Let $T$ be a self-adjoint invertible operator on $\mathcal{H}$ with a continuous spectrum, means the spectral measure is nonatomic. For which class of invertible operators $V$( with continuous spectrum) other than unitary $VTV^{\*}$ also have continuous spectrum?



Invariance of spectrum under conjugation
In any infinite-dimensional Hilbert space, the only such operators $V$ that work for all $T$ are scalar multiples of the identity.


Suppose $V$ is not a scalar multiple of a unitary. Then there are linearly independent vectors $v$, $w$ such that $V^\* V v = w$.
Let $T$ be a self-adjoint invertible operator such that $Tw = v$. Then $VTV^\* (Vv) = V T w = V v$, so $V T V^\*$ does not have continuous spectrum.
All that remains is to show that such $T$ can be chosen with continuous spectrum, which is not difficult.


