Invertible opeartors and diagonal operators

Question

Let $T$ be an invertible operator on an infinite dimensional Hilbert space $H$. Is there any diagonal Operator $N$ on $H$ and a unitary operator $W$ such that $WN=T$?

I know there is diagonal operator $I$ and invertible operator $T$ such that $T(I)=T$. But what about unitary equivalence?

Answer 1

This is not even true on finite-dimensional spaces. Then $T=WN$ implies that $T$ has orthogonal columns.