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It's easy to see it can't be true in finite dimension. Also, it can hold for operators on infinite-dimensional vector spaces over $\mathbb C$, as seen here: Can $AB-BA=I$ hold if both $A$ and $B$ are operators on an infinitely-dimensional vector space over $\mathbb C$?. What happens if both $A$ and $B$ are bounded linear operators on a Banach space?

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adrija
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    the point is that it cannot happen for continuous or, equivalently, bounded linear operators. – Thomas Sep 09 '15 at 20:03
  • what's wrong with the example in the answer of the link you gave in the question? –  Sep 10 '15 at 06:23
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    You can find very nice answers here: http://math.stackexchange.com/questions/54397/the-identity-cannot-be-a-commutator-in-a-banach-algebra – Etienne Sep 15 '15 at 15:25

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