3,115 research outputs found
The Commutant of Multiplication by z on the Closure of Rational Functions in
For a compact set a finite positive Borel measure
on and 1 \le t < \i, let be the set of rational
functions with poles off and let be the closure of
in For a bounded Borel subset let \area_{\mathcal D} denote the area (Lebesgue) measure
restricted to and let H^\i (\mathcal D) be the weak-star closed
sub-algebra of L^\i(\area_{\mathcal D}) spanned by bounded and analytic
on for some compact subset We show that if contains no non-trivial
direct summands, then there exists a Borel subset
whose closure contains the support of and there exists an isometric
isomorphism and a weak-star homeomorphism from onto such that for all
Consequently, we obtain some structural decomposition
theorems for \rtkmu.Comment: arXiv admin note: text overlap with arXiv:2212.1081
Invertibility in Weak-Star Closed Algebras of Analytic Functions
For a compact subset and a positive finite Bore1
measure supported on let be the weak-star closure in
of rational functions with poles off We show that if
has no non-trivial summands and then is invertible in if and only if Chaumat's
map for and applied to is bounded away from zero on the envelope
with respect to and The result proves the conjecture
posed by J. Dudziak in 1984.Comment: arXiv admin note: text overlap with arXiv:2212.1081
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