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Spectral properties of the operator equation $$AX+XB=Y$$. (English) Zbl 0826.47013
The main purpose of this paper is to study the operator $$\tau_{A, B}= AX+ XB$$ acting on the space $$\mathcal L$$ of bounded linear operators $$X$$. The spectral equality $$\sigma(\tau_{A, B})= \sigma(A)+ \sigma(B)$$ has been proved in three cases:
1. $$A$$ and $$B$$ generate eventually norm continuous $$C_0$$- semigroups.
2. One of the operators is bounded;
3. $$A$$ and $$B$$ generate $$C_0$$-semigroups one of which is holomorphic.
For commuting operators $$A$$, $$B$$ if one of them is bounded it has been proved that $$\sigma(A+ B)\subset \sigma(A)+ \sigma(B)$$.

##### MSC:
 47A62 Equations involving linear operators, with operator unknowns 47A10 Spectrum, resolvent 47D06 One-parameter semigroups and linear evolution equations
##### Keywords:
$$C_ 0$$-semigroups
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