13

Prove that $\sigma(AB) \backslash \{0\} = \sigma(BA)\backslash \{0\} $. Where $A,\ B$ are bounded operators in Banach space and $\sigma$ denotes spectrum.

t.b.
  • 78,116
Philipp G. Sinicyn
  • 891

1 Answers1

24

This is a classic (I learned this from Appendix A, A.1 of G.K. Pedersen, $C^{\ast}$-algebras and their automorphism groups, see also exercise 4.1.3 in his Analysis Now — thanks, w(ild)3life):

Let $\mathcal A$ be a $\mathbb{C}$-algebra with a unit (here $\mathcal{A}$ is the algebra of bounded linear operators on your Banach space). Then $\sigma(AB) \smallsetminus \{0\} = \sigma(BA) \smallsetminus \{0\}$.

If $\lambda \notin \sigma(AB) \cup \{0\}$ then there is $C$ such that $$ C(\lambda - AB) = 1 = (\lambda-AB)C. $$ Then verify that $\lambda^{-1}(1 + BCA)$ is the inverse of $(\lambda-BA)$ so that $\lambda \notin \sigma(BA) \cup \{0\}$: $$ (1 + BCA)(\lambda-BA) = \lambda = (\lambda-BA)(1+BCA), $$ as a straightforward computation shows.

Later:

There's a nice mnemonic on how to guess the inverse, also addressed in-depth in this MO-thread by Bill Dubuque:

Recall the geometric series $(1-q)^{-1} = 1 + q + q^2 + \cdots$, so formally $$\begin{align*} (1-BA)^{-1} &= 1+ BA + BABA + BABABA + \cdots \\ &= 1 + B(1+ AB+ ABAB + \cdots)A \\ &= 1 + B(1-AB)^{-1}A \\ & = 1+BCA \end{align*}$$ with $C = (1-AB)^{-1}$. Similarly with $(\lambda-AB)^{-1}$.

Community
  • 1
t.b.
  • 78,116
  • I'm pretty sure I've seen this problem here before, but I couldn't find it. – t.b. Nov 05 '11 at 16:07
  • 2
    You may also be interested in having a look at this MO thread started by Bill Dubuque. – t.b. Nov 05 '11 at 16:08
  • Is a $\mathbb C = C^*$? Never seen that notation. – JT_NL Nov 05 '11 at 16:56
  • 2
    @Jonas: A $\mathbb{C}$-algebra is an algebra over the field of complex numbers (as in "$k$-algebra over a commutative ring $k$"): a complex vector space equipped with a bilinear and associative multiplication. The spectrum of an element of a unital $\mathbb{C}$-algebra is $$\sigma(A) = {\lambda \in \mathbb{C} : \lambda - A \text{ is not invertible}}.$$ – t.b. Nov 05 '11 at 16:59
  • How to prove that $\lambda^{-1}(\lambda - BCA)$ is the inverse of $\lambda - BA$ – Philipp G. Sinicyn Nov 05 '11 at 17:07
  • @Philipp: Sorry, there were some silly typos $$(1+BCA)(\lambda-BA) = \lambda - BA + B[C(\lambda - AB)]A = \lambda$$ and similarly for the other side. – t.b. Nov 05 '11 at 17:24
  • 3
    This is also exercise E.4.1.3 (page 133) of Pederson's Anaylsis NOW (his hint is the answer). – wildildildlife Nov 06 '11 at 00:58
  • @wildildildlife: thanks, I forgot about that, so maybe I learned it there... Anyway, I learned it from Pedersen :) – t.b. Nov 06 '11 at 01:03