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**Example text**

C) The norm IIP(1)fll is a monotone increasing function of 1for any fixed f E 2,. Such projection operators play an important role in the theory of selfadjoint operators. 111. Spectral Theory of Bounded Self-Adjoint Operators A. Initial Formulation of the Spectral Problem The discussion of bounded operators is considerably simpler than that of the unbounded ones because the former can be readily extended so that they are defined over the entire Hilbert space. We assume therefore in this part that every operator is linear and defined over the entire space.

The conclusion is DBt = DB and Bt = B. (c) An important differential operator in Q2 is defined formally by the equation Df = i -df. 39) This formal definition leads to a number of different operators depending on the nature of the interval of Q 2 and on the specifications for the domain of the operator. , equal to the integral of its derivativef '(x), moreover the functionf'(x) must also belong to 2 2 . We shall examine the operator D separately for the case of a finite interval typified by Q2[0, 11, the infinite interval 2,[- co, + co], and the semiinfinite interval Q2[0,a ] .

18) From Theorem 5 it follows that every element of the form Bf is in the domains of H and Z + H 2 , therefore also in the domain of H 2 . ij as its domain. Then ( H - iZ)(C + iB) = ( H - iZ)(H + iZ)(Z+ H2)-' = I, as we were to prove. The operator C iB = ( H - iZ)-' is a special case of the resolvent operator R, which is defined as follows + R, = ( H - zZ)-'; an operator which is bounded for every nonreal value of z. Theorem 6. The bounded operators B and C defined by a self-adjoint operator H are permutable with H in the following sense: The elements Bh and Ch are always in the domain D,, of H.