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# Wydawnictwa / Czasopisma IMPAN / Studia Mathematica / Wszystkie zeszyty

## Elementary operators on Banach algebras and Fourier transform

### Tom 173 / 2006

Studia Mathematica 173 (2006), 149-166 MSC: 47B48, 47B47, 42B10. DOI: 10.4064/sm173-2-3

#### Streszczenie

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 $\mathbb R^{2n}$.

#### Autorzy

• Miloš ArsenovićMatematički Fakultet
Studentski trg 16–18
11000 Beograd, Serbia and Montenegro
e-mail
• Dragoljub KečkićMatematički Fakultet