We consider elementary operators $x\mapsto {\sum }_{j=1}^n{a}_{j}x{b}_j$, acting on a unital Banach algebra, where $a_j$ and $b_j$ are separately commuting families of generalized scalar elements. We give an ascent estimate and a lower bound estimate for such an operator. Additionally, we give a weak variant of the Fuglede-Putnam theorem for an elementary operator with strongly commuting families $a_j$ and $b_j$, i.e. $a_j=a_j^{\text{'}}+i{a}_j^{\text{'}\text{'}}$ ($b_j=b_j^{\text{'}}+i{b}_j^{\text{'}\text{'}}$), where all $a_j^{\text{'}}$ and $a_j^{\text{'}\text{'}}$ ($b_j^{\text{'}}$ and $b_j^{\text{'}\text{'}}$) commute. The main tool is an L¹ estimate of the Fourier transform of a certain class of $C_{cpt}^{\infty }$ functions...

We discuss boundedness and compactness properties of the embedding $M_{\Lambda }^1\subset L^1\left(\mu \right)$, where $M_{\Lambda }^1$ is the closed linear span of the monomials $x^{{\lambda }_n}$ in $L^1\left([0,1]\right)$ and $\mu$ is a finite positive Borel measure on the interval $[0,1]$. In particular, we introduce a class of “sublinear” measures and provide a rather complete solution of the embedding problem for the class of quasilacunary sequences $\Lambda$. Finally, we show how one can recapture some of Al Alam’s results on boundedness and the essential norm of weighted composition operators from $M_{\Lambda }^1$...

The present paper is devoted to the study of the “quality” of the compactness of the trace operator. More precisely, we characterize the asymptotic behaviour of entropy numbers of the compact map $t{r}_{\Gamma }:B_{p₁,q}^s(ℝⁿ,w_{\varkappa }^{\Gamma })\to L_{p₂}\left(\Gamma \right)$, where Γ is a d-set with 0 < d < n and $w_{\varkappa }^{\Gamma }$ a weight of type $w_{\varkappa }^{\Gamma }\left(x\right)dist{(x,\Gamma )}^{\varkappa }$ near Γ with ϰ > -(n-d). There are parallel results for approximation numbers.