Symmetrisable operators :Part III Hilbert space operators Symmetrisable by bounded operators

The fact that the most general symmetrisable operators in Hilbert Space do not possess a number of the desirable properties of such operators in unitary spaces makes it necessary to look for a more restricted class of operators. There are two reasons for our particular choice. In the first place many of the conditions introduced in the course of Part II concerned reltionships between the domain of the symmetrising operatorHand the domain and range of the symmetrisable operatorA. These conditions are now all automatically satisfied. The other reason is that the construction used in section 4 to relate symmetrisable operators to certain symmetric operators clearly required that eitherHorH−1was bounded. The case ofH−1bounded has already been dealt with in section 9 and shown to be fairly simple. The case in which H is bounded is clearly of considerable complexity, since we have already seen (example in proof of Theorem 10.6.) that the con |H| the bound ofHby