Shamir–Duduchava factorization of elliptic symbols

This paper considers the factorization of elliptic symbols which can be represented by matrix-valued functions. Our starting point is a Fundamental Factorization Theorem, due to Budjanu and Gohberg [2]. We critically examine the work of Shamir [15], together with some corrections and improvements as proposed by Duduchava [6]. As an integral part of this work, we give a new and detailed proof that certain sub-algebras of the Wiener algebra on the real line satisfy a sufficient condition for a right standard factorization. Moreover, assuming only the Fundamental Factorization Theorem, we provide a complete proof of an important result from Shargorodsky [16], on the factorization of an elliptic homogeneous matrixvalued function, useful in the context of the inversion of elliptic systems of multidimensional singular integral operators in a half-space.