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I am taking a course on operator theory this semester and this question was left as an exercise in the course. I am afraid that I will not be able to solve this question by myself.

Let H be a Hilbert Space. Show that if T is an isometry , then $Im(T)= Ker(I-T^{*}T)$ and also that $(I-TT^{*})$ is the orthogonal projection on $Ker(T^{*})$.

Now, I tried taking an element in each $Im(T)$ and $Ker(I-T^{*}T) $ and then tried showing that it exists in latter and I failed in both.

Where should I use the property that $T$ is an isometry and how?

Unfortunately, I could not make much progress on the 2nd part also.

Do you mind offering some help?

1 Answers1

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If $T$ is an isometry then $T^*T=I,$ and $\operatorname{im}(T)$ is not equal to $\ker(I-T^*T)$ but to $\ker(I-TT^*)$:

  • $\operatorname{im}(T)\subset\ker(I-TT^*)$ because $(I-TT^*)T=T(I-T^*T)=T0=0.$
  • $\ker(I-TT^*)\subset\operatorname{im}(T)$ because if $(I-TT^*)x=0$ then $x=TT^*(x)\in\operatorname{im}(T).$

$P:=TT^*$ is an orthogonal projection (since $P^*=P$ and $P^2=P$), hence $I-P$ is the orthogonal projection onto $\operatorname{im}(I-P)=(\ker(I-P))^\bot=$$(\operatorname{im}T)^\bot=\ker(T^*).$

Anne Bauval
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