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Erschienen in:
2018 | OriginalPaper | Buchkapitel
Spectral regularity of a C*-algebra generated by two-dimensional singular integral operators
verfasst von : Harm Bart, Torsten Ehrhardt, Bernd Silbermann
Erschienen in: The Diversity and Beauty of Applied Operator Theory
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Given a bounded simply connected domain $${U} \subset {\mathbb{C}}$$ having a Lyapunov curve as its boundary, let $$\mathcal{L}({L}^{2}(U))$$ stand for the $$(\mathbb{c}^\ast)$$ -algebra of all bounded linear operators acting on the Hilbert space $$\mathcal{L}^{2}(U)$$ with Lebesgue area measure. We show that the smallest C*-subalgebra $$\mathcal{A}$$ of $$\mathcal{L}({L}^{2}(U))$$ containing the singular integral operator $$(S_Uf)(z)\;=\;-\frac{1}{\pi}{\int\limits_{U}}\frac{f(w)}{(z-w)^2}dA(w),$$ along with its adjoint $$(S^*_Uf)(z)\;=\;-\frac{1}{\pi}{\int\limits_{U}}\frac{f(w)}{(z-w)^2}dA(w)$$ all multiplication operators $$aI, a \in\; C(\overline{U})$$ , and all compact operators on $$\mathcal{L}^{2}(U)$$ , is spectrally regular. Roughly speaking the latter means the following: if the contour integral of the logarithmic derivative of an analytic $$\mathcal{A}$$ -valued function f is vanishing (or is quasi-nilpotent), then f takes invertible values on the inner domain of the contour in question.