We study relations between spectra of two operators that are connected to
each other through some intertwining conditions. As application we obtain new
results on the spectra of multiplication operators on B(\cl H) relating it to
the spectra of the restriction of the operators to the ideal C2β of
Hilbert-Schmidt operators. We also solve one of the problems, posed in
[B.Magajna, Proc. Amer. Math. Soc, 141 2013, 1349-1360] about the positivity of
the spectrum of multiplication operators with positive operator coefficients
when the coefficients on one side commute. Using the Wiener-Pitt phenomena we
show that the spectrum of a multiplication operator with normal coefficients
satisfying the Haagerup condition might be strictly larger than the spectrum of
its restriction to C2β.Comment: 12 pages, v2: corrected some typos, to appear in Methods of Funct.
Anal. Topo