Some Properties Of Quaternionic Matrix And Quaternionic Operator

In this thesis, the definition of quadratic numerical range of a quaternionic matrix is introduced. The relations between the set of left eigenvalues and quadratic numerical range of a given quaternionic matrix are discussed. Furthermore, the conclusion of the set of the left eigenvalues of 2?2 or higher order quaternionic matrix can be considered as the localization of quadratic numerical range is proved. In addition, the closure of quaternionic matrix’s similarity orbit is described.Note that the definition of spherical spectrum of quaternionic operator is different from the spectrum of a bounded operator acting on complex Hilbert space. In this thesis, we also discuss the similarity invariant and the upper semi-continuity of spherical spectrum, and the spherical spectrum of the direct sums of quaternionic operators, and give some properties of the spherical spectrum of quaternionic operator with form of the sum of compact operator and quasinilpotent operator.The results obtained in this thesis are helpful to the discussion of the left eigenvalue of quaternionic matrix and its related problems, and also helpful to the study of other properties of quaternionic operator.