Related papers: Commuting Contractive Families

A bounded operator $u$ on $X$ is called rigid when there is an increasing sequence of positive integers $(n_k)_{k\geq 1}$, such that for every $x$ in $X$ we have $\lim_{k \rightarrow +\infty} u^{n_k} x = x$. For any $r$ in $[0,1]$, we…

We show that the celebrated Lomonosov theorem cannot be improved by increasing the number of commuting operators. Specifically, we prove that if T is the operator on l_1 without a non-trivial closed invariant subspace constructed by…

The Collatz map (or the $3n{+}1$-map) $f$ is defined on positive integers by setting $f(n)$ equal to $3n+1$ when $n$ is odd and $n/2$ when $n$ is even. The Collatz conjecture states that starting from any positive integer $n$, some iterate…

A general form of contractive idempotent functionals on coamenable locally compact quantum groups is obtained, generalising the result of Greenleaf on contractive measures on locally compact groups. The image of a convolution operator…

We find an explicit tetrablock isometric dilation for every member $(A_\alpha, B, P)$ of a family of tetrablock contractions indexed by a parameter $\alpha$ in the closed unit disc (only the first operator of the tetrablock contraction…

For a family of continuous functions $f_1,f_2,\dots \colon I \to \mathbb{R}$ ($I$ is a fixed interval) with $f_1\le f_2\le \dots$ define a set $$ I_f:=\big\{x \in I \colon \lim_{n \to \infty} f_n(x)=+\infty\big\}.$$ We study the properties…

The operator function (A,B)\to\tr f(A,B)(K^*)K, defined on pairs of bounded self-adjoint operators in the domain of a function f of two real variables, is convex for every Hilbert Schmidt operator K, if and only if f is operator convex. As…

Consider the two-parameter family of real analytic maps $F_{a,b}:x \mapsto x+ a+{b\over 2\pi} \sin(2\pi x)$ which are lifts of degree one endomorphisms of the circle. The purpose of this paper is to provide a proof that for any closed…

Let $A$ be a group acting by automorphisms on the group $G.$ \textit{The commuting graph $\Gamma(G,A)$ of $A$-orbits} of this action is the simple graph with vertex set $\{x^{A} : 1\ne x \in G \}$, the set of all $A$-orbits on $G\setminus…

Let $H(\mathbb{C})$ be the set of all entire functions endowed with the topology of uniform convergence on compact sets. Let $\lambda,b\in\mathbb{C}$, let $C_{\lambda,b}:H(\mathbb{C})\to H(\mathbb{C})$ be the composition operator…

A finite family $\mathrsfs{F}$ of subsets of a finite set $X$ is union-closed whenever $f,g\in\mathrsfs{F}$ implies $f\cup g\in\mathrsfs{F}$. These families are well known because of Frankl's conjecture. In this paper we developed further…

For $0<r<1$, let us consider the following annulus: \[ \mathbb A_r= \{ z\in \mathbb C\, : \, r<|z|<1 \}. \] A Hilbert space operator $T$ for which $\overline{\mathbb A}_r$ is a spectral set is called an $\mathbb A_r$-\textit{contraction}.…

Taking into account that Rold\'an et al.'s ample spectrum contractions have managed to extend and unify more than ten distinct families of contractive mappings in the setting of metric spaces, in this manuscript we present a first study on…

A distributed algorithm is described for finding a common fixed point of a family of m>1 nonlinear maps M_i : R^n -> R^n assuming that each map is a paracontraction and that at least one such common fixed point exists. The common fixed…

An n-tuple (n \geq 2), T = (T_1, \ldots, T_n), of commuting bounded linear operators on a Hilbert space \mathcal{H} is doubly commuting if T_i T_j^* = T_j^* T_i for all $1 \leq i < j \leq n$. If in addition, each T_i \in C_{\cdot 0}, then…

Lueders theorem states that two observables commute if measuring one of them does not disturb the measurement outcomes of the other. We study measurements which are described by continuous positive operator-valued measurements (or POVMs)…

We investigate the problem of pointwise convergence of the family of non-linear integral operators: \begin{equation} L_\lambda(f,x) = \int_a^b \sum_{m=1}^N f^m(t) K_{\lambda ,m}(x,t) dt, \end{equation} where $\lambda $ is a real parameters,…

We introduce and study adhesive spaces. Using this concept we obtain a characterization of stable Baire maps $f:X\to Y$ of the class $\alpha$ for wide classes of topological spaces. In particular, we prove that for a topological space $X$…

We give several applications of a recent theorem of the second author, which solved a conjecture of the first author with Hay and Neal, concerning contractive approximate identities; and another of Hay from the theory of noncommutative peak…

We prove that there exists a constant $c_0$ such that for any $t \in \mathbb{N}$ and any $n\geq c_0 t$, if $A \subset S_n$ is a $t$-intersecting family of permutations then$|A|\leq (n-t)!$. Furthermore, if $|A|\ge 0.75(n-t)!$ then there…