By O.U. Schmidt

ISBN-10: 0716704315

ISBN-13: 9780716704317

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By hypothesis, W= {0} and hence Y = Tcflr (II) => (III): Suppose that Y e Inv T is such that Y ~ {0} and cr(TjY) = 0. Since cr(TjY) is compact, Yc VT. Then TjY is bounded and cr(TjY) ~ 0. This, however, contradicts the assumption on cr(TjY). (III) => (IV): Let Z e SMb(T). To see that Z e SM(T), let Y e Inv T be such that cr(TjY)ccr(TjZ). Since cr(TjZ) is compact, so is cr(TjY). 19) and, by hypothesis, cr(TjW) = 0 implies that W= {0}. Consequently, Y = TcVT and hence Yc Z. Therefore, Z e SM(T). (IV)=> (1): Evidently, {O} is aT-bounded spectral maximal space and hence {0} is spectral maximal, by (IV).

PROOF. 6. X;)CG;, x0,x1 e Inv T with i=O,l. X;)CG; implies X; e X(T,G;), it follows that X; c X(T ,G), i=O, 1. 10) X(T,G1 ) = :::(T,G1)(£JX(T,0). 10). 8) provides a spectral decomposition of X by T into a linear sum of a T-bounded spectral maxima1 space and a spectral maximal space ofT. 8), then T has the 1-SDP. 1, for n = 1. Next, we investigate for some further properties of the T-bounded spectral maximal spaces of a closed T with the 1-SDP. 11. LEMMA. Given T with the 1-SDP, let F c [be compact.

By hypothesis, W= {0} and hence Y = Tcflr (II) => (III): Suppose that Y e Inv T is such that Y ~ {0} and cr(TjY) = 0. Since cr(TjY) is compact, Yc VT. Then TjY is bounded and cr(TjY) ~ 0. This, however, contradicts the assumption on cr(TjY). (III) => (IV): Let Z e SMb(T). To see that Z e SM(T), let Y e Inv T be such that cr(TjY)ccr(TjZ). Since cr(TjZ) is compact, so is cr(TjY). 19) and, by hypothesis, cr(TjW) = 0 implies that W= {0}. Consequently, Y = TcVT and hence Yc Z. Therefore, Z e SM(T). (IV)=> (1): Evidently, {O} is aT-bounded spectral maximal space and hence {0} is spectral maximal, by (IV).

### Abstract Theory of Groups by O.U. Schmidt

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