Spectral synthesis in the kernel of a convolution operator on weighted spaces

Abstract:

Weighted spaces of analytic functions on a bounded convex domain $D\subset \mathbb C^p$ are treated. Let $U =\{ u_n\} _{n=1}^\infty$ be a monotone decreasing sequence of convex functions on $D$ such that $u_n(z)\longrightarrow \infty$ as $\operatorname {dist}(z,\partial D) \longrightarrow 0$. The symbol $H(D,U)$ stands for the space of all $f\in H(D)$ satisfying $|f(z)|\exp (-u_n(z))\longrightarrow 0$ as $\operatorname {dist}(z,\partial D)\longrightarrow 0$, for all $n\in \mathbb N$. This space is endowed with a locally convex topology with the aid of the seminorms $p_n(f)=\sup _{z\in D}|f(z)|\exp (-u_n(z))$, $n=1, 2, \dots$. Clearly, every functional $S\in H^*(D)$ is a continuous linear functional on $H(D,U)$, and the corresponding convolution operator $M_S : f\longrightarrow S_w(f(z+w))$ acts on $H(D,U)$. All elementary solutions of the equation \[ M_S[f]=0, \tag{*} \] i.e., all solutions of the form $z^\alpha e^{\langle a,z\rangle }$, $\alpha \in \mathbb Z_+^p$, $a\in \mathbb C^p$, belong to $H(D,U)$. It is shown that the system $E(S)$ of elementary solutions is dense in the space of solutions of equation $(*)$ that belong to $H(D,U)$.