If U and T are normal operator which commute with each other then U+T is normal

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. $$$$ = $$$$

Help needed

• 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! – 5xum Nov 14 '18 at 8:30

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.