3,115 research outputs found

### The Commutant of Multiplication by z on the Closure of Rational Functions in $L^t(\mu)$

For a compact set $K\subset \mathbb C,$ a finite positive Borel measure $\mu$
on $K,$ and 1 \le t < \i, let $\text{Rat}(K)$ be the set of rational
functions with poles off $K$ and let $R^t(K, \mu)$ be the closure of
$\text{Rat}(K)$ in $L^t(\mu).$ For a bounded Borel subset $\mathcal D\subset
\mathbb C,$ let \area_{\mathcal D} denote the area (Lebesgue) measure
restricted to $\mathcal D$ and let H^\i (\mathcal D) be the weak-star closed
sub-algebra of L^\i(\area_{\mathcal D}) spanned by $f,$ bounded and analytic
on $\mathbb C\setminus E_f$ for some compact subset $E_f \subset \mathbb
C\setminus \mathcal D.$ We show that if $R^t(K, \mu)$ contains no non-trivial
direct $L^t$ summands, then there exists a Borel subset $\mathcal R \subset K$
whose closure contains the support of $\mu$ and there exists an isometric
isomorphism and a weak-star homeomorphism $\rho$ from $R^t(K, \mu) \cap
L^\infty(\mu)$ onto $H^\infty(\mathcal R)$ such that $\rho(r) = r$ for all
$r\in\text{Rat}(K).$ 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 $K\subset \mathbb C$ a compact subset and $\mu$ a positive finite Bore1
measure supported on $K,$ let $R^\infty (K,\mu)$ be the weak-star closure in
$L^\infty (\mu)$ of rational functions with poles off $K.$ We show that if
$R^\infty (K,\mu)$ has no non-trivial $L^\infty$ summands and $f\in R^\infty
(K,\mu),$ then $f$ is invertible in $R^\infty (K,\mu)$ if and only if Chaumat's
map for $K$ and $\mu$ applied to $f$ is bounded away from zero on the envelope
with respect to $K$ and $\mu.$ The result proves the conjecture $\diamond$
posed by J. Dudziak in 1984.Comment: arXiv admin note: text overlap with arXiv:2212.1081

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