Functional analytic characterizations of the Gelfand-Shilov spaces Sαβ

Let P denote the differentiation operator i d/dx and %plane1D;4AC; the operator of multiplication by x in L₂(ℝ). With suitable domains the operators P and %plane1D;4AC; are self-adjoint. In this paper, characterizations of the space S_α^β of Gelfand and Shilov are derived in terms of the operators P and %plane1D;4AC;. The main result is that S_α=D^ω(|Q|^1/α)∩ D^∞(P), S^β = D^∞(%plane1D;4AC;)D^ω(|P|^1/β) and S_α^β = D^ω(|%plane1D;4AC;|^1/β) ∩ ∩ D^ω (|P|^1/β. Here D^∞(·) denote the C^∞ - 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 = _1/2^1/2. That is, the corresponding space of generalized functions can be identified with the space of generalized functions introduced by De Bruijn in ZBr1].