Advertisement
Published in:
2018 | OriginalPaper | Chapter
Spectral regularity of a C*-algebra generated by two-dimensional singular integral operators
Authors : Harm Bart, Torsten Ehrhardt, Bernd Silbermann
Published in: The Diversity and Beauty of Applied Operator Theory
Publisher: Springer International Publishing
Activate our intelligent search to find suitable subject content or patents.
Select sections of text to find matching patents with Artificial Intelligence. powered by
Select sections of text to find additional relevant content using AI-assisted search. powered by
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.