Spectral synthesis in the kernel of a convolution operator on weighted spaces. (English. Russian original) Zbl 1196.32007

Yulmukhametov, R. S.

St. Petersbg. Math. J. 21, No. 2, 353-363 (2010); translation from Algebra Anal. 21, No. 2, 264-279 (2009).

Summary: 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 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 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 \[ (*)\qquad M_S[f]=0, \] 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 \( H(D,U)\).

MSC:

32A50	Harmonic analysis of several complex variables
42A85	Convolution, factorization for one variable harmonic analysis
47B40	Spectral operators, decomposable operators, well-bounded operators, etc.

Keywords:

weighted spaces of analytic functions; convolution operator; spectral synthesis

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