By Francis Joseph Murray
The description for this publication, An advent to Linear variations in Hilbert area. (AM-4), might be forthcoming.
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Extra resources for An introduction to linear transformations in Hilbert space
12 is a Hilbert space. Proof'. From the previous def':1n1tions, it is readily seen that Postulates A and B are satisf'ied. To show Postulate C we notice that s 1 , the sequence of' elements ~ 1 , ~2 , • • • where ~°' = ! c5cx, 1 , 6°'' 2 , • • • l ls a denumerable orthonormal set. Thus given n, we take the f'irst n of' the ~a 's, and Lemma 2 of' the preceding section tells us that these are linearly independent. a 1,a2 , ••• l ror which there ls an N such that lf' n ~ N, ~ = o. •. l then f'n € U(S 1 ) and ff'-f'nf E.
If' EiEj = o, Ej = Ej-EiEj = (1-Ei)Ej. s 3 and 4 above that mj is in m1 mj or mj in m1. Thus mj ls orthogonal to ml. Obviously the range of' E 1+ • • • +~ must be in 2l( m 1 u • • • u 71tn )'. On the other hand 1f' f'E 2l ( m 1u • • • u mn), 1t is readily established that f' = f' 1 + • • • +f'n, where f' 1 E m 1 , • • • , f'n E mn. Now (E 1+ ••• + ~)(f' 1 + ••• +f'n) = f' 1+ ••• +f'n since E1 f'j = E1 Ejf'j = 9, if' 1 j. Thus f' 1+ ••• +f'n is in the range of' E 1 + • • • +Fn_ and U( m 1u • • • u mn) is included in this range..
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).