Annals of Mathematical Sciences and Applications

Let $G$ be a topological group and $E$ a quasi-complete space. We study approximation properties of almost periodic vectors of continuous, equicontinuous representations $\pi : G \to \mathcal{B}(E)$. We extend an approximation theorem of Weyl and Maak from isometric Banach space representations to representations on quasi-complete spaces. We prove that if $\pi$ is almost periodic, then $E$ has a generalized direct sum decomposition $E = \oplus_{ \theta \in \widehat{G}} E_\theta$, where each $E_\theta$ is linearly spanned by finite-dimensional, $\pi$-invariant subspaces. We show that on left translation invariant, quasi-complete subspaces of $L^p(G)$ ($G$ locally compact, $1 \leq p \lt \infty$), the left regular representation is almost periodic if and only if $G$ is compact.

We give an elementary proof of the Myers–Steenrod theorem, stating that the group of isometries of a connected Riemannian manifold $M$ is a Lie group acting smoothly on $M$. Our proof follows the approach of Chu and Kobayashi, but replacing their use of a theorem of Palais with a topological condition detecting when a locally compact subspace of $M$ is an embedded integral manifold of a given $k$-plane distribution.

Let $G$ be a locally compact group. In this paper, we study various invariant subspaces of the duals of the algebras $A_M(G)$ and $A_{cb}(G)$ obtained by taking the closure of the Fourier algebra $A(G)$ in the multiplier algebra $MA(G)$ and completely bounded multiplier algebra $M_{cb}A(G)$ respectively. In particular, we will focus on various functorial properties and containment relationships between these various invariant subspaces including the space of uniformly continuous functionals and the almost periodic and weakly almost periodic functionals.

$\def\Lip{\mathrm{Lip}}\def\Lipb{\mathrm{Lip^b}}\def\Liploc{\mathrm{Lip^{loc}}}\def\Lippt{\mathrm{Lip^{pt}}}$Let $\Lip(X)$, $\Lipb(X)$, $\Liploc(X)$ and $\Lippt(X)$ be the vector spaces of Lipschitz, bounded Lipschitz, locally Lipschitz and pointwise Lipschitz (real-valued) functions defined on a metric space $(X, d_X)$, respectively. We show that if a weighted composition operator $Tf = h \cdot f \circ \varphi$ defines a bijection between such vector spaces preserving Lipschitz constants, local Lipschitz constants or pointwise Lipschitz constants, then $h = \pm 1/\alpha$ is a constant function for some scalar $\alpha \gt 0$ and $\varphi$ is an $\alpha$-dilation.

We prove that every $m \times n$ row-stochastic (RS) matrix can be written as a convex combination of $n^m$ many $\lbrace 0, 1 \rbrace$–RS matrices. In the special cases of $2 \times 3$ and $3 \times 3$ RS matrices, the proofs are given constructively. Algorithms for computing the convex decompositions of row-stochastic matrices are provided.

We survey the recent development of approximate amenability and pseudo-amenability in Banach algebras, concentrating on the relationship between the two notions. It has been conjectured that a Banach algebra with a bounded approximate identity is approximately amenable if and only if it is pseudo-amenable. So far we only have partial affirmative results concerning the conjecture. Some open problems are also posted.

Inertial-based methods have the drawback of not preserving the Fejér monotonicity of iterative sequences, which may result in slower convergence compared to their corresponding non-inertial versions. To overcome this issue, Mu and Peng [Stat. Optim. Inf. Comput. 3 (2015), 241–248; $\href{ https://mathscinet.ams.org/mathscinet-getitem?mr=3393305}{MR3393305}$] suggested an alternating inertial method that can recover the Fejér monotonicity of even subsequences. In this paper, we propose a modified version of the forward-backward algorithm with alternating inertial and relaxation effects to solve an inclusion problem in real Hilbert spaces. The weak and linear convergence of the presented algorithm is established under suitable and mild conditions on the involved operators and parameters. Furthermore, the Fejér monotonicity of even subsequences generated by the proposed algorithm with respect to the solution set is recovered. Finally, our tests on image restoration problems demonstrate the superiority of the proposed algorithm over some related results.

In this note we refine some of the results of [$\href{https://mathscinet.ams.org/mathscinet/relay-station?mr=2818926}{18}$] on the gradient-projection algorithm in the infinite-dimensional Hilbert space setting by weakening the conditions imposed on the choices of the parameters in [$\href{https://mathscinet.ams.org/mathscinet/relay-station?mr=2818926}{18}$, Theorems 4.2, 4.3 and 5.2]. In addition, we also show that the relaxed gradient-projection algorithm has a sublinear rate of convergence.

Let $G$ be a locally compact Abelian group and $A(G)$ its Fourier Algebra. In this paper, in the setting of the Fourier algebra $A(G)$ of $G$, we present some tools to study the sets of synthesis and Ditkin sets and, to illustrate the use of them, we give a large number applications of the tools introduced.