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Let $H$ be a separable Hilbert space. Let $a,b: H\to H$ be bounded linear operators.

$a$ and $b$ are called orthogonal, if $a^*b=ab^*=0$. It is easy to see that this means that the support and image spaces of such operators are orthogonal, hence $\|a+b\|=\max(\|a\|,\|b\|)$.

My question is, how can one prove that "almost orthogononality" implies this equation approximately? More precisely, I would like to prove the following statement.

For all $\delta>0$, there exists $\eta>0$ with the following property: Whenever $a,b: H\to H$ are bounded operators such that $\|a\|,\|b\|\leq 1$ and $\|a^*b\|,\|ab^*\|\leq\eta$, then $$\|a+b\|\leq\delta+\max(\|a\|,\|b\|).$$

I have often heard of this statement before, but until now I have never seen a proof. I would suspect that the proof has to be pretty elementary, but somehow every one of my attempts fails. I would be very thankful, if someone could show me how to do it or give me a reference, since this statement would come in very handy for my own research.

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The most general result involving norms of sums of almost orthogonal operators is the Cotlar-Stein almost-orthogonality lemma, which can be applied to your problem almost directly.

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