On the characterization of Gelfand-Shilov-Roumieu spaces

Generalized $\mathbf{m}$-Gelfand-Shilov-Roumieu vector spaces $\mathcal{S}_{\mathbf{m}}(\mathbf{X})$ are introduced. Here $\mathbf{m} = (m^{(1)},...,m^{(n)})$, $\mathbf{X}=(X_{1},...,X_{n})$ and $m^{(1)},...,m^{(n)}$ are sequences of positive real numbers and $X_{1},...,X_{n}$ are operators in a Hilbert space. Conditions are given on the sequences $m^{(1)},...,m^{(n)}$ and on the operators $X_{1},...,X_{n}$ so that the equality $S_{\mathbf{m}}(\mathbf{X}) = S_{m^{(1)}}(X_{1})\cap ... \cap{S}_{m^{(n)}}(X_{n})$ is valid. As a corollary we obtain a new proof of a characterization theorem for classical Gelfand-Shilov spaces.