Let $A$ and $B$ be bounded linear operators on a Hilbert space $H$. Suppose $AB$ is compact.
Must $A$ or $B$ be compact?
I suppose that the answer is "no", but I cannot find any counterexample.
Must $A$ or $B$ be compact if $AB$ is compact?
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functional-analysis
1 Answers
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Find an example where $A$ and $B$ are infinite-dimensional projections (and thus not compact) but $AB = 0$.
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0sweet !!!!!!!!!!! – 2017-01-23
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0Elegant example. – 2017-01-23
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0Thank you! It is a really nice example) – 2017-01-23