Samuel multiplicities and Browder Spectrum of Operator Matrices

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 $H\oplus K$.