Fast orbital convergence reveals more hypercyclic vectors

We construct a purely unrectifiable set of finite $\mathcal H^1$-measure in every infinite dimensional separable Banach space $X$ whose image under every $0\neq x^*\in X^*$ has positive Lebesgue measure. This demonstrates completely the failure of the Besicovitch-Federer projection theorem in infinite dimensional Banach spaces.

Let X be an infinite dimensional separable Banach space.There exists a hypercyclic operator on X which is equal to the identity operator on an infinite dimensional closed subspace of X.

Existence of Hypercyclic Operators on Topological Vector Spaces

Common hypercyclic vectors for the conjugate class of a hypercyclic operator

Hypercyclic operators on countably dimensional spaces

This paper is devoted to the study of operators satisfying the condition $$ ||A||\, = \max \{ \rho (AB):||B||\, = 1\} , $$ where ρ stands for the spectral radius; and Banach spaces in which all operators satisfy this condition. Such spaces are called V−spaces. The present paper contains partial solutions of some of the open problems posed in the first part of the paper. The main results: (1) Each subspace of l p (1 < p < ∞) is a V−space. (2) For each infinite dimensional Banach space X there exists an equivalent norm |||·||| on X such that the space (X, |||·|||) is not a V−space. (3) Let X be a separable infinite dimensional Banach space with a symmetric basis. If X has the V-property, then X is isometric to l p , 1 < p < ∞.

Hypercyclic orbits intersect subspaces in wild ways

In this paper, a criterion for the existence of large linear algebras consisting, except for zero, of one-to-one operators on an infinite dimensional Banach space is provided. As a consequence, it is shown that every separable infinite dimensional Banach space supports a commutative infinitely generated free linear algebra of operators all of whose nonzero members are one-to-one. In certain cases, the assertion holds for nonseparable Banach spaces.

Equilibria in a large production economy with an infinite dimensional commodity space and price dependent preferences

Hypercyclic bilinear operators on Banach spaces

If E is the nuclear space s of rapidly decreasing sequences and F is any infinitedimensional Banach space with Schauder basis, then for every 0-neighborhood U in E there exists an absolutely convex 0-neighborhood Vc U in E such that /~v is norm-isomorphic to F (see [6]). This result of Saxon was improved later on by Valdivia [8] proving its validity when E is an arbitrary nuclear space and F any infinite-dimensional separable Banach space. In the present paper we bring up these results into the context of 2-nuclearity. Namely, we will prove that for certain sequence spaces 4, a Mackey space a can be found satisfying the following condition: If F is any infinite-dimensional Banach space with Schauder basis, for every 0-neighborhood U in othere is an absolutely convex 0-neighborhood V in osuch that ffv is norm-isomorphic to F. As a consequence we prove an embedding theorem of k-nuclear spaces into some product of any given infinite-dimensional Banach space with Schauder basis. This research, supported by the University of Extremadura, was carried out during the author's visit to the University of Kaiserslautern (F.R.G.).

On the lack of exact controllability for mild solutions in Banach spaces

In this paper, we first prove some random fixed point theorems for random nonexpansive operators in Banach spaces. As applications, some random approximation theorems for random 1-set-contraction or random continuous condensing mappings defined on closed balls of a separable Banach space, or on separable closed convex subsets of a Hilbert space or on spheres of infinite dimensional separable Banach spaces are established. Our results are generalizations, improvements or stochastic versions of the corresponding results of Bharucha-Reid (1976), Lin (1988, 1989), Lin and Yen (1988), Massatt (1983), Sehgal and Waters (1984) and Xu (1990).

We show that operators on a separable infinite dimensional Banach space $X$ of the form $I +S$, where $S$ is an operator with dense generalised kernel, must lie in the norm closure of the hypercyclic operators on $X$, in fact in the closure of the mixing operators.

We prove that there exists a Lipschitz function froml1 into ℝ2 which is Gâteaux-differentiable at every point and such that for everyx, y el1, the norm off′(x) −f′(y) is bigger than 1. On the other hand, for every Lipschitz and Gâteaux-differentiable function from an arbitrary Banach spaceX into ℝ and for everye > 0, there always exist two pointsx, y eX such that ‖f′(x) −f′(y)‖ is less thane. We also construct, in every infinite dimensional separable Banach space, a real valued functionf onX, which is Gâteaux-differentiable at every point, has bounded non-empty support, and with the properties thatf′ is norm to weak* continuous andf′(X) has an isolated pointa, and that necessarilya e 0.