Symbol calculus and Fredholmness for a Banach algebra of convolution type operators with slowly oscillating and piecewise continuous data
✍ Scribed by Maria Amélia Bastos; António Bravo; Yuri Karlovich
- Publisher
- John Wiley and Sons
- Year
- 2004
- Tongue
- English
- Weight
- 358 KB
- Volume
- 269-270
- Category
- Article
- ISSN
- 0025-584X
No coin nor oath required. For personal study only.
✦ Synopsis
Abstract
A symbol calculus for the smallest Banach subalgebra 𝒜~[SO,PC]~ of the Banach algebra ℬ︁(L^n^~p~(ℝ)) of all bounded linear operators on the Lebesgue spaces L^n^~p~(ℝ) (1 < p < ∞, n ≥ 1) which contains all the convolution type operators W~a,b~ = __a__ℱ^−1^__b__ℱ with a ∈ [SO, PC]^n×n^ and b ∈ [SO~p~, PC~p~]^n×n^ is constructed. Here [SO, PC]^n×n^ means the C*‐algebra generated by all slowly oscillating (SO) and all piecewise continuous (PC) n × n matrix functions, and [SO~p~, PC~p~]^n×n^ is a Fourier multiplier analogue of [SO, PC]^n×n^ on L~p~(ℝ). As a result, a Fredholm criterion for the operators A ∈ 𝒜~[SO,PC]~ is established. The study is based on the compactness of the commutators AW~a,b~ − W~a,b~A where A ∈ 𝒜~[SO,PC]~, a ∈ SO, and b ∈ SO~p~, on the Allan‐Douglas local principle, and on the two projections theorem. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)