On the characterization of Gelfand-Shilov-Roumieu spaces

TL;DR

This paper introduces generalized Gelfand-Shilov-Roumieu spaces using sequences and operators in Hilbert spaces, providing conditions for their characterization and offering a new proof for classical Gelfand-Shilov spaces.

Contribution

It defines generalized vector spaces with sequences and operators, establishing conditions for their equality and characterizing classical Gelfand-Shilov spaces.

Findings

Established conditions for space equality involving sequences and operators.

Provided a new proof for the characterization of classical Gelfand-Shilov spaces.

Extended the understanding of Gelfand-Shilov-Roumieu spaces in Hilbert spaces.

Abstract

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.