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The classical Rouché Theorem from complex function theory has been generalized to a result involving Fredholm operator valued functions (see [9] and [6]). The setting in question is (generally) a non-commutative one, so actually there are two versions of the generalization. In this paper, a unifying enhanced variant is presented involving the notion of the spectral radius. Its proof is based on a maximum principle for the spectral radius. An example is given showing that the enhanced Rouché Theorem presented here is a genuine improvement over the earlier version of the result. [ABSTRACT FROM AUTHOR]

Abstract: Using families of irreducible Hilbert space representations as a tool, the theory of analytic Fredholm operator valued function is extended to a -algebra setting. This includes a -algebra version of Rouché’s Theorem known from complex function theory. Also, criteria for spectral regularity of -algebras are developed. One of those, involving the (generalized) Calkin algebra, is applied to -algebras generated by a non-unitary isometry. [Copyright &y& Elsevier]

Abstract: A new class of Banach algebra valued functions is identified for which the logarithmic residue with respect to a Cauchy domain vanishes (if and) only if the functions take invertible values in . Trace conditions and the extraction of elementary factors of the type play an important role. The class contains the Fredholm operator valued functions and the Banach algebra valued functions possessing a simply meromorphic resolvent as special instances. An example is given to show that new ground is covered and a long standing open problem is discussed from a fresh angle. [Copyright &y& Elsevier]