By Thomas Hunt Morgan

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1 Rectangular and Polar Decomposition of the Spectrum 39 As a simple application of the symbols and the polar symbols, we are going to explain the relation between nonnormal operators and normal operators and prove the following rectangular decomposition property of the hyponormal operators, which resembles that of normal ones. , for a complex number z = x+iy, IT z = x x Suppose that X,Y E IT z = y. ' y are self-adjoint. £(~) It is evident that IT (a. (X+iY)) c a(Y). 6. 1. If X and Yare self-adjoint operators Y E S~ U S~, then ITx(aja(X+iY)) = a(X).

5 Completely Non-normal Operators. Suppose of ~ A E which reduces AM eM, and ~(~) A, M is a closed linear subspace i . , A*M eM. In addition, assume that the restriction of normal. Then A to M is called a normal subspace of M, A. AIM, A is is said to be comEletel:l: non-normal i f there are no non-zero normal subspaces. 8. spaaes M and ~ N Let of A E ~(~) . There must be linear sub- suah that ~ = M eN where N is a normal subspaae of A and the restriation of to M, AIM, is aompletely non-normal. PROOF.

We have s: ~~~lIfnll = 2 k~lflgk(x)1 dfl(X) < E E B, fleE) f(x) 2 = fl(M) 2 =. such that Since, for any natural num- 49 limllf -fll = 0 n"'''' n which proves the assertion. , every a R(t) E for all £(~) the vector-valued function E~, is measurable, then R(t) function (or operator- £(~)-valued R(·) is a projection for all t E M, If for defined in is called measurable. jection-valued function. ~o c~l c~2 t E M. R(t)a M In addition, if then we call R(·) a pro- For example, suppose that c··· c~n c··· is a sequence of finite-dimensional spaces in ~, where and suppose that FOCF1C ...