Elementary operators on Banach algebras and Fourier transform

Miloš Arsenović¹ ; Dragoljub Kečkić¹

¹Matematički Fakultet Univerzitet u Beogradu Studentski trg 16–18 11000 Beograd, Serbia and Montenegro

Studia Mathematica, Tome 173 (2006) no. 2, pp. 149-166

Résumé

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 $\mathbb R^{2n}$.

Détail

DOI : 10.4064/sm173-2-3

Keywords: consider elementary operators mapsto sum jxb acting unital banach algebra where separately commuting families generalized scalar elements ascent estimate lower bound estimate operator additionally weak variant fuglede putnam theorem elementary operator strongly commuting families j j where commute main tool estimate fourier transform certain class cpt infty functions mathbb