Let X be a Banach space. An operator Te B(X) is called nilpotent if T0 for some N. An operator T E B(X) is called quasi-nilpotent if (T) {0). The spectral mapping theorem implies that nilpotent operators are quasi-nilpotent. Prove the following statements: (a) If dim X oo, then quasi-nilpotent operators are nilpotent. Hint: use the Jordan canonical form (b) For 0 lol