Volume 229, Nº 5 (2018)

We consider the action of the quadratic form of the Laplace operator and its extensions in subspaces of linear combinations of the “transverse” and “longitudinal” functions with the fixed orbital momentum with respect to the coordinate origin. In the statement of the problem, it is required that the extensions obtained, after the transfer back to the space of vector functions, can be represented as simple limit expressions with two coefficients. We study the behavior of these coefficients with respect to the initial choice of the linear subspace. Bibliography: 5 titles.

Answering a question of S. R. Treil (2004), for every δ, 0 < δ < 1, we construct examples of contractions whose characteristic function F ∈ H^∞(ε →ε_∗) satisfies the conditions ‖F(z)x‖ ≥ δ‖x‖ and dim ε_∗ ⊝ F(z)ε = 1 for every z ∈ ????, x ∈ ε, but is not left invertible. In addition, we show that the condition where is the trace class of operators, which is sufficient for the left invertibility of an operatorvalued function F satisfying the estimate ‖F(z)x‖ ≥ δ‖x‖ for every z ∈ ????, x ∈ ε, with some δ > 0 (Treil, 2004), is necessary for the left invertibility of an inner function F such that dim ε_∗ ⊝ F(z)ε < ∞ for some z ∈ ????.

By using the fixed-point approach due to D. Rutsky, the problem of ideals in H^∞ is solved for “data sequences” in H^∞(l¹) (instead of H^∞(l²) in the traditional setting). Bibliography: 7 titles.

We consider spaces of entire functions with systems of weighted estimates. The case of twoterm weight sequences consisting of radial and nonradial components is studied. Under some assumptions on the weight sequence, we obtain a complete description of generators in these spaces. We apply this result to the problem of normal solvability of systems of convolution equations in the Roumieu spaces of ultradifferentiable functions and, as a particular case, in Gevrey classes.

Let a function f be holomorphic in the unit ball ????ⁿ, continuous in the closed ball \( {\overline{\mathbb{B}}}^n \) , and let f(z) ≠ 0, z ∈ ????ⁿ. Assume that |f| belongs to the α-Hölder class on the unit sphere Sⁿ, 0 < α ≤ 1. The present paper is devoted to the proof of the statement that f belongs to the α/2-Hölder class on \( {\overline{\mathbb{B}}}^n \).