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Register a new userA review of some recent work on hypercyclicity
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Authors
C. T. J. Dodson
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Even linear operators on infinite-dimensional spaces can display interesting dynamical properties and yield important links among functional analysis, differential and global geometry and dynamical systems, with a wide range of applications. In particular, hypercyclicity is an essentially infinite-dimensional property, when iterations of the operator generate a dense subspace. A Frechet space admits a hypercyclic operator if and only if it is separable and infinite-dimensional. However, by considering the semigroups generated by multiples of operators, it is possible to obtain hypercyclic behaviour on finite dimensional spaces. This article gives a brief review of some recent work on hypercyclicity of operators on Banach, Hilbert and Frechet spaces.
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This note reviews some of the recent work on biharmonic conformal maps (see cite{OC}, Chapter 11, for a detailed survey). It will be focused on biharmonic conformal immersions and biharmonic conformal maps between manifolds of the same dimension and their links to isoparametric functions and Yamabe type equations, though biharmonic morphisms (maps that preserve solutions of bi-Laplace equations), generalized harmonic morphisms (maps that pull back germs of harmonic functions to germs of biharmonic functions), and biharmonic conformal and Riemannian submersions will also be touched.
We provide with criteria for a family of sequences of operators to share a frequently universal vector. These criteria are variants of the classical Frequent Hypercyclicity Criterion and of a recent criterion due to Grivaux, Matheron and Menet where periodic points play the central role. As an application, we obtain for any operator T in a specific class of operators acting on a separable Banach space, a necessary and sufficient condition on a subset $Lambda$ of the complex plane for the family {$lambda$T : $lambda$ $in$ $Lambda$} to have a common frequently hypercyclic vector. In passing, this permits us to easily exhibit frequent hypercyclic weighted shifts which do not possess common frequent hypercyclic vectors. We also provide with criteria for families of the recently introduced operators of C-type to share a common frequently hypercyclic vector. Further, we prove that the same problem of common $alpha$-frequent hypercyclicity may be vacuous, where the notion of $alpha$-frequent hypercyclicity extends that of frequent hypercyclicity replacing the natural density by more general weighted densities. Finally, it is already known that any operator satisfying the classical Frequent Universality Criterion is $alpha$-frequently universal for any sequence $alpha$ satisfying a suitable condition. We complement this result by showing that for any such operator, there exists a vector x which is $alpha$-frequently universal for T , with respect to all such $alpha$.
We review some history and some recent results concerning Toeplitz determinants and their applications. We discuss, in particular, the crucial role of the two-dimensional Ising model in stimulating the development of the theory of Toeplitz determinants.
It is noted that some results of the authors research papers cited in the comprehensive review paper The Science of Sungrazers, Sunskirters, and Other Near-Sun Comets by G. H. Jones et al. (2018) have been factually misrepresented. The purpose of this note is to offer (i) revisions in order to correct the numerous errors and inaccuracies and (ii) additions where the text is confusing.
In this short note, we first consider some inequalities for comparison of some algebraic properties of two continuous algebra-multiplications on an arbitrary Banach space and then, as an application, we consider some very basic observations on the space of all continuous algebra-multiplications on a Banach space.
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