Finite sums of idempotents and logarithmic residues on connected domains (Q1891857)

Let \(D\) be a bounded Cauchy domain in the complex plane \(\mathbb{C}\). Moreover let \(f\) be an analytic Banach algebra valued function which is defined on an open neighbourhood of \(\overline {D}\) and invertible on \(\partial D\). We know, that if the logarithmic residue of a complex-valued function with respect to \(D\) vanishes, that is \({1\over {2\pi i}} \int-{\partial D} {{f' (t)} \over {f(t)}} dt=0\), then \(f= f(z)\) has no zeros on \(D\) (compare with argument principle). An other interpretation of this result is that \(f\) takes only invertible values on \(D\). There are known generalizations of this result, for example, to functions \(f\) having values in a commutative Banach algebra or whose values are Fredholm operators. On the other hand there exist counterexamples basing on non-connected domains of definition for \(f\). In the present paper the author constructs counterexamples for connected Cauchy domains. For this reason he proves the following Theorem. Let \(D\) be an bounded and connected Cauchy domain in the complex plane \(\mathbb{C}\) and let \(t_1, \dots, t_n\) be distinct points. Further suppose that \(p_1, \dots, p_n\) are non-zero idempotents in a Banach algebra \(A\). Then there exists an analytic \(A\)-valued function \(f\) defined on \(\mathbb{C}\) which is not invertible and \(t_1, \dots, t_n\) and whose left and right logarithmic residues are the sum of the given idempotents: \[ {\textstyle {1\over {2\pi i}}} \int_{\partial D} f'(t) f^{-1} (t) dt= {\textstyle {1\over {2\pi i}}} \int_{\partial D} f^{-1} (t) f'(t) dt= \sum_{k=1}^n p_k. \]