# Questions regarding essentially normal operators in BDF theory

I am reading the paper "UNITARY EQUIVALENCE MODULO THE COMPACT OPERATORS AND EXTENSIONS OF C*-ALGEBRAS" by BDF and I seem to have some misunderstandings.

In the followng I don't understand why in the last line it says that every operator in $$E$$ is essentially normal? I thought the decision if an operator is esentially normal is either if $$\pi(T)$$ is normal in the Calkin algebra (with $$\pi$$ being the quotient map) or $$T^{*}T-TT^{*}$$ is compact, but how does $$\phi$$ tell us anything about the essential normal property?

$$\space$$

Also the following part confuses me. It says in the end that $$S-S'$$ compact (on the same argument as before that $$\phi(S-S')=0$$) which I am not sure about since also the identity $$I$$ and some operator $$T$$ is part of $$E$$, not only the compact operators.

$$T^*T-TT^*$$ is compact by design. Since $$\phi$$ is a $$*$$-homomorphism onto a commutative algebra, $$\phi(T^*T-TT^*)=\phi(T)\phi(T^*)-\phi(T^*)\phi(T)=0.$$ Which means that $$T^*T-TT^*\in\ker\phi=K (H)$$.
For the second part, you get $$S-S'\in \ker\phi=K (H)$$ so $$R=S-S'\in K (H)\subset E'$$. Then $$S=R+S'\in E'$$.