Solving Linear Operator Equations

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Abstract
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).

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