Lower norm of an operator ($\nu (T)$)

For a densely defined operator,The symmetric lower norm $\mu (T)=\min \{\nu (T),\nu (T^{\ast })\}$ equals $\Vert T^{-1}{\Vert }^{-1}$ when $T$ is invertible and is zero otherwise.

A closed operator is Fredholm when it has closed range and finite-dimensional kernel and cokernel. It is upper semi-Fredholm when only closed range and finite kernel are required, and lower semi-Fredholm when only closed range and finite cokernel are required.

Several essential spectra distinguish failure of upper and lower semi-Fredholm properties. For $T=A-zI$, these failures are detected at zero by the positive self-adjoint operators $T^{\ast }T$ and $T{T}^{\ast }$.

The discrete spectrum consists of isolated eigenvalues of finite algebraic multiplicity.