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D С Lay

Proof ЩТ) is closed since d{T)<oo, and n(T)>0 since а(Г)= oo Thus T is m state I3 or III3, and I3 is possible only if 6{T)=0 Part (a) follows immediately from Theorem 2 3, and is not a new result In part (b) we use the fact that а(Т)=со implies dimyr(T)n^(rO>0 (cf Lemma 3 4 of [17]) Lemma 2 4 then implies that, for Я m a deleted neighborhood of 0, d{À -T) = 0 and п(Я - Г) > 0,1 e, Д - Г is m state I3 The statement that d{T) < n{T) (with n{T) possibly infinite) when Ô{T) = q<oo is a consequence of Theorem 4 5 of [8] and the fact that m this paper the only operators T we examine are such that 9{T^)-\-M{T) = X Part (c) of this theorem, like part (a), was included for the sake of completeness From (2 6) of Theorem 2 3 it follows immediately that for Я m some deleted neighborhood of 0 the operators Я - T are all in the same state either I^, I3, Ш^, or III3 Our shght improvement on this well- known result IS to ehmmate the possibility that the operators Я - T are in state II This is a consequence of Theorem 2 9 and the fact that 0 cannot be a pole oîR^{T) when a{T) = 00

3 . 4 . Theorem. Suppose that а(Г) = oo and d(T) = 00

( a ) Ifô{T) < 00, then T is in state II3 or III3 andX-T is in state I3 for all Я in a deleted neighborhood of 0

( b ) // «5(T) = 00, but n{T)< CO and ЩТ) is closed, then either X-T is in state III3 and has a closed range for all X in a neighborhood of 0, or T is in state III3 and X-T is in state 111^ for all X in a deleted neighborhood of 0

Proof Certainly T is not one-to-one and m{T) Ф X, so that T is in state II3 or III3 State II3 IS impossible if and only if T has closed range The rest of part (a) follows from the argument given for part (b) of Theorem 3 3 The statements m (b) follow immediately from Theorem 2 3 and are not new

The chart below summarizes the results of the four theorems in this section Given knowledge about the ascent, descent, nullity and defect of a linear operator which has properties (2 1) and (2 2), one can tell from the chart what is known about the operators Г and Я - T for Яша neighborhood of 0 For example, the phrase "deleted neighborhood m state I3" means that Я - T is in state I3 for all Я in a deleted neighborhood of 0

Certain combinations of ascent, descent, nullity and defect are impossible Twelve (shaded) squares are eliminated because è{T) = 0 if and only iîd{T) = 0 Four additional squares are crossed out since n{T)^d{T) whenever a(7) is finite (cf Theorem 3 1), and d(T)Sn{T) whenever Ô{T) is finite (cf Theorem 3 3 (b))

The additional hypothesis that ЩТ) is closed appears in two squares of the chart The lower "half of each of these squares is blank, indicating a lack of information about the situation when the range of T may not be closed As might be expected, one square is labeled "no information" None of the ideas developed in this paper provide any insight into the situation when a{T) = ô{T) = n{T) = diT)= CO There exist quasimlpotent operators with infinite ascent, descent, nulhty and defect On the other hand it may happen that for all Я in some open subset of (7(T), (x{X-T) = ô{X~T) = n{X-T) = d{X-T)= CO (The author has not investigated this situation )