Search Results

Let $T:L^2(\mathbb{R}^n,\mathbb{R}^n)\rightarrow L^2(\mathbb{R}^n,\mathbb{R}^n)$ be a linear operator of the form: for all $x\in \mathbb{R}^n$ $$ T_{\kappa}(f)(x):=\int_{u\in\mathbb{R}^n} \, f(u)\kapp …

Let $E=C([0,1]^m,\mathbb{R}^n)$ where $K=[0,1]^m$ where $E$ has the compact convergence topology. Recall that for a function $f:[0,1]^m\rightarrow [0,1]^m$ the associated composition operator $C_f$ i …

Let $(X,d,x)$ be a separable pointed metric space and let $\mathcal{F}(X)$ be its Arens-Eells (also called its Lipschitz-Free space; in the case where $X$ is Banach) space. We view the $1$-Wasserstei …

Let $W$ be the Wiener measure on $C_0(\mathbb{R})$ and let $T\in L(C_0(\mathbb{R}),C_0(\mathbb{R}))$ be a hypercylic operator; i.e. there exists some $f \in C_0(\mathbb{R})$ such that $\{T^n(f)\}_{n=1 …

The Wiener Tauberian Theorem gives condition on an $f\in L^1(\mathbb{R})$ such that the "induced 1-parameter family" $\{T_b(f)\}_{b\in \mathbb{R}}$ has a dense span in $L^1(\mathbb{R})$; where $T_b$ i …

Let $b\in \mathbb{R}\neq 0$, and consider the translation operators: $$ \begin{align} T_b:C(\mathbb{R}) & \rightarrow C(\mathbb{R})\\ f &\mapsto f(\cdot + b). \end{align} $$ *Are there known characte …

Let $\{T_i\}_{i \in I}$ be a family of hypercylic operators on a separable Banach space $X$. From the transitivity theorem, we know that $HC(T_i)$, the set of vectors $x \in X$ with $\{T_i^n(x):n \in …

Let $\nu$ be a finite Borel measure on $\mathbb{R}^n$ and define the shift operator $T_a$ on $L^p_{\nu}(\mathbb{R}^n)$ by $f\to f(x+a)$ for some fixed $a\in \mathbb{R}^n-\{0\}$. Suppose moreover that …

Let $B$ be a separable Banach space, $L:B \to B$ be a hypercyclic operator, $k>0$, $I_B$ the identity on $B$, and define $L_k: =k (I_B - L)$. When is $L_k$ hypercyclic on $B$? Can anything else be …

Fix $N>1$. Let $f\in C^{\infty}(\mathbb{R},\mathbb{R})$ be such that the composition operator via $$ \begin{aligned} C_f:C^{\infty}(\mathbb{R},\mathbb{R}) &\rightarrow C^{\infty}(\mathbb{R},\mathbb{ …

Let $\{X_i\}_{i \in \mathbb{N}} $ be a sequence of $n$-dimensional linear subspaces of the separable Hilbert space $H$ and let $\{\phi_i\}_{i \in I}$ be a sequence of continuous injective linear maps …

Background Let $(U_t)_{t \in \mathbb{R}}$ be the (translation) $C_0$-group on $L^1(\mathbb{R})$ defined by $$ U_t(f)(x) = f(x-t) \quad \text{for almost every } x \in \mathbb{R} $$ (for $t \in \mathb …

$\DeclareMathOperator\C{C}\DeclareMathOperator\HC{HC}$Definitions: Let $T:X\rightarrow X$ be a bonded linear operator on a separable (infinite-dimensional) Banach space and define the sets: $ \HC …

It is well-known that the weighted backshift operator $B_{\lambda}:\ell^p \rightarrow \ell^p$ is hypercyclic (with $\lambda>1$); that is, there exists a dense set of sequences $X\subseteq \ell^p$ for …

Let $H(\mathbb{C})$ be the space of holomorphic functions on the complex plane. Then it is well-known that for $a\neq 0$, the translation operator $$ t_a(f)\triangleq f(x)\mapsto f(x+a), $$ is topolo …