On the
-comparison algebra of a class of singular Sturm-Liouville expressions on the real line
Abstract
In this article we study a
-comparison algebra in the sense of [C2] with generators related to the ordinary differential expression H on the full real line R where,with constants
, (1.1) (Error rendering LaTeX formula) More precisely, the algebra, called
, is generated by the multiplications
, by functions
and the (singular integral) operators(Error rendering LaTeX formula),and their adjoints. Here
, the inverse positive square root of the unique self-adjoint realization H of the expression (1.1), in the Hilbert space
.(We use the same notation for both, (1.1) and its realization.) The case of
was discussed earlier in [Tg1], even for all n-dimensional problem. The commutators are compact and the Fredholm properties of operators in
are determined by a complex-valued symbol on a symbol space homeomorphic to that of the usual Laplace comparison algebra on
, although the symbol itself is calculated by different formulas.
![C<sup>*</sup>](http://siba-ese.unile.it/plugins/generic/latexRender/cache/71d3c6dc470d20fd5afb0700aba9fe94.png)
![𝛼≥ 0, 𝛽∈ R](http://siba-ese.unile.it/plugins/generic/latexRender/cache/c5ad7116cad64316d4c248c03b1d4b60.png)
![\mathbf A](http://siba-ese.unile.it/plugins/generic/latexRender/cache/982c0ac0ca870a4b6408355a2cbbc469.png)
![a( M) : u( x) → a(x)u(x)](http://siba-ese.unile.it/plugins/generic/latexRender/cache/39e89de317f91ee688e76588d990b811.png)
![a(x)∈ C([-∈fty,+∈fty])](http://siba-ese.unile.it/plugins/generic/latexRender/cache/4bf69a9089e497f2af147698034eceac.png)
![Λ = H<sup>-1/2</sup>](http://siba-ese.unile.it/plugins/generic/latexRender/cache/e0a4c9f86fa4e054c09ea7cd3cbba629.png)
![\mathbf H = L<sup>2</sup> ( R )](http://siba-ese.unile.it/plugins/generic/latexRender/cache/27b9b3939fe8b92bdc98b90f64d149c9.png)
![𝛽 < 𝛼+ 1](http://siba-ese.unile.it/plugins/generic/latexRender/cache/b028797945b18673080312cafecaffd2.png)
![\mathbf A](http://siba-ese.unile.it/plugins/generic/latexRender/cache/982c0ac0ca870a4b6408355a2cbbc469.png)
![R<sup>n</sup>](http://siba-ese.unile.it/plugins/generic/latexRender/cache/8709922046aaf17601461c6edd5c0c49.png)
DOI Code:
10.1285/i15900932v11p93
Full Text: PDF