Surjectivity of certain adjoint operators and applications

This paper is an extension and generalization of some previous works, such as the study of M. Benalili and A. Lansari. Indeed, these authors, in their work about the finite co-dimension ideals of Lie algebras of vector fields, restricted their study to fields $$X_0$$ X 0 of the form $$X_0=\sum _{i=1}^{n}( \alpha _i \cdot x_i+\beta _i\cdot x_i^{1+m_i}) \frac{\partial }{\partial x_i}$$ X 0 = ∑ i = 1 n ( α i · x i + β i · x i 1 + m i ) ∂ ∂ x i , where $$\alpha _i, \beta _i $$ α i , β i are positive and $$m_i$$ m i are even natural integers. We will first study the sub-algebra U of the Lie-Fréchet space E, containing vector fields of the form $$Y_0 = X_0^+ + X_0^- + Z_0$$ Y 0 = X 0 + + X 0 - + Z 0 , such as $$ X_0\left( x,y\right) =A\left( x,y\right) =\left( A^{-}\left( x \right) ,A^{+}\left( y\right) \right) $$ X 0 x , y = A x , y = A - x , A + y , with $$A^-$$ A - (respectively, $$ A^+ $$ A + ) a symmetric matrix having eigenvalues $$ \lambda < 0$$ λ < 0 (respectively, $$\lambda >0 $$ λ > 0 ) and $$Z_0$$ Z 0 are germs infinitely flat at the origin. This sub-algebra admits a hyperbolic structure for the diffeomorphism $$\psi _{t*}=(exp\cdot tY_0)_*$$ ψ t ∗ = ( e x p · t Y 0 ) ∗ . In a second step, we will show that the infinitesimal generator $$ad_{-X}$$ a d - X is an epimorphism of this admissible Lie sub-algebra U. We then deduce, by our fundamental lemma, that $$U=E$$ U = E .

Every Separable Complex Fréchet Space with a Continuous Norm is Isomorphic to a Space of Holomorphic Functions

Extending a result of Mashreghi and Ransford, we prove that every complex separable infinite-dimensional Fréchet space with a continuous norm is isomorphic to a space continuously included in a space of holomorphic functions on the unit disc or the complex plane, which contains the polynomials as a dense subspace. As a consequence, we deduce the existence of nuclear Fréchet spaces of holomorphic functions without the bounded approximation.

The Composition Operation on Spaces of Holomorphic Mappings

We discuss the continuity of the composition on several spaces of holomorphic mappings on open subsets of a complex Banach space. On the Fréchet space of entire mappings that are bounded on bounded sets, the composition turns out to be even holomorphic. In such a space, we consider linear subspaces closed under left and right composition. We discuss the relationship of such subspaces with ideals of operators and give several examples of them. We also provide natural examples of spaces of holomorphic mappings where the composition is not continuous.

Differentiability of the evolution map and Mackey continuity

We solve the differentiability problem for the evolution map in Milnor’s infinite-dimensional setting. We first show that the evolution map of each {C^{k}}-semiregular Lie group G (for {k\in\mathbb{N}\sqcup\{\mathrm{lip},\infty\}}) admits a particular kind of sequentially continuity – called Mackey k-continuity. We then prove that this continuity property is strong enough to ensure differentiability of the evolution map. In particular, this drops any continuity presumptions made in this context so far. Remarkably, Mackey k-continuity arises directly from the regularity problem itself, which makes it particular among the continuity conditions traditionally considered. As an application of the introduced notions, we discuss the strong Trotter property in the sequentially and the Mackey continuous context. We furthermore conclude that if the Lie algebra of G is a Fréchet space, then G is {C^{k}}-semiregular (for {k\in\mathbb{N}\sqcup\{\infty\}}) if and only if G is {C^{k}}-regular.

A study on a class of modified Bessel-type integrals in a Fréchet space of Boehmians

In this paper, an attempt is being made to discuss a class of modified Bessel- type integrals on a set of generalized functions known as Boehmians. We show that the modified Bessel-type integral, with appropriately defined convolution products, obeys a fundamental convolution theorem which consequently justifis pursuing analysis in the Boehmian spaces. We describe two Fréchet spaces of Boehmians and extend the modifid Bessel-type integral between the diferent spaces. Furthermore, a convolution theorem and a class of basic properties of the extended integral such as linearity, continuity and compatibility with the classical integral, which provide a convenient extention to the classical results, have been derived

Distributional Chaoticity of C0-Semigroup on a Frechet Space

This paper is mainly concerned with distributional chaos and the principal measure of C 0 -semigroups on a Frechet space. New definitions of strong irregular (semi-irregular) vectors are given. It is proved that if C 0 -semigroup T has strong irregular vectors, then T is distributional chaos in a sequence, and the principal measure μ p ( T ) is 1. Moreover, T is distributional chaos equivalent to that operator T t is distributional chaos for every ∀ t > 0 .