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?
2026-04-12 17:06:36.1776013596
Can $AB-BA=I$ hold if $A$ and $B$ are bounded linear operators on a Banach space?
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