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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
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