# Journal of Operator Theory

Volume 72, Issue 2, Fall 2014  pp. 475-485.

On the characterization of Gelfand-Shilov-Roumieu spaces

Authors:  Mihai Pascu
Author institution: Institute of Mathematics "Simion Stoilow" of the Romanian Academy, RO-014700 Bucharest, Romania, and "Petroleum-Gas" University of Ploiesti, Bd. Bucuresti, 39, Ploiesti, Romania

Summary:  Generalized $\mathbf{m}$-Gelfand--Shilov--Roumieu vector spaces $\mathcal{S}_{\mathbf{m}}(\mathbf{X})$\break are introduced. Here $\mathbf{m}\!=\!(m^{(1)},\dots, m^{(n)})$, $\mathbf{X}\!=\!(X_{1}, \dots, X_{n})$ and $m^{(1)},\dots, m^{(n)}$ are sequences of positive real numbers, while $X_{1},\dots,X_{n}$ are operators in a Hilbert space. Our definition extends ter Elst's definition of Gevrey vector spaces \cite{TE2}. Conditions are given on the sequences $m^{(1)},\dots,m^{(n)}$ and on the operators $X_{1},\dots,X_{n}$ so that the equality ${S}_{\mathbf{m}}(\mathbf{X})={S}_{m^{(1)}}(X_{1})\cap\dots \cap{S}_{m^{(n)}}(X_{n})$ is valid. As a corollary we obtain a proof of a characterization theorem for Gelfand-Shilov-Roumieu spaces.

DOI: http://dx.doi.org/10.7900/jot.2013jun04.2009
Keywords:  Gelfand-Shilov-Roumieu vectors, Heisenberg group