JournalszaaVol. 35, No. 3pp. 243–265

Riesz-Like Bases in Rigged Hilbert Spaces

  • Giorgia Bellomonte

    Università degli Studi di Palermo, Italy
  • Camillo Trapani

    Università degli Studi di Palermo, Italy
Riesz-Like Bases in Rigged Hilbert Spaces cover
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Abstract

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

Cite this article

Giorgia Bellomonte, Camillo Trapani, Riesz-Like Bases in Rigged Hilbert Spaces. Z. Anal. Anwend. 35 (2016), no. 3, pp. 243–265

DOI 10.4171/ZAA/1564