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Page 487
Operators with Closed Range It was observed in Lemma 2.8 that the closure of the range of an operator U e B ( X , Y ) consists of those vectors y such that y * UX = 0 implies y * y = 0. Or , in other words , UX = { y \ U * y * = 0 ...
Operators with Closed Range It was observed in Lemma 2.8 that the closure of the range of an operator U e B ( X , Y ) consists of those vectors y such that y * UX = 0 implies y * y = 0. Or , in other words , UX = { y \ U * y * = 0 ...
Page 488
It follows from the definition of U * that every element in its range satisfies the stated condition . Q.E.D. 3 LEMMA . If the adjoint of an operator U in B ( X , Y ) is one - to - one and has a closed range , then UX = Y. PROOF .
It follows from the definition of U * that every element in its range satisfies the stated condition . Q.E.D. 3 LEMMA . If the adjoint of an operator U in B ( X , Y ) is one - to - one and has a closed range , then UX = Y. PROOF .
Page 489
since the range of U * is closed , ** U * y * for some y * e Y * . If z * is the restriction of y * to 3 , then x * U * z * . Hence , the range of U * is also closed . It follows from the previous lemma that U x = UX 3 .
since the range of U * is closed , ** U * y * for some y * e Y * . If z * is the restriction of y * to 3 , then x * U * z * . Hence , the range of U * is also closed . It follows from the previous lemma that U x = UX 3 .
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Contents
Preliminary Concepts | 1 |
B Topological Preliminaries | 10 |
quences | 26 |
Copyright | |
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