\section{$B(H)$}
\label{section:hilbert-endomorphism}

\begin{definition}[$B(H)$]
\label{definition:hilbert-endomorphism}
    Let $H$ be a Hilbert space, then $B(H) = L(H; H)$ is the algebra of all bounded linear operators on $H$. 
\end{definition}


% 1. Every non-trivial ideal of B(H) contains the finite-rank operators.
% 2. If H is separable, then the only non-trivial closed idela of B(H) are the compact operators.