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In Biberdorf, E. A. and Väth, M., On the spectrum of orthomorphisms and Barbashin operators, Z. Anal. Anwendungen 18, 1999(4), 12-31 it is shown that that even in the more general case of an orthomorphism, the essential spectrum (for various definitions of “essential spectrum”) is the same as the spectrum and coincides with the “essential range” of the operator (for an appropriate definition of ”essential range”). Unsurprisingly, in case of a multiplication operator, the essential range of the operator is the essential range of the mulitplication function.