By Xia

Spectral research of linear operators has consistently been one of many extra lively and demanding fields of operator thought, and of in depth curiosity to many operator theorists. Its devel opments are likely to be heavily on the topic of yes vital difficulties in modern arithmetic and physics. within the final twenty years, many new theories and engaging effects were came upon. Now, during this path, the fields are possibly wider and deeper than ever. This e-book is dedicated to the research of hyponormal and semi-hyponormal operators. the most effects we will current are these of the writer and his collaborators and associates, in addition to a few pertaining to comparable themes. to some degree, hyponormal and semi-hyponormal opera tors are "close" to common ones. even supposing these periods of operators comprise general operators as a subclass, what we're drawn to are, evidently, nonnormal operators in these sessions. With the well-studied general operators in hand, we cer tainly desire to understand the homes of hyponormal and semi-hypo common operators which resemble these of ordinary operators. yet extra vital than that, the investigations may be concen trated at the phenomena which simply take place within the nonnormal cases.

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**Extra resources for Spectral Theory of Hyponormal Operators**

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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 ...