Distance between Unilateral shift and invertible operators.

I want to prove that the distance between unilateral shift and normal operators is $1$. But I need to prove that $d(S,\operatorname{Inv}(L(H))= 1$, where $H$ is a Hilbert space.

Does anyone have any ideas or hints?

T.Y

Solution 1:

Let $T$ be any surjective operator. Let $\{e_1,e_2,\ldots\}$ be an orthonormal basis and $S$ the unilateral shift for that basis. Then $e_1$ is orthogonal to the range of $S$. As $T$ is surjective, there exists $y\in H$ with $Ty=e_1$. So \begin{align} \|(S-T)y\|^2&=\langle Sy,Sy\rangle +\langle Ty,Ty\rangle-2\text{Re}\,\langle Sy,Ty\rangle\\[0.3cm] &=\|Sy\|^2+\|e_1\|^2-2\text{Re}\,\langle Sy,e_1\rangle\\[0.3cm] &=\|y\|^2+1. \end{align} Then $$ \frac {\|(S-T)y\|}{\|y\|}=\sqrt {1+\frac1 {\|y\|^2}}. $$ This shows that $\|S-T\|>1$.

Taking $T$ to be an arbitrarily small scalar multiple of the identity we can get the norm of $S-T $ close to $1$ as we want. So the distance from $S$ to the surjective operators is $1$.