Geodesic
Solving Linear Operator Equations
Canadian journal of mathematics, Tome 26 (1974) no. 6, pp. 1384-1389

Voir la notice de l'article provenant de la source Cambridge University Press

Let be a complex Banach space and the algebra of bounded operators on . M. Rosenblum's theorem [13; 12] (also discovered by M. G. Kreĭn, cf. [9]) states that (if A, B are fixed bounded operators) the spectrum of the operator on defined by = AX – XB is contained in σ (A) – σ(B) = {α – β : α∊σ(A), β∊σ(B)}. In particular, the condition σ(A) ∩ σ(B) = Ø implies that for each Y ∊ there is a unique X ∊ such that AX – XB = Y. This does not completely settle the question of solvability of the equation AX – XB = Y: for example, if A is the backward unilateral shift and B = 0, then the equation has a solution (for any Y) even though σ(B) ⊆ σ(A). 
Davis, Chandler; Rosenthal, Peter. Solving Linear Operator Equations. Canadian journal of mathematics, Tome 26 (1974) no. 6, pp. 1384-1389. doi: 10.4153/CJM-1974-132-6
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