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This note contains reflections on a type of factorization that came up in the work of analytic operator valued functions by I.C. Gohberg and E.I. Sigal (cf. [14] , [10] , [11]). The determining characteristic of the factorizations in question is that they are obtained by the extraction of elementary factors of the form I − P + λ P where P is an idempotent in the underlying Banach algebra. It will be demonstrated here that there is a fundamental difference between the finite dimensional (matrix) case and the situation where infinite dimensional Banach algebras are taken into consideration. [ABSTRACT FROM AUTHOR]

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]

We give another proof of the Szeg\H{o}–Widom Limit Theorem. This proof relies on a new Banach algebra method that can be directly applied to the asymptotic computation of the Toeplitz determinants. As a by-product, we establish an interesting identity for operator determinants of Toeplitz operators, namely if $a\_1,\backslash dots,a\_R$ are certain matrix valued functions defined on the unit circle, then $\backslash det\; \backslash B(e^\{T(a\_1)\}\backslash cdots\; e^\{T(a\_R)\}\; T(e^\{-a\_R\}\backslash cdots\; e^\{-a\_1\})\backslash B)\; =\backslash det\backslash B(\; T(e^\{\backslash ta\_1\}\backslash cdots\; e^\{\backslash ta\_R\})\; e^\{-T(\backslash ta\_R)\}\backslash cdots\; e^\{-T(\backslash ta\_1)\}\backslash B)$ where $\backslash ta\_r(e^\{i\backslash theta\})\backslash eg\; a\_r(e^\{-i\backslash theta\})$ [ABSTRACT FROM AUTHOR]