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Let $T$ be a multiplication operator on $L^2([a, b ])$. Find necessary and sufficient conditions for $T$ to be a projection. let g be a fixed function in $L^2([a,b])$, and $T(f(x))=g(x)f(x)$.

Mnifldz
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saeid sahragard
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1 Answers1

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T is a projection, iff $T^2 = T$. So $g^2(x)f(x) = g(x)f(x)$ for each $f\in L^2([a,b])$. Thus it must be $g^2(x)-g(x)=0$ (which means $g(x)\in\{0,1\}$) almost everywhere.

Stephan Kulla
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  • Shouldn't the first sentence be "iff $T^2 = T$ and $T = T^*$? If an idempotent operator is not self-adjoint, then I don't think it's a projection. – william_grisaitis Mar 24 '23 at 17:13
  • @william_grisaitis Due to https://en.wikipedia.org/wiki/Projection_(linear_algebra) you are referring to "orthogonal projections" – Stephan Kulla Apr 17 '23 at 20:50