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
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.