Riesz-like bases in rigged Hilbert spaces

TL;DR

This paper extends classical basis concepts to rigged Hilbert spaces, introducing Riesz-like bases via unbounded operators and analyzing their properties and implications for the structure of these spaces.

Contribution

It generalizes Riesz basis notions to rigged Hilbert spaces using unbounded operators, linking basis properties to the structure of triplet Hilbert spaces.

Findings

Riesz-like bases are characterized via unbounded operators in rigged Hilbert spaces.

If the operator has a bounded inverse, the rigged Hilbert space is equivalent to a triplet of Hilbert spaces.

The paper provides a framework for understanding bases in the context of rigged Hilbert spaces.

Abstract

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