The operator theoretica l approach has the advantage of a compact presenta tion of results simultaneously for wide classes of diffraction problems and space settings and gives a different and deeper understanding of the solut ion procedures. \;

Th e main objective is to demonstrate how diffraction problems guide us to operator factorisation conce pts and how useful those are to develop and to sim plify the reasoning in the applications. \;

In eight widely independent sections we shall address the following questions: \;

How can we consider the classical Wiener-Hopf procedure as an operator fa ctorisation (OF) and what is the profit of that in terpretation? \; \; \; \;&nb sp\; \; \; \;

What are the characteristics of Wiener-Hopf operat ors occurring in Sommerfeld half-plane problems an d their features in terms of functional analysis?

What are the most relevant metho ds of constructive matrix factorisation in Sommerf eld problems?

How does OF appear generally in linear boundary value and transmissio n problems and why is it useful to think about thi s question? \; \; \; \; \;

What are adequate choic es of function(al) spaces and symbol classes in or der to analyse the well-posedness of problems and to use deeper results of factorisation theory?

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Where do we need other kinds of operator rel ations beyond OF?

What are very p ractical examples for the use of the preceding ide as\, e.g.\, in higher dimensional diffraction prob lems? \;

Historical remarks and corresponding references are provided at the end of each section.

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