The notions of Bessel sequence, Riesz-Fischer sequence and Riesz basis are generalized to a rigged Hilbert space D[t] ⊂ H ⊂ D^×[t^×]. A Riesz-like basis, in particular, is obtained by considering a sequence {ξ_n} ⊂ D which is mapped by a one-to-one continuous operator T : D[t] → 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.
Bellomonte, G., Trapani, C. (2016). Riesz-like bases in rigged Hilbert spaces. ZEITSCHRIFT FUR ANALYSIS UND IHRE ANWENDUNGEN, 35(3), 243-265 [10.4171/ZAA/1564].
Riesz-like bases in rigged Hilbert spaces
BELLOMONTE, Giorgia;TRAPANI, Camillo
2016-01-01
Abstract
The notions of Bessel sequence, Riesz-Fischer sequence and Riesz basis are generalized to a rigged Hilbert space D[t] ⊂ H ⊂ D^×[t^×]. A Riesz-like basis, in particular, is obtained by considering a sequence {ξ_n} ⊂ D which is mapped by a one-to-one continuous operator T : D[t] → 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.File | Dimensione | Formato | |
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