Compact Normal Operators, Commutativity Up To A λ For A Pair Of Operators And Operator Completion Problems

In this article, we study compact normal operators, commutativity up to a A for a pair of operators and operator completion problems, to which much attention is paid by many scholars in the field of operator theory. We divide this article into three chapters.It is well known that linear compact operators on a Hilbert space form one of the most important class of bounded operators and have many excellent properties. And compact normal operators have some more excellent properties, such as its spectral representation, functional calculus and so on. So it is very important to discover conditions under which a compact operator is normal. Recently. M. Sadkane presented three necessary and sufficient conditions under which a n-by-n complex matrix is normal. And n-by-n complex matrices can be considered as the class of linear compact operators on Hilbert space Cn. In the first chapter of this article, from the angle of operator theory, we give three characterizations of a compact normal operator and our method is different from M. Sadkane's.Commutation relations between self-adjoint operators in a complex Hilbert space play an important role in the interpretation of quantum mechanical observables and analysis of their spectra. And such relations have been extensively studied in the mathematical literature. An interesting, related aspect concerns the commutativity up to a factor for a pair of operators which is very important in the field of both quantum mechanics and mathematics. Sequential measurements are also very important in quantum mechanics and we may study it by the method of operator theory. In the second chapter of this article, by the method of decomposing an operator according to its structure, we research that non-commutation relation and sequential measurements. We get some important properties of them and answer the question that was raised by S. Gudder and G. Nagy. We also prove that AB = \BA if and only if one of the following conditions holds(i) there exists a Bii 0, 1 i m, = 1 and B = diag(B11, ..., Bmm, B2); (ii) BH - 0, for all 1 i m, and both A and B are compatible.Operator completion problems provide an excellent mechanism to understand matrix structure and operator theory more deeply. And operator completion problems arise in a variety of applications, such as statistics (e.g. entropy methods for missing data), chemistry (the molecular conformation problem), systems theory, discrete optimization(relaxation methods), data compression, etc.. as well as in operator theory and within matrix theory (e.g. determinantal inequalities). In recent ten years, many scholars have studied some classes of operator completion problems, for example, Hong-Ke Du. Cai-Xing Gu, Katsutoshi Takahashi, Jian-Lian Chui. Jin-Chuan Hou, L. Rodman, Spotkosky. In the third chapter of this article, we study some classes of operator completion problems of and We characterize the intersection of the spectraof and , respectively. We give a necessary and sufficient conditionunder which the operator is invertible for all X B( ). And we provethat ( I - A, B) is not right invertible }, is closed, dinxN and dimR(PB) < }, P is the projection from H onto R(A - ).