Research On Some Questions In Operator Theory

In this paper, some problems in operator theory are discussed. At the meantime, some new conclusions on frame theory, linear preserving problem-s, and generalized quantum gates are obtained. The main contents are as follows.In Chapter1, firstly, we study the convex combinations of orthonormal bases for a Hilbert Space. Next, we introduce the concepts of frame and effective sequence, and then the images of the orthonormal basis for Hilbert space under different operators are mainly investigated. By using the previ-ous results and considering the special properties of the orthonormal basis, we show that the identity for Parseval frames holds by the mean of operator theory. Finally, we give a characterization of effective sequences.In Chapter2, we discuss the linear separable and additive separable maps between tensor product spaces. Consequently, we give the definition of strongly separable operators, and then discuss the relationships between strongly separable operators and rank-one preservers.In Chapter3, we first introduce the concepts of minimum modulus and surjectivity modulus of Banach space operators, and then characterize the surjective additive mapping preserving the minimum modulus of Banach s-pace operators, finally, we show the uniqueness of minimum modulus on B(H).In Chapter4, allowable generalized quantum gates and its realization condition are discussed, some properties of restricted allowable generalized quantum gