Abstract
The paper deals with a local study of the Banach algebra A [SO, PC] generated by the convolution type operators W a b = a F −1bF with data a E ∈ [SO,PC] n× n and b∈ [SO p ,PC] p n× n which act on the Lebesgue space L np (ℝ) (1 < p < ∞, n ≥ 1). Here [SO,PC] n× n means the C*-algebra generated by slowly oscillating (SO)and piecewise continuous(PC) n × n matrix functions, and SOBA11020200787 is a Fourier multiplier analogue onL p (ℝ)of [SO,PC] n× n The work is based on the study of Fourier multiplier analogue SO p ofSOon the characterization of the multiplicative linear functionals of slowly oscillating functions, and on the compactness of the commutators AW a, b — W a, b A, where A ∈ A PC] a ∈ SOandb ∈ SO p. Making use of the Allan-Douglas local principle we construct homomorphisms of A [SO PC]onto local Banach algebras and establish a Fredholm criterion for operators A ∈ A [SO PC] in terms of the invertibility of their images in the local algebras.
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Bastos, M.A., Bravo, A., Karlovich, Y.I. (2004). Convolution Type Operators with Symbols Generated by Slowly Oscillating and Piecewise Continuous Matrix Functions. In: Gohberg, I., Wendland, W., Ferreira dos Santos, A., Speck, FO., Teixeira, F.S. (eds) Operator Theoretical Methods and Applications to Mathematical Physics. Operator Theory: Advances and Applications, vol 147. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7926-2_24
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