Commutativity of spectrum on algebras with unit

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Let \mathcal A be a \mathbb C-algebra with a unit, where \mathcal A is the algebra of bounded linear operators on a Banach space. Then \sigma(ab) \setminus \{0\} = \sigma(ba) \setminus \{0\}.

Definition: The spectrum of an element a \in A is the set \sigma(a) = \{\lambda \in \mathbb C | a - \lambda \not \in A^{-1}\}

where a^{-1} \subset a is the set of invertible elements.

For matrices, we obtain the eigenvalue set. For continuous functions, we obtain the image.

If \lambda \not \in \sigma(ab) \cup \{0\} then there is a c s.t. c(\lambda - ab) = 1 = (\lambda - ab)c.

Then one can verify that \lambda^{-1}(1 + bca) is the inverse of (\lambda - ba) so that \lambda \not \in \sigma(ba) \cup \{0\}:

(1 + bca)(\lambda - ba) = \lambda = (\lambda - ba)(1 + bca).

This inverse can be guessed by an analogy with the geometric series (1 - x)^{-1} = 1 + x + x^2 + \cdots as explained here.

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