Corollay of Theorem 4.1.3 from Arveson's Book

Question

from book A Short Course on Spectral Theory , William Arveson

Corollary of Theorem 4.1.3 says

Every Toeplitz operator $T_\phi,\phi \in L^\infty$, satisfies

$\displaystyle\inf\{\|T_{\phi}+K\| \colon K\in \mathcal{K} \}=\|T_{\phi}\|=\|\phi\|_\infty.$

if $P_n=S^n S^{*n}$. don't know how deduce

$\|P_n(A+K)P_n\|=\|S^{*n}(A+K)S^n\|$

thanks

Answer 1

The shift $S$ is an isometry; algebraically, that's $S^*S=I$. So, for any $T$, $$ \|ST\|^2=\|T^*S^*ST\|=\|T^*T\|=\|T\|^2. $$ And $$ \|TS^*\|^2=\|TS^*ST^*\|=\|TT^*\|=\|T^*\|^2=\|T\|^2. $$ Now apply each $n$ times.