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DOI: 10.46698/r2980-5208-7458-m Space of Holomorphic Functions of Polynomial Growth as Local Algebra
Abstract:
Let \(G\) be a domain in the complex plane, star-shaped with respect to the point 0, \(H^{-\infty}(G)\) be the space of holomorphic functions in \(G\) of polynomial growth near the boundary of \(G\). The Duhamel product \(\ast\) is introduced in it. This product is used in operational and operator calculus, in the spectral theory, in the problem of the spectral multiplicity of a linear operator, in boundary value problems. It is shown that \(H^{-\infty}(G)\) with it is a unital topological algebra. The integration operator \(J(f)(z)=\int\nolimits_0^z f(t)\,dt\) acts linearly and continuously in \(H^{-\infty}(G)\). It is proved that all linear continuous operators in \(H^{-\infty}(G)\) that commute with \(J\), are represented as \(S_g(f)=f\ast g\), where \(g\) is a fixed function from \(H^{-\infty}(G)\). In the case where \(G\) is strictly star-shaped with respect to zero, a criterion for the invertibility of an element of the algebra \(H^{-\infty}(G)\) and a criterion for the operator \(S_g\) to have the continuous linear inverse are proved. It is shown that every nonzero operator from the commutator subgroup \(J\) is a composition of the power of the operator \(J\) and some isomorphism from the aforementioned commutator subgroup. In the proving of \(\ast\)-invertibility the Neumann series is used, usually applied in Banach spaces. In non-normable locally convex spaces of functions it was previously used by L. Berg, N. Wigley, and M. T. Karaev. All closed ideals of the algebra \((H^{-\infty}(G),\ast)\), closed invariant subspaces and cyclic vectors of \(J\) in \(H^{-\infty}(G)\) are described. From the obtained results it follows that the operator \(J\) is unicellular and the algebra \((H^{-\infty}(G),\ast)\) is local. The only maximal ideal in it is the set of all \(\ast\)-irreversible elements.
Keywords: Duhamel product, integration operator, space of holomorphic functions of polynomial growth
Language: Russian
For citation: Ivanova, O. A. and Melikhov, S. N. Space of Holomorphic Functions of Polynomial Growth as Local Algebra, Vladikavkaz Math. J., 2025, vol. 27, no. 1, pp. 44-55 (in Russian). DOI 10.46698/r2980-5208-7458-m
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