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 $H\oplus K$.
Categories: math.FA