By Per-Olov Lowdin
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Extra info for Advances in quantum chemistry. / Volume 4
Now we come to the main theorem of this section, known as Wecken’s lemma : For every bounded, self-adjoint operator H there is a positive operator T such that T 2 = H Z and this T is permutable with every bounded operator permutable with H. 44) O < H2 < I . We introduce the operator C = I - HZ,thus 0 < C < I. 46) for all n. 42) also follows by induction with the aid of the equation Yn+l- Y,,= +[ Y,,’ - Y,Z-,] = $( Y,, - Y,,-I)( Y,, + Y,,-i). 47) Beta A. Lengyel 40 Here we make use of the fact that the product of positive operators is positive.
2) If At has an inverse, then (At)-' = (3) If the operator Att exists, its adjoint is At. (4) If A is a closed operator with domain everywhere dense, the operator At has the same properties. Beta A. Lengyel 48 C. 9) ( A f ,g ) = (f,A d holds for everyfand g in D. This definition implies that the adjoint operator A t exists and is an extension of A . Thus D' 2 D is everywhere dense and the existence of A" is also assured. Since Att is a closed extension of A whose adjoint is again A', it is sufficient to consider closed symmetric operators only.
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. The new transformation is called a closed extension of T. Its domain is still not necessarily a closed linear manifold.