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**Example text**

The domain of this operator can be extended to include the set of sequences {x,} for which xlkxk)2 is convergent. This set forms a linear manifold M which contains L. A linear transformation T, whose domain D is not closed, can frequently be extended by the following continuity argument: If there is a limit point f of D which is not in D, it may happen that, wheneverf,+f, the sequence Tf,is a convergent sequence which tends to an element g. Then we assign f to the domain of the extended operator T, and say that Tf = g.

In symbols: A 3 B implies Bt 3 A t . The question immediately arises, what symmetric operators can be extended so that they become self-adjoint? In order to discuss this question and related other questions we have to digress and introduce the concept of closure. The next section is devoted to this topic. Exercises 1. n. set {q,}in &,define a linear operator H formally by the equations H q , = n q , ,n = 1,2,3, ... Now specify the domain of H first so that H is symmetric, but not self-adjoint; then extend the domain of it so as to obtain a self-adjoint operator.

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. This theorem asserts that if such an operator is bounded away from zero then it Bela A. Lengyel 58 can be inverted and the inverse is again a bounded self-adjoint operator.