Given a measurable set $A\subset {ℝ}^n$ of positive measure, it is not difficult to show that $|A+A|=\left|2A\right|$ if and only if $A$ is equal to its convex hull minus a set of measure zero. We investigate the stability of this statement: If $\left(\right|A+A|-|2A\left|\right)/\left|A\right|$ is small, is $A$ close to its convex hull? Our main result is an explicit control, in arbitrary dimension, on the measure of the difference between $A$ and its convex hull in terms of $\left(\right|A+A|-|2A\left|\right)/\left|A\right|$.

Let $G$ be the semidirect product $V⋊K$ where $K$ is a connected semisimple non-compact Lie group acting linearly on a finite-dimensional real vector space $V$. Let $\pi$ be a unitary irreducible representation of $G$ which is associated by the Kirillov-Kostant method of orbits with a coadjoint orbit of $G$ whose little group is a maximal compact subgroup of $K$. We construct an invariant symbolic calculus for $\pi$, under some technical hypothesis. We give some examples including the Poincaré group. ...

We consider the equation $\mathrm{d}y\left(t\right)/\mathrm{d}t=(A+B(t\left)\right)y\left(t\right)$ $(t\ge 0)$, where $A$ is the generator of an analytic semigroup ${\left({\mathrm{e}}^{At}\right)}_{t\ge 0}$ on a Banach space $𝒳$, $B\left(t\right)$ is a variable bounded operator in $𝒳$. It is assumed that the commutator $K\left(t\right)=AB\left(t\right)-B\left(t\right)A$ has the following property: there is a linear operator $S$ having a bounded left-inverse operator $S_l^{-1}$ such that $\parallel S{\mathrm{e}}^{At}\parallel$ is integrable and the operator $K\left(t\right)S_l^{-1}$ is bounded. Under these conditions an exponential stability test is derived. As an example we consider a coupled system of parabolic equations. ...

CONTENTS1. Introduction...................................................................................................... 52. Notation and basic terminology........................................................................... 73. Definition and basic properties of the $L_{p,q}$ spaces................................. 114. Integral representation of bounded linear functionals on $L_{p,q}\left(B\right)$........ 235. Examples in $L_{p,q}$ theory...................................................................................

For any number field $K$ with non-elementary $3$-class group ${\mathrm{Cl}}_3\left(K\right)\simeq C_{3^e}\times C_3$, $e\ge 2$, the punctured capitulation type $\varkappa \left(K\right)$ of $K$ in its unramified cyclic cubic extensions $L_i$, $1\le i\le 4$, is an orbit under the action of $S_3\times S_3$. By means of Artin’s reciprocity law, the arithmetical invariant $\varkappa \left(K\right)$ is translated to the punctured transfer kernel type $\varkappa \left(G_2\right)$ of the automorphism group $G_2=\mathrm{Gal}({\mathrm{F}}_3^2\left(K\right)/K)$ of the second Hilbert $3$-class field of $K$. A classification of finite $3$-groups $G$ with low order and bicyclic commutator quotient $G/G^{\text{'}}\simeq C_{3^e}\times C_3$, $2\le e\le 6$, according to the algebraic...

CONTENTSIntroduction................................................................................................................................... 5 Notations.......................................................................................................................... 5§ 1. Preliminaries........................................................................................................................ 6 1. Right invertible operators.....................................................................................................

Denote by spanf₁,f₂,... the collection of all finite linear combinations of the functions f₁,f₂,... over ℝ. The principal result of the paper is the following. Theorem (Full Clarkson-Erdős-Schwartz Theorem). Suppose ${\left({\lambda }_j\right)}_{j=1}^{\infty }$ is a sequence of distinct positive numbers. Then $span1,x^{\lambda ₁},x^{\lambda ₂},...$ is dense in C[0,1] if and only if ${\sum }_{j=1}^{\infty }\left({\lambda }_j\right)/({\lambda }_j²+1)=\infty$. Moreover, if ${\sum }_{j=1}^{\infty }\left({\lambda }_j\right)/({\lambda }_j²+1)<\infty$, then every function from the C[0,1] closure of $span1,x^{\lambda ₁},x^{\lambda ₂},...$ can be represented as an analytic function on z ∈ ℂ ∖ (-∞, 0]: |z| < 1 restricted to (0,1). This result improves an...

