Recall that the Calkin algebra, is the quotient $B(H)/B\_0(H)$, where $H$ is a Hilbert space and $B(H)$ and $B\_0(H)$ are the algebra of bounded and compact operators on $H$. 


Let $H$ be separable and $Q:B(H)\rightarrow B(H)/B\_0(H)$ be a natural quotient map. Is it true for quotient norm that 
$$\|Q(T)\|=\lim\_n\|T(I-P\_n)\|$$
where $P\_n$ is the projection on the span of the first $n$ elements in an orthonarmal basis. 



About the quotient norm in the Calkin algebra
Yes, because $(P\_n)$ is an approximate unit for the ideal of compact operators. If $C$ is any compact operator then $$\|T(I - P\_n)\| \leq \|(T-C)(I-P\_n)\| + \|C(I-P\_n)\| \leq \|T - C\| + \|C - CP\_n\|,$$ and $\|C - CP\_n\| \to 0$ as $n \to \infty$, so the norm in the Calkin algebra is achieved by taking the limit $\lim \|T(1 - P\_n)\|$.


