Research On Quadratic Numerical Range Of Bounded Linear Operators

Research on Quadratic Numerical Range of Bounded LinearOperatorsZhang JingjieAbstract The study of numerical range started in 1918-1919 by Toeplitz and Hausdorff when they first proved that W(A) is always convex, since then, the study of numerical range theory had been one of the most active research areas. The subject is related and has applications to many different branches of pure and applied science such as operator theory, functional analysis, C"-algebras, Banach algebras, matrix norms, inequalities, numerical analysis, perturbation theory, matrix polynomials, systems theory, quantum physics, etc. In recent years, the quadratic numerical range, one of the most important generalizations of the numerical range, was put forward in the course of people studying the spectral theory of the block operator matrix to the need of the development of some branches mentioned above. As we all know, one of the main aim of the spectral theory research is to find out the location of the spectrum, it is just because of the quadratic numerical range gives a better information about the localization of the spectrum than the numerical range and that provoke my interest to study the quadratic numerical range. The aim of this paper is to make a deep investigate in this area, the main content as follows:The first part of this paper pays the emphasis on the quadratic numerical range. At first we get an equivalence characteristic of the numerical range: W(A) = \JPn?pn W(PnA\En). From the definition of the quadratic numerical range, we can see that the quadratic numerical range depend on the space decomposition. From contrast, we can see that the quadratic numerical range gives a better information about the localization of the spectrum than the numerical range, perhaps just because of this, the quadratic numerical range of an operator need not to be convex, and even that the quadratic numerical range of an operator need not connected, then we give a condition under which the quadratic numerical range of an operator is not connected.In the second part: the emphasis is put on the n-numerical range. After we define the n-numerical range of bounded linear operators on a Hilbert space, we find that the n-numerical range have a series properties very similar to that of the quadratic numerical range. At the same time, we prove that under certain conditions, Wn(A) C W^A and that when H is finite dimensional and dimTi = n, we have a (A) = Wn(A). Therefore, it is nature to guess that when H is an infinite dimensional Hilbert space, for any space decomposition Dn € ?? the followingequality hold: a(A] = f}DnfD WDn (A).In the last part: the emphasis is put on the relations between the numerical range and operator completion problem. We first give a new proof of lemma 1 in [17]. At the same time, we find the example which was given in [17] to explain the author's assertion is wrong. Through analyzing of the cause, we conclude a more general theorem. On the other hand, benefit from the facts above and according to the properties of the numerical range, we study the operator spectrum perturbation problem, in the last, we give some better proofs on some theorems which simplify the original proof much.