J.C Bourin has established the pinching theorem. The theroem is as following: Let $A\in B(H)$ be an operator with $\Bbb D \subset W_e(A)$,where $\Bbb D$ is the open unit disc, $W_e(A)$ is the essential numerical range of $A$ and let $\{X_n\}_{n=1}^{\infty}$ be a sequence of operators such that $\sup_n||X_n||<1$. Then we have a pinching
$P(A)=\oplus_{n=1}^{\infty}X_n$.
There is one corollary.
Cor:Let $A$ be an operator with $\Bbb D \subset W_e(A)$.For any contraction $X$, there is a sequence $\{U_n\}$ of unitary operators such that $U_n^{*}AU_n\rightarrow X$ in the weak operator topology.
How to construct the sequence of unitary operators?
According to the pinching thereom, there exists a standard decomposition of $H$ for which $A=(A_{i,j})$ with diagonal blocks $A_{i,i}\cong X_i $ for all $i$, where $A_{i,i}\cong X_i $ means that $A_{i,i}$ is unitarily equivalent to $X_i$. If we let $X_n=X$, we can conclude that $A_{i,i}\cong X$, but how to prove that there is a sequence $\{U_n\}$ of unitary operators such that $U_n^{*}AU_n\rightarrow X$ in the weak operator topology?
By replacing $X$ with $\frac n{n+1}X$ you may assume that $\|X\|<1$. By the Theorem, there exists a projection $E$ with $X=EAE$. Now you can use that the unitaries are wot-dense in the unit ball.