arXiv subjects
Publications and source records attributed to Edward S. Sichel.
When finding an original proof to a known result describing expansions on compact metric spaces as surjective isometries, we reveal that relaxing the condition of compactness to total boundedness preserves the isometry property and nearly that of surjectivity. While a counterexample is found showing that the converse to the above descriptions do not hold, we are able to characterize boundedness in terms of specific expansions we call anticontractions.
We give a simple, straightforward proof of the non-hypercyclicity of an arbitrary normal operator $A$ (bounded or unbounded) in a complex Hilbert space as well as of the collection $\left\{e^{tA}\right\}_{t\ge 0}$ of its exponentials, which, under a certain condition on the spectrum of $A$, is the $C_0$-semigroup generated by it. We also establish non-hypercyclicity for symmetric operators.
We study the chaoticity of bounded and unbounded weighted backward shifts in the space $c_0(\mathbb{N})$ of vanishing sequences via a novel straightforward approach based on a newly found sufficient condition for linear chaos and show that their extensions to the space $c(\mathbb{N})$ of convergent sequences are not even hypercyclic. Thus, we furnish bounded and unbounded linear chaotic operators in $c(\mathbb{N})$ in a different way: as conjugates to the weighted backward shifts in $c_0(\mathbb{Z}_+)$ via a homeomorphic isomorphism between the two spaces.