关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解

Stochastic equations in infinite dimensions

Zeros of orthogonal polynomials

Preface 1. Fourier series and Fourier transforms 2. Distributions 3. Elliptic equations (fundamental theory) 4. Initial value problems (Cauchy problems) 5. Evolution equations 6. Hyperbolic equations 7. Semi-linear hyperbolic equations 8. Green's functions and spectra Supplementary remarks Guide to the literature Bibliography Symbols Index.

The theory of partial differential equations

For the last decade there has been a generalized trend in mathematics on the search for large algebraic structures (linear spaces, closed subspaces, or infinitely generated algebras) composed of mathematical objects enjoying certain special properties. This trend has caught the eye of many researchers and has also had a remarkable influence in real and complex analysis, operator theory, summability theory, polynomials in Banach spaces, hypercyclicity and chaos, and general functional analysis. This expository paper is devoted to providing an account on the advances and on the state of the art of this trend, nowadays known as lineability and spaceability.

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Linear subsets of nonlinear sets in topological vector spaces

Distributional chaos for strongly continuous semigroups is studied and characterized. It is shown to be equivalent to the existence of a distributionally irregular vector. Finally, a sufficient condition for distributional chaos on the point spectrum of the generator of the semigroup is presented. An application to the semigroup generated in $L^2(R)$ by a translation of the Ornstein-Uhlenbeck operator is also given.

Distributional chaos for strongly continuous semigroups of operators

The chaotic and hypercyclic behavior of the C
 0-semigroups of operators generated by a perturbation of the Ornstein-Uhlenbeck operator with a multiple of the identity in $${L^2(\mathbb {R}^N)}$$
 is investigated. Negative and positive results are presented, depending on the signs of the real parts of the eigenvalues of the matrix appearing in the drift of the operator.

Hypercyclic Semigroups Generated by Ornstein-Uhlenbeck Operators

We study frequent hypercyclicity in the context of strongly continuous semigroups of operators. More precisely, we give a criterion (sufficient condition) for a semigroup to be frequently hypercyclic, whose formulation depends on the Pettis integral. This criterion can be verified in certain cases in terms of the infinitesimal generator of semigroup. Applications are given for semigroups generated by Ornstein-Uhlenbeck operators, and especially for translation semigroups on weighted spaces of $p$-integrable functions, or continuous functions that, multiplied by the weight, vanish at infinity.

Frequently hypercyclic semigroups

Trotter-Kato theorems for bi-continuous semigroups and applications to Feller semigroups

We study frequent hypercyclicity in the context of strongly continuous semigroups of operators. More precisely, we give a criterion (sufficient condition) for a semigroup to be frequently hypercyclic, whose formulation depends on the Pettis integral. This criterion can be verified in certain cases in terms of the in- finitesimal generator of semigroup. Applications are given for semigroups gener- ated by Ornstein-Uhlenbeck operators, and especially for translation semigroups on weighted spaces of p-integrable functions, or continuous functions that, multiplied by the weight, vanish at infinity.

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Elisabetta M. Mangino

Papers

Distributional chaos for strongly continuous semigroups of operators

Hypercyclic Semigroups Generated by Ornstein-Uhlenbeck Operators

Frequently hypercyclic semigroups

Trotter-Kato theorems for bi-continuous semigroups and applications to Feller semigroups

Frequently hypercyclic semigroups