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

By Francis Joseph Murray

The description for this publication, An creation to Linear adjustments in Hilbert area. (AM-4), can be forthcoming.

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AfJ we remarked before Definition 6, in §1, a linear transformation is closed. THEOREM IV. If T is a continuous additive transformation, whose domain is dense and with bound C, (Cf. Chapter II, §;, Theorem I), then T• (and T*) is a linear transformation with the same bound as T. PROOF: [T] exists by Theorem II of Chapter II,§;. Since [T P = T •, we may suppose that T = [T] and has domain the full space. By Theorem III of this section, T • has domain dense. a. since ~· is linear manifold. Thus Theorem II of Chapter II, §;, implies that T• is linear 1f it is continuous.

If' {h,kl is a:rry pair of' ~,$ ~ 2 , it can be expressed as the sum of' an element of' ! • by Theorem VI of' Chapter II, §5. Thus given h and k there is a unique f' in the domain of' T and a g in the domain of' T' such that {h,kJ {f',Tf'J+{T•g,gJ = {f',Tf'J+{-T*g,g! or such that h = f'-T*g k = Tf'+g. k = e, this means that to every H, there in the domain of' T such that In particular if' is an f' h = (1+T*T)f'. This f' is unique, since if' there were two distinct we would have two resolutions of' fh,91.

H is closed symmetric and H* is symmetric, then H is self'-adjoint. If' the domain of' a symmetric transf'ormation H is the f'ull space, H is self'-adjoint. A synnnetric linear transf'ormation is self'-adjoint. If' The f'irst sentence is a consequence of' Lemma 1 • If' H is closed symmetric and H* is synnnetric, we obtain by Lemma 1 and Corollary 2 of' Theorem III of' the preceding section that H C H* C (H*)* = H. The third statement f'ollows f'rom Lemma 1 of' this section since a transf'ormation with domain the f'ull space can have no proper extension.

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