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Zapiski Nauchnykh Seminarov LOMI, 1989, Volume 178, Pages 166–183
(Mi znsl4682)
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This article is cited in 2 scientific papers (total in 2 papers)
Invariant subspaces of multiplication by $z$ of $E^p$ in a multiply connected domain
D. V. Yakubovich
Abstract:
Let $G$ be a multiply connected domain with boundary
$\Gamma_0\cup\dots\cup\Gamma_s$ where $\Gamma_j$ are closed piecewice $C^2$-smooth curves.
A subspace $Z$ in Hardy–Smirnov class $E^p(G)$, $1\leqslant p\leqslant\infty$,
is called invariant if $zf(z)\in Z$ for $f\in Z$. Define domains
$V_j$ by $\Gamma_j=\partial V_j$, $\mathbb{C}\setminus G=V_0\cup\dots\cup V_s$;
suppose that $V_0$ is unbounded. For an invariant subspace $Z$ in $E^p(G)$ the function
$\chi_Z\in L^\infty(\Gamma_{int})$, $\Gamma_{int}{\stackrel{def}=}\Gamma_1\cup\dots\cup\Gamma_s$ is defined
by the equalities
$\mathcal{H}_j{\stackrel{def}=}\mathrm{clos}_{L^P(\Gamma_j)}\{x(\Gamma_j):x\in Z\}=(\overline{\chi}_Z\mid\Gamma_j)\cdot E^p(V_j)$,
$|\chi_Z|\equiv1$ a.e. on $\Gamma_j$ for $j\geqslant1$ $(\chi_Z\mid\Gamma_j\equiv0\ if\ \mathcal{H}_j=L^p(\Gamma_j))$.
THEOREM 1. (i) Let $Z$ be an invariant subspace in $E^p(G)$
such that $GCD(Z)=1$. Then
$$
Z=\{x: \varphi\cdot x\in E_0^{1,\infty}(V_j), j\geqslant1\}.
$$
Here $\varphi$ is measurable function on $\Gamma_{int}$, $\varphi\equiv0$ or $|\varphi|\geqslant1$,
a.e. on each $\Gamma_j: L_0^{1,\infty}(\Gamma_j)=\{f\in L^{1/2}(\Gamma_j): m\{|f|\}>A\}=o(A^{-1})$,
$A\to+\infty$ ($m$ is the Lesbegue measure),
$E_0^{1,\infty}(V_j)=E^{1/2}(V_j)\cap L_0^{1,\infty}(\Gamma_j)$,
and $GCD(Z)$ is common least divisor of inner
parts of functions in $Z$.
(ii) If the inequality $d\,\omega_{V_j}\leqslant cd\,\omega_G\mid\Gamma_j$ holds
forharmonic measures for $j\geqslant1$, then
$$
Z=\{x: \chi_z x\mid\Gamma_j\in E^p(V_j),\ \rho\cdot x\in L_0^{1,\infty}(\Gamma_{int})\}
$$
for a measurable function $\rho$ on $\Gamma_{int}$.
THEOREM 2. Let $\Gamma_j$ be analytic, $\tau_j$ be conformal mappings of
$V_j$ onto the unit disk ($j\geqslant1$). Suppose $Z$ is invariant subspace
in $E^2(G)$, $GCD(Z)=1$. There еxist outer $g_j\in E^2(V_j)$,
inner $\theta_j$ in $V_j$, $m_j\in\mathbb{Z}$ such that $|g_j|^2=\mathrm{Re}\,(\tau_j\theta_j v_j)+1$ a.e.
on $\Gamma_j$ for some $v_j\in E_0^{1,\infty}(V_j)$ and
$$
Z=\{x: x\mid\Gamma_j\in(\chi\mid\Gamma_j)E^2(V_j),\ |xg_j^{-1}|\in L^2(\Gamma)\ for\ j\geqslant1\}.
$$
Here $\chi\in L^\infty(\Gamma_{int})$ is defined by $\chi\mid\Gamma_j=\tau_j\theta_jg_j/\overline{g}_j$.
Conversely, every $g_j$, $\theta_j$, $m_j$ satisfying the above conditions give rise
to an invariant subspace $Z$ such that $GCD(Z)=1$ and $\chi_z=\chi$.
This generalizes the results of Hitt and Sarason [5,6].
Citation:
D. V. Yakubovich, “Invariant subspaces of multiplication by $z$ of $E^p$ in a multiply connected domain”, Investigations on linear operators and function theory. Part 18, Zap. Nauchn. Sem. LOMI, 178, "Nauka", Leningrad. Otdel., Leningrad, 1989, 166–183; J. Soviet Math., 61:2 (1992), 2046–2056
Linking options:
https://www.mathnet.ru/eng/znsl4682 https://www.mathnet.ru/eng/znsl/v178/p166
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