Sentence:  Such an operator is obviously the zero operator ; for since it is <WSD>nilpotent</WSD> , the minimal polynomial for this operator is of the form * * f for some * * f ; but then since the operator is diagonalizable , the minimal polynomial cannot have a repeated root ; hence * * f and the minimal polynomial is simply x , which says the operator is 0 . 
Meaning of the word marked with <WSD> tags:  equal to zero when raised to a certain power