Riesz-Like Bases in Rigged Hilbert Spaces

Abstract

The notions of Bessel sequence, Riesz-Fischer sequence and Riesz basis are generalized to a rigged Hilbert space $\mathcal{D}[t]\subset \mathcal{H}\subset {\mathcal{D}}^{\times }[t^{\times }]$. A Riesz-like basis, in particular, is obtained by considering a sequence $\{{\xi }_n\}\subset \mathcal{D}$ which is mapped by a one-to-one continuous operator $T:\mathcal{D}[t]\to \mathcal{H}[\Vert \cdot \Vert ]$ into an orthonormal basis of the central Hilbert space $\mathcal{H}$ of the triplet. The operator $T$ is, in general, an unbounded operator in $\mathcal{H}$. If $T$ has a bounded inverse then the rigged Hilbert space is shown to be equivalent to a triplet of Hilbert spaces.