Let G be an infinite, compact abelian group and let Λ be a subset of its dual group Γ. We study the question which spaces of the form $C_{\Lambda }\left(G\right)$ or $L{¹}_{\Lambda }\left(G\right)$ and which quotients of the form $C\left(G\right)/C_{\Lambda }\left(G\right)$ or $L¹\left(G\right)/L{¹}_{\Lambda }\left(G\right)$ have the Daugavet property. We show that $C_{\Lambda }\left(G\right)$ is a rich subspace of C(G) if and only if $\Gamma \setminus {\Lambda }^{-1}$ is a semi-Riesz set. If $L{¹}_{\Lambda }\left(G\right)$ is a rich subspace of L¹(G), then $C_{\Lambda }\left(G\right)$ is a rich subspace of C(G) as well. Concerning quotients, we prove that $C\left(G\right)/C_{\Lambda }\left(G\right)$ has the Daugavet property if Λ is a Rosenthal set, and that $L{¹}_{\Lambda }\left(G\right)$ is a poor subspace of L¹(G)...

We characterise the boundedness of the $H^{\infty }$ calculus of a sectorial operator in terms of dilation theorems. We show e. g. that if $-A$ generates a bounded analytic $C_0$ semigroup $\left(T_t\right)$ on a UMD space, then the $H^{\infty }$ calculus of $A$ is bounded if and only if $\left(T_t\right)$ has a dilation to a bounded group on $L^2([0,1],X)$. This generalises a Hilbert space result of C.LeMerdy. If $X$ is an $L^p$ space we can choose another $L^p$ space in place of $L^2([0,1],X)$.

The vertex algebra ${𝒲}_{1+\infty ,c}$ with central charge $c$ may be defined as a module over the universal central extension of the Lie algebra of differential operators on the circle. For an integer $n\ge 1$, it was conjectured in the physics literature that ${𝒲}_{1+\infty ,-n}$ should have a minimal strong generating set consisting of $n^2+2n$ elements. Using a free field realization of ${𝒲}_{1+\infty ,-n}$ due to Kac–Radul, together with a deformed version of Weyl’s first and second fundamental theorems of invariant theory for the standard representation...

In the following text for a discrete finite nonempty set $K$ and a self-map $\varphi :X\to X$ we investigate interaction between different entropies of generalized shift ${\sigma }_{\varphi }:K^X\to K^X$, ${\left(x_{\alpha }\right)}_{\alpha \in X}\mapsto {\left(x_{\varphi \left(\alpha \right)}\right)}_{\alpha \in X}$ and cellularities of some Alexandroff topologies on $X$.

Let $G$ be any group containing an infinite elementary amenable subgroup and let $2\&lt;p\&lt;\infty$. We construct an exhaustion of ${\ell }^{p}G$ by closed invariant subspaces which all intersect trivially a fixed non-trivial closed invariant subspace. This is an obstacle to ${\ell }^p$-dimension and gives an answer to a question of Gaboriau.

We prove that on ${ℝ}^N$, there is no n-supercyclic operator with 1 ≤ n < ⌊(N + 1)/2⌋, i.e. if ${ℝ}^N$ has an n-dimensional subspace whose orbit under $T\in \left({ℝ}^N\right)$ is dense in ${ℝ}^N$, then n is greater than ⌊(N + 1)/2⌋. Moreover, this value is optimal. We then consider the case of strongly n-supercyclic operators. An operator $T\in \left({ℝ}^N\right)$ is strongly n-supercyclic if ${ℝ}^N$ has an n-dimensional subspace whose orbit under T is dense in $\mathbb{P}ₙ\left({ℝ}^N\right)$, the nth Grassmannian. We prove that strong n-supercyclicity does not occur non-trivially...

A subset of a product of topological spaces is called $n$-thin if every its two distinct points differ in at least $n$ coordinates. We generalize a construction of Gruenhage, Natkaniec, and Piotrowski, and obtain, under CH, a countable $T_3$ space $X$ without isolated points such that $X^n$ contains an $n$-thin dense subset, but $X^{n+1}$ does not contain any $n$-thin dense subset. We also observe that part of the construction can be carried out under MA.