The multiplicative property characterizes $\ell_p$ and $L_p$ norms

Sharp value for the norm of composition and multiplicative operators between two different Grand Lebesgue spaces

We calculate in this paper the norm of composition, multiplicative and product operator, generated by multiplicative and measurable argument transformation between two different ordinary Lebesgue-Riesz and Grand Lebesgue spaces. We set ourselves the aim to obtain the exact expression for the norm of the considered operators by means of building of appropriate examples.

On the spaces of bounded and compact multiplicative Hankel operators

A multiplicative Hankel operator is an operator with matrix representation $M(α) = \{α(nm)\}_{n,m=1}^\infty$, where $α$ is the generating sequence of $M(α)$. Let $\mathcal{M}$ and $\mathcal{M}_0$ denote the spaces of bounded and compact multiplicative Hankel operators, respectively. In this note it is shown that the distance from an operator $M(α) \in \mathcal{M}$ to the compact operators is minimized by a nonunique compact multiplicative Hankel operator $N(β) \in \mathcal{M}_0$, $$\|M(α) - N(β)\|_{\mathcal{B}(\ell^2(\mathbb{N}))} = \inf \left \{\|M(α) - K \|_{\mathcal{B}(\ell^2(\mathbb{N}))} …

Bounded multiplicative Toeplitz operators on sequence spaces

In this paper, we study the linear mapping which sends the sequence $x=(x_n)_{n \in \mathbb{N}}$ to $y=(y_n)_{n \in \mathbb{N}}$ where $y_n = \sum_{k=1}^\infty f(n/k)x_k$ for $f: \mathbb{Q}^+ \to \mathbb{C}$. This operator is the multiplicative analogue of the classical Toeplitz operator, and as such we denote the mapping by $\mathscr{M}_f$. We show that for $1 \leq p \leq q \leq \infty$, if $f \in \ell^r(\mathbb{Q}^+)$, then $\mathscr{M}_f:\ell^p \to \ell^q$ is bounded where $\frac{1}{r} = 1 - \frac{1}{p} + \frac{1}{q} $. Moreover, for the cases when $p=1$ with any $q$, $p=q$, and $q=\infty$ …

Multiplicative norm convergence in Banach lattice $f$-algebras

A net $(x_α)$ in an $f$-algebra $E$ is called multiplicative order convergent to $x\in E$ if $\lvert x_α-x\rvert u\oc 0$ for all $u\in E_+$. This convergence has been investigated and applied in a recent paper by Aydın \cite{AAydn}. In this paper, we study a variation of this convergence for Banach lattice $f$-algebras. A net $(x_α)$ in a Banach lattice $f$-algebra $E$ is said to be multiplicative norm convergent to $x\in E$ if $\lvert \lvert x_α-x\rvert u\rVert\to 0$ for each $u\in E_+$. We study on this concept and investigate its relationship with the other convergences, and also we introdu…

Approximately multiplicative maps from weighted semilattice algebras

We investigate which weighted convolution algebras $\ell^1_ω(S)$, where $S$ is a semilattice, are AMNM in the sense of Johnson (JLMS, 1986). We give an explicit example where this is not the case. We show that the unweighted examples are all AMNM, as are all $\ell^1_ω(S)$ where $S$ has either finite width or finite height. Some of these finite-width examples are isomorphic to function algebras studied by Feinstein (IJMMS, 1999). We also investigate when $(\ell^1_ω(S),{\bf M}_2)$ is an AMNM pair in the sense of Johnson (JLMS, 1988), where ${\bf M}_2$ denotes the algebra of 2-by-2 complex matric…

关于在晶格中扩张系统的循环性：乘法序列、多项式、狄利克雷型空间和代数

On the Cyclicity of Dilated Systems in Lattices: Multiplicative Sequences, Polynomials, Dirichlet-type Spaces and Algebras

本文探讨了在晶格中扩张系统的循环性，分析了乘法序列、多项式、狄利克雷型空间和代数的完备性及性质。