Abstract
This survey article (also containing a few new results) is devoted to a consideration of various properties of some well known classes of operators, such as decomposable operators, generalized scalar operators, well-bounded operators and AC-operators, for example, when attention is restricted to the particular (but important) setting of Fourier $p$-multiplier operators on the real line group. The aim is to highlight some relevant phenomena and to pose several natural questions which occur in this setting.
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