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we had question in our end-semester exam. State true or false with explanation. If U and T are normal operator which commute with each other then U+T is normal.

On inner product space. It is not given that U and T are operators on finite dimensional inner product space.

The statement does not hold for infinite dimensional inner product space.(as told by our instructor)

I cannot think for counter example for this statement. i.e. two operators s.t. TT*=TT , UU*=U*U and UT=TU but U+T is not normal. Also I Ddon't know how to construct T$^*$ given T on infinite dimensional inner product space.

here T* is adjoint operator. i.e. $<T(x),y>$ = $<x,T^*(y)>$

Help needed

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  • $\begingroup$ Please read this before posting: math.meta.stackexchange.com/questions/5020/… to better format your questions in future. I fixed the formatting of this first question, but you should not expect others to do this for you! $\endgroup$ – 5xum Nov 14 '18 at 8:30
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If $U$ and $T$ are normal operators on a Hilbert space, then, by a famous theorem of Fuglede:

$UT=TU$ implies $UT^*=T^*U$ and $TU^*=U^*T$.

Therefore we have that $U+T$ is normal.

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