Invariant subspaces of multiplication by $z$ of $E^p$ in a multiply connected domain

D. V. Yakubovich

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 18, Tome 178 (1989), pp. 166-183

Résumé

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].