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Resolution of the Identity Given a subspace M in 8 and the corresponding projection operator E, we say that M reduces the bounded, self-adjoint operator H if EH = HE. When M reduces H , then MI also reduces H because the projection operator of MI is Z - E, and this commutes with H whenever E does. f be an element of M which reduces H . Then Ef =f, Hf = HEf = EHf, therefore Hf E M. Similarly, if g E MI, then Hg E MI. n. system, the Bela A. lengyel 32 matrix will be of the form 0 0 The zeros indicate that all elements in the corresponding block vanish.

If H is a finite matrix which may be written in the diagonal form, what are the matrices T, H - amd H'? 2. Show that if Hf = Lof then E, f = 0 for 1< l o ,and the expression llE,f112 = ( E , f , f ) has a discontinuity at 1 = L o . 3. Show that if mZ< H < MI, then EM = Z. Is Em = O ? E. , in the following manner : m < l , < I , < . .

What is the domain of the operator A’ ? Is this operator self-adjoint? 2. What is the domain of the operator B2 over Q2,? Is this operator self-adjoint ? 3. Prove that the operator D on Q2[0,a33 is a closed symmetric operator. 4. Show that the operator d2/dx2with a proper definition of its domain is self-adjoint in all three types of 2, spaces discussed. E. An Auxiliary Theorem about Bounded Self-Adjoint Operators In the proof of the spectral theorem for general self-adjoint operators we shall need a theorem concerning bounded self-adjoint operators.