Elementary operators on Banach algebras and Fourier transform
Volume 173 / 2006
Studia Mathematica 173 (2006), 149-166
MSC: 47B48, 47B47, 42B10.
DOI: 10.4064/sm173-2-3
Abstract
We consider elementary operators $x\mapsto\sum_{j=1}^na_jxb_j$, 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. $a_j=a_j'+ia_j''$ ($b_j=b_j'+ib_j''$), where all $a_j'$ and $a_j''$ ($b_j'$ and $b_j''$) commute. The main tool is an $L^1$ estimate of the Fourier transform of a certain class of $C_{\rm cpt}^\infty$ functions on $\mathbb R^{2n}$.
