self-adjoint. In this paper, characterizations of the space Saß of Gelfand and Shilov are derived in terms of the operators P and %plane1D;4AC;. The main result is that Sa=D¿(|Q|1/a)n D8(P), Sß = D8(%plane1D;4AC;)D¿(|P|1/ß) and Saß = D¿(|%plane1D;4AC;|1/ß) n n D¿ (|P|1/ß. Here D8(·) denote the C8 - and the analyticity domain of the operator between brackets. In ZBuRe], Burkill et al. introduce the test function space T. Our results imply that T = 1/21/2. That is, the corresponding space of generalized functions can be identified with the space of generalized functions introduced by De Bruijn in ZBr1].