Associated with $T=U|T|$ (polar decomposition) in ${\cal{L}}({\bf H})$ is a related operator $\skew3\tilde{T} = |T|^{\frac{1}{2}}U|T|^{\frac{1}{2}}$, called the Aluthge transform of $T$. In this paper we study some connections between $T$ and $\skew3\tilde{T}$, including the following relations; the single valued extension property, an analogue of the single valued extension property on $W^{m}(D, {\bf H})$, Dunford's property $(C)$ and the property $(\beta)$.