U-Operators

Motivated by recent work on the rate of growth of frequently hypercyclic entire functions due to Blasco, Grosse-Erdmann and Bonilla, we investigate conditions to ensure that the differentiation operator is chaotic or frequently hypercyclic on generalized weighted Bergman spaces of entire functions studied by Lusky, whenever the differentiation operator is continuous. As a consequence we partially complete the knowledge of possible rates of growth of frequently hypercyclic entire functions for the differentiation operator.

Infinite order differential operators in spaces of entire functions

We describe continuous linear operators acting in a countable inductive limit of weighted Frchet spaces of entire functions of several complex variables and commuting in these spaces with systems of partial differentiation and translation operators. Under the made assumptions, the commutants of the systems of differentiation and translation operators coincide. They consist of convolution operators defined by an arbitrary continuous linear functional on . At that, we do not assume that the set of the polynomials is dense in . In the space topological dual to , we introduce the natural multiplication. Under this multiplication, the algebra is isomorphic to the aforementioned commutant with the usual multiplication, which is the composition of the operators. This isomorphism is also topological if is equipped by the weak topology, while the commutant is equipped by the weak operator topology. This implies that the set of the polynomials of the differentiation operators is dense in the commutant with topology of pointwise convergence. We also study the possibility of representing an operator in the commutant as an infinite order differential operator with constant coefficients. We prove the immediate continuity of linear operators commuting with all differentiation operators in a weighted (LF)-space of entire functions isomorphic via Fourier-Laplace transform to the space of infinitely differentiable functions compactly supported in a real multi-dimensional space.

Sampling in spaces of entire functions of exponential type in [formula omitted

Conditions are discovered which ensure that the space of entire functions can be represented as the sum of an ideal in the space of entire functions and the kernel of a convolution operator. In this way conditions for the multipoint de la Vallée-Poussin problem to have a solution are found.Bibliography: 14 titles.

Invariant subspace problem for classical spaces of functions

The study of the hypercyclicity of an operator is an old problem in mathematics; it goes back to a paper of Birkhoff in 1929 proving the hypercyclicity of the translation operators in the space of all entire functions with the topology of uniform convergence on compact subsets. This article studies the hypercyclicity of translation operators in some general reproducing kernel Hilbert spaces of entire functions. These spaces are obtained by duality in a complex separable Hilbert space H by means of an analytic H-valued kernel. A link with the theory of de Branges spaces is also established. An illustrative example taken from the Hamburger moment problem theory is included.

In the paper, we consider the problem on eigenvalues of a loaded differential operator of the first order with a periodic boundary condition on the interval [–1; 1], that is, equation contains a load at the point (–1) and the function of bounded variation (t), with the condition Φ(−1) = Φ(1) = 1 . A characteristic determinant of spectral problem is constructed for the considered loaded differentiation operator, which is an entire analytical function on the spectral parameter. On the basis of the characteristic determinant formula, conclusions are proved about the asymptotic behavior of the spectrum and eigenfunctions of the loaded spectral problem for the differentiation operator, the characteristic determinant of which is an entire analytic function of the spectral parameter l. A theorem on the location of eigenvalues on the complex plane l is formulated, where the regular growth of an entire analytic function is indicated. A theorem is proved on the asymptotics of the zeros of an entire function, that is, the eigenvalues of the original considered spectral problem for a loaded differential operator of differentiation, and the asymptotic properties of an entire function with distribution of roots are studied.

In this paper, we investigate the boundedness of composition operators defined on a quasi-Banach space continuously included in the space of smooth functions on a manifold. We prove that the boundedness of a composition operator strongly limits the behavior of the original map, and it provides an effective method to investigate the properties of composition operators using the theory of dynamical system. Consequently, we prove that only affine maps can induce bounded composition operators on any quasi-Banach space continuously included in the space of entire functions of one variable if the function space contains a nonconstant function. We also prove that any polynomial automorphisms except affine transforms cannot induce bounded composition operators on a quasi-Banach space composed of entire functions in the two-dimensional complex affine space under several mild conditions.

We prove the impossibility of expressing the space of entire functions on any infinite dimensional complex Banach space with a Schauder basis E as a countable union of spaces of entire functions that are bounded on countable open covers of E .

