## An introduction to linear transformations in Hilbert space by Francis Joseph Murray PDF

By Francis Joseph Murray

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COROLLARY. Let T be such that T • exists. Then (a) A transformation T' is C T' i f and only if. for evecy f in the domain of T and evecy g in the domain of T'' (f,T 1 g)+(Tf,g) = o. (b) A transformation T' is C T*, if and only if for evecy f in the domain of T and evecy g in the domain of' T • , (f,T'g) = (Tf,g). THEOREM III. , 7Jl(D) = n. Then [Ta] exists if and only if T' (or T*) has domain dense. By Definitions 4 and 5, we see that [Ta] exists if and only 7Jl('I) is the graph of a transformation.

By Lemma 5, (1-E 1 )+E 2 is a projection if' and only if' E 2 (1-E 1 ) = o or E 2 = E2E 1 • These two results ilIIply the f'irst statment of' the Lemma. If E 2 = E 2E 1 , Lemma 4 ilIIplies 7n 2 = 7n 1 7n 2 or m2 C 7n 1 • Since E 1-E 2 = E 1-E 2E 1 = (1-E 2 )E 1 Lemmas 3 and 4 imply that the range of' E 1- E2 is m1• '1t2 LEMMA 7. Let E 1 , E 2 , . • • be a sequence of' mutually orthogonal projections, with range m1, m2, . . respectively. , Ef =ii_~ r~= 1 Ecf whenever this limit exists). Then E is a projection with range me m, u m2u ••• ) (where u denotes the logical sum).

D, T*T is self-adjoint. If T' denotes contraction of T, with domain the domain of T*T, then [T' J = ·r. PROOF. By Theorem VII, (1+T*T)- 1 is self'-adjoint. By LeilllllB. 7 of the preceding section, 1+T*T is self-adjoint. If' in LeilllllB. 5, we let H1 = 1+T*T, H2 = • 1 , we obtain that T*T is self-adjoint. It remains to prove our statement· concerning T 1 • Since T 1 C T, we must have [T ' ] C T. g,Tgl of ~ which is orthogonal to all (f',Tf1 for which T*Tf can be defined. (Cf. Corollary 1 , to Theorem VI of Chapter II, §5).