In this paper, we shall apply this concept to the ring of linear operators on Banach space X and prove that if a linear operator T on a Banach space X is defined in all of X and has finite descent and ascent then there is a Drazin inverse T~D satisfying TT~DT~k=T~k, T~DTT~D= T~D, and T~DT=TT~D for some nonegative integer k.

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C~*—algebra; Operator space; Ternary ring of operators; Universal free product; Full amalgamated free product; TRO—homomorphism;*—homomorphism

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ring of operators：算子環(huán)

ring of numbers 數(shù)環(huán) | ring of operators 算子環(huán) | ring of quotients 商環(huán)