We prove results of existence and approximation of the solutions of the convolution equations in spaces of entire functions of exponential type on infinite dimensional spaces. In particular we obtain: let E be a complex, quasi-complete and dual nuclear locally convex space and Ω \Omega a convex balanced open subset of E; let H ( Ω ) \mathcal {H} (\Omega ) be the space of the holomorphic functions on Ω \Omega , equipped with the compact open topology and H ′ ( Ω ) \mathcal {H}’(\Omega ) its strong dual; let F H ′ ( Ω ) \mathcal {F} \mathcal {H}’(\Omega ) denote the image of H ′ ( Ω ) \mathcal {H}’(\Omega ) through the Fourier-Borel transform F \mathcal {F} ; equip this space F H ′ ( Ω ) \mathcal {F} \mathcal {H}’(\Omega ) with the image of the topology of H ′ ( Ω ) \mathcal {H}’(\Omega ) via the map F \mathcal {F} . Then, “every nonzero convolution operator on F H ′ ( Ω ) \mathcal {F} \mathcal {H}’(\Omega ) is surjective” and “every solution of the homogeneous equation is limit of exponential-polynomial solutions". Our results are more generally valid when E is a Schwartz bornological vector space with the approximation property. Previous results in Fréchet-Schwartz and Silva spaces are thus extended to domains that are not Fréchet or D.F.-spaces.

In this paper, we consider the question on study of zeros of an entire function of one class, which coincides with quasi-polynomials of exponential type. Eigenvalue problems for some classes of differential operators on a segment are reduced to a similar problem. In particular, the studied problem is led by the eigenvalue problem for a linear differential equation of the third order with regular boundary value conditions in the space W^3_2(0, 1). The studied entire function is adequately characteristic determinant of the spectral problem for a third-order linear differential operator with periodic boundary value conditions. An algorithm to construct a conjugate indicator diagram of an entire function of one class is indicated, which coincides with exponential type quasi-polynomials with comparable exponents according to the monograph by A.F. Leontyev. Existence of a countable number of zeros of the studied entire function in each series is proved, which are simultaneously eigenvalues of the above-mentioned third-order differential operator with regular boundary value conditions. We determine distance between adjacent zeros of each series, which lies on the rays perpendicular to sides of the conjugate indicator diagram, that is a regular hexagon on the complex plane. In this case, zero is not an eigenvalue of the considered operator, that is, zero is a regular point of the operator. Fundamental difference of this work is finding the corresponding eigenfunctions of the operator. System of eigenfunctions of the operator corresponding in each series is found. Adjoint operator is constructed.

A vector x x in a linear topological space X X is called universal for a linear operator T T on X X if the orbit { T n x : n ≥ 0 } \{ {T^n}x:n \geq 0\} is dense in X X . Our main result gives conditions on T T and X X which guarantee that T T will have universal vectors. It applies to the operators of differentiation and translation on the space of entire functions, where it makes contact with Pólya’s theory of final sets; and also to backward shifts and related operators on various Hilbert and Banach spaces.

Let $\Phi (z) = \Sigma _0^\infty {\beta _j}{z^j}$ have radius of convergence $r\;(0 < r < \infty )$ and no singularities other than poles on the circle $|z| = r$. A complete solution is obtained for the infinite order differential equation $( \ast )\;\Sigma _0^\infty {\beta _j}{u^{(j)}}(z) = g(z)$. It is shown that $(\ast )$ possesses a solution if and only if the function g has a polynomial expansion in terms of the Appell polynomials generated by $\Phi$. The solutions of $( \ast )$ are expressed in terms of the coefficients which appear in the Appell polynomial expansions of g. An alternate method of solution is obtained, in which the problem of solving $( \ast )$ is reduced to the problem of finding a solution, within a certain space of entire functions, of a finite order linear differential equation with constant coefficients. Additionally, differential operator techniques are used to study Appell polynomial expansions.

If f is an entire function in the complex plane such thatwhere 0 ≤ α &lt; 1, and all the derivatives of f at 0 are integers, then it is easy to show that f is a polynomial (see e.g. Straus [10]). The best possible result of this type was proved by Pólya [9]. The main aim of this paper is to prove two analogous results for harmonic functions defined in the whole of the Euclidean space Rn, where n ≥ 2 (i.e. entire harmonic functions).