Elementary Operator Subspace Angle, And The Effect Algebra

Operator theory is a very and important research field in the functional analysis. Ever since von Neumann,Hilbert etc.established the operator theory at the beginning of 20 centuries,the operator theory has already quicklied got to develop and seep through mathematics of each branch and become a through long not wane the research topic.Its research contents involves to many branches of the pure mathematics and the applied mathematics such as geometry theory,operator perturbation theory,matrix analysis,approximation theory,noncommutative ring theory,noncommutative probability theory,mathematical physics,and many others. The research of this thesis focuses on elementary operators,angles between two closed subspaces and effect algebras on a Hilbert space.This article is divided into three chapters.In recent years,all of them are active research topics in the operator theory and the quantum information theory.By the research toward these topics, one can make to the inside relation of the operator structure to change of more clear, these research to the operator theory do not only have important theories meaning and apply value,but also strengthenned the research foundation of the operator theory from the certain meaning.In Chapter 1,by using the elementary ways,the norm(norm-attainable) relations between the elementary operators with length n and the restrictions of these operators on the unitary group are researched.Secondly,by using the technique of operator blocks and operator polar decomposition,the relations between the reduced minimal numerical ranges of operators and these maximal munerical ranges are established. Subsequently,some properties of the reduced minimal numerical ranges of Aluthge transforms are given.Finally,the dominant operator pair is defined and some properties of the dominant operator pair are attained.In Chapter 2,by using space decomposition and block operator matrices,the relations between the angle of two closed subspaces and norms,spectra,the reduced minimal numerical ranges,ranges closedness of operators are researched.In Chapter 3,by using familiar technique of operator blocks,the quantum operations and generatized quantum gates are researched.An affirmative answer of a Gudder question is given and some results of Gudder are extended to the infinite dimensional subspaces.The results from the thesis consist of the following statements.1.By using the elementary tectonic ways,the norm(norm-attainable) of the elementary operator with length 2 and the norm(norm-attainable) of the restriction of this operator on the unitary group are consistent.2.The norm(norm-attainable) of the elementary operator with length n and the norm(norm-attainable) of the restriction of this operator on the unitary group are consistent. 3.The reduced minimal numerical range of operator is defined and researched. The relations between the reduced minimal numerical ranges of operators and the maximal numerical ranges of these operators are attained.Subsequently,some properties of the reduced minimal numerical ranges of Althuge transforms are given.4.The dominant operator pair is defined and reseached.Subsequently,some results about the dominant operator pair are attained.5.Using the technique of block operators,the fine characterizations on the angles between two closed subspaces are given and the relations between these angles and norms,spectra,range,and many others are researched.6.By reseaching the properties of the quantum sequantial products,an affir-mative answer of a Gudder question is given.7.Using the method of the spectral theory of operators,the quantum operations and generalized quantum gates are reseached and some results of Gudder are extended to the infinite dimentional subspaces.