Bounds of a self-adjoint transformation

As usual we can say a little more about the special case of self-adjoint transformations than in the general case. We consider, for any self-adjoint transformation , the sets of real numbers and It is clear that . If, for every , we write , then and , so that every number in occurs also in and consequently . We write and we say that is the lower bound and is the upper bound of the self-adjoint transformation . If we recall the definition of a positive transformation, we see that is the greatest real number for which and is the least real number for which . Concerning these numbers we assert that

Half the proof is easy. Since it is clear that both and are dominated by . To prove the reverse inequality, we observe that the positive character of the two linear transformations and implies that both and are positive, and, therefore, so also is their sum . Since implies , the assertion is trivial in this case; in any other case we may divide by and obtain the result that . In other words, whence , and the proof is complete.

We call the reader’s attention to the fact that the computation in the main body of this proof could have been avoided entirely. Since both and are positive, and since they commute, we may conclude immediately ( Section: Functions of transformations ) that their product is positive. We presented the roundabout method in accordance with the principle that, with an eye to the generalizations of the theory, one should avoid using the spectral theorem whenever possible. Our proof of the fact that the positiveness and commutativity of and imply the positiveness of was based on the existence of square roots for positive transformations. This fact, to be sure, can be obtained by so-called "elementary" methods, that is, methods not using the spectral theorem, but even the simplest elementary proof involves complications that are purely technical and, for our purposes, not particularly useful.