Algebraic Reflexivity of Non-Canonical Isometries on Lipschitz Spaces

Let Lip([0,1]) be the Banach space of all Lipschitz complex-valued functions f on [0,1], equipped with one of the norms: fσ=|f(0)|+f′L∞ or fm=max|f(0)|,f′L∞, where ·L∞ denotes the essential supremum norm. It is known that the surjective linear isometries of such spaces are integral operators, rather than the more familiar weighted composition operators. In this paper, we describe the topological reflexive closure of the isometry group of Lip([0,1]). Namely, we prove that every approximate local isometry of Lip([0,1]) can be represented as a sum of an elementary weighted composition operator and an integral operator. This description allows us to establish the algebraic reflexivity of the sets of surjective linear isometries, isometric reflections, and generalized bi-circular projections of Lip([0,1]). Additionally, some complete characterizations of such reflections and projections are stated.

Automorphism groups and isometries for cyclic orbit codes

<p style='text-indent:20px;'>We study orbit codes in the field extension <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{F}_{q^n} $\end{document}</tex-math></inline-formula>. First we show that the automorphism group of a cyclic orbit code is contained in the normalizer of the Singer subgroup if the orbit is generated by a subspace that is not contained in a proper subfield of <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{F}_{q^n} $\end{document}</tex-math></inline-formula>. We then generalize to orbits under the normalizer of the Singer subgroup. In that situation some exceptional cases arise and some open cases remain. Finally we characterize linear isometries between such codes.</p>

DETECTING STEINER AND LINEAR ISOMETRIES OPERADS

We study the indexing systems that correspond to equivariant Steiner and linear isometries operads. When G is a finite abelian group, we prove that a G-indexing system is realized by a Steiner operad if and only if it is generated by cyclic G-orbits. When G is a finite cyclic group, whose order is either a prime power or a product of two distinct primes greater than 3, we prove that a G-indexing system is realized by a linear isometries operad if and only if it satisfies Blumberg and Hill’s horn-filling condition. We also repackage the data in an indexing system as a certain kind of partial order. We call these posets transfer systems, and develop basic tools for computing with them.

Near-isometries of the unit sphere

UDC 517.5We approximate ε -isometries of the unit sphere in ℓ 2 n and ℓ ∞ n by linear isometries.