Let $${\mathcal B}_{p,w}$$ B p , w be the Banach algebra of all bounded linear operators acting on the weighted Lebesgue space $$L^p(\mathbb {R},w)$$ L p ( R , w ) , where $$p\in (1,\infty )$$ p ∈ ( 1 , ∞ ) and w is a Muckenhoupt weight. We study the Banach subalgebra $$\mathfrak {A}_{p,w}$$ A p , w of $${\mathcal B}_{p,w}$$ B p , w generated by all multiplication operators aI ( $$a\in \mathrm{PSO}^\diamond $$ a ∈ PSO ⋄ ) and all convolution operators $$W^0(b)$$ W 0 ( b ) ( $$b\in \mathrm{PSO}_{p,w}^\diamond $$ b ∈ PSO p , w ⋄ ), where $$\mathrm{PSO}^\diamond \subset L^\infty (\mathbb {R})$$ PSO ⋄ ⊂ L ∞ ( R ) and $$\mathrm{PSO}_{p,w}^\diamond \subset M_{p,w}$$ PSO p , w ⋄ ⊂ M p , w are algebras of piecewise slowly oscillating functions that admit piecewise slowly oscillating discontinuities at arbitrary points of $$\mathbb {R}\cup \{\infty \}$$ R ∪ { ∞ } , and $$M_{p,w}$$ M p , w is the Banach algebra of Fourier multipliers on $$L^p(\mathbb {R},w)$$ L p ( R , w ) . For any Muckenhoupt weight w, we study the Fredholmness in the Banach algebra $${\mathcal Z}_{p,w}\subset \mathfrak {A}_{p,w}$$ Z p , w ⊂ A p , w generated by the operators $$aW^0(b)$$ a W 0 ( b ) with slowly oscillating data $$a\in \mathrm{SO}^\diamond $$ a ∈ SO ⋄ and $$b\in \mathrm{SO}^\diamond _{p,w}$$ b ∈ SO p , w ⋄ . Then, under some condition on the weight w, we complete constructing a Fredholm symbol calculus for the Banach algebra $$\mathfrak {A}_{p,w}$$ A p , w in comparison with Karlovich and Loreto Hernández (Integr. Equations Oper. Theory 74:377–415, 2012) and Karlovich and Loreto Hernández (Integr. Equations Oper. Theory 75:49–86, 2013) and establish a Fredholm criterion for the operators $$A\in \mathfrak {A}_{p,w}$$ A ∈ A p , w in terms of their symbols. A new approach to determine local spectra is found.
Mediterranean Journal of Mathematics – Springer Journals
Published: Aug 3, 2017
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