Let $H$ be a infinite dimensional Hilbert space and $T: H \to H $ be a bounded normal star cyclic operator with star cyclic vector $e$ ( $T$ will be called star cyclic with star cyclic vector $e$ if $\overline{ span\{T^m T^{*n} e : m,n \in \mathbb N_0\} } =H$ ), then prove or disprove $||TUe||=||Te||$ for any unitary operator $U:H \to H$.
Please help me in this problem, I have counter example in finite dimensional case but not in infinite dimensional case. Either help me proving this or disprove this by giving a counter example.
Thanks in advanced