Distribution of spectrum in a direct sum decomposition of operators into normal and completely non-normal parts

Abstract

We discuss the distribution of spectra of a direct sum decomposition of an arbitrary operator into normal and completely non normal parts. We utilize the fact that any given operator 𝑇∈𝐵(𝐻) can be decomposed into a direct summand 𝑇=𝑇1⊕𝑇2 with 𝑇1 and 𝑇2 are the normal and completely non normal parts respectively. This canonical decomposition is preferred to other forms of decomposition such as Polar and Cartesian decompositions because these two do not transfer certain properties (for instance the spectra, numerical range, and numerical radius) from the original /decomposed operator to the constituent parts. This is presumably done since these parts are simpler to deal with.