  we show that the definitions of some classes of semi-Fredholm operators,
which use the language of algebra and first introduced by X. Fang in [8], are
equivalent to that of some well-known operator classes. For example, the
concept of shift-like semi-Fredholm operator on Hilbert space coincide with
that of upper semi-Browder operator. For applications of Samuel multiplicities
we characterize the sets of $\bigcap_{C\in
B(K,\,H)}\sigma_{ab}(M_{C}),\bigcap_{C\in
  B(K,\,H)}\sigma_{sb}(M_{C})$ and $\bigcap_{C\in
  B(K,\,H)}\sigma_{b}(M_{C}),$ respectively, where $M_{C}=({array}{cc}A&C 0&B
{array})$ denotes a 2-by-2 upper triangular operator matrix acting on the
Hilbert space $H\oplus K$.

Functional Analysis