Logarithmic residues, Rouché's theorem, and spectral regularity: The \(C^{\ast}\)-algebra case
The paper extends the theory of analytic Fredholm operator valued functions to \(C^*\)-algebras. An element \(a\) of a unital \(C^*\)-algebra \(\mathcal B\) is called Fredholm if it is invertible modulo elements in \(\mathcal C(\mathcal B)\), where \(\mathcal C(\mathcal B)\) denotes the closure of the ideal consisting of all finite rank elements of \(\mathcal B\). The authors review and extend the results on Fredholm elements in [\textit{R. Hagen}, \textit{S. Roch} and \textit{B. Silbermann}, \(C^*\)-algebras and numerical analysis. New York, NY: Marcel Dekker (2001; Zbl 0964.65055)]. In the case where \(\mathcal B\) is \(L(H)\) for some Hilbert space \(H\), the aforementioned theory reduces to the well-known theory of Fredholm operators. The authors investigate Fredholm valued analytic functions \(f:D\to \mathcal B\), where \(D\) is a non-empty connected open subset of \(\mathbb C\). If \(f(\lambda)\) is invertible for at least one \(\lambda\in D\), then the set of all \(\lambda\in D\) where \(f\) is not invertible has no accumulation point in \(D\). At such points, the principal part of the Laurent expansion of \(f\) is finite and its coefficients are finite rank elements of \(\mathcal B\). For a bounded Cauchy domain \(\Delta\subset \overline \Delta \subset D\), it is shown that the logarithmic residue \(\frac{1}{2\pi \mathrm{i}} \int_{\partial \Delta} f'(\lambda) f(\lambda)^{-1}\mathrm{d}\lambda\) is a finite sum of finite rank idempotents in \(\mathcal B\), given that \(f\) is invertible on \(\partial \Delta\). The authors proceed to prove the following analogue of Rouché's theorem: If \(g\) is a \(\mathcal B\)-valued function on \(D\) and \(\max_{\lambda\in\partial \Delta} \left\| \left( g(\lambda) - f(\lambda) \right) f(\lambda)^{-1} \right\| < 1\), then \(g\) is a Fredholm function on \(\Delta\), it is invertible on \(\partial \Delta\) and the trace of the logarithmic residues of \(f\) and \(g\) are equal. The paper concludes with some examples; the example of \(C^*\)-algebras generated by one non-unitary isometry is treated in detail.
abstract analytic Fredholm valued function
scientific article; zbMATH DE number 6112934
scientific article