Spectral Analysis in Hilbert Space
From the last result of the preceding chapter, a good deal of nontrivial information can be obtained about self-adjoint operators on Hilbert space, incomparably more than is available for operators in Banach spaces, other than those of special compactness properties, or for arbitrary (non-self-adjoint) operators on Hilbert space. This result fails, however, to give a structure theorem for a self-adjoint operator on a Hilbert space, i.e., a result giving a simple explicit form for the operator, within the isomorphism appropriate to Hilbert space (unitary equivalence). What one would like, of course, is an analog in Hilbert space to the diagonalization of a self-adjoint operator in a finite-dimensional unitary space. Simple examples such as the operation of multiplication by x, acting on L2(0,1), show that a direct analog does not exist: there need be no eigenvalues or eigenspaces whatever. However, this example points the way to an appropriate analog, which is that in which invariant subspaces are decomposed, not into a discrete direct sum of eigenspaces, but into a continuous direct integral of infinitesimal eigenspaces, any one of which corresponds to a point of measure zero in the formation of the direct integral.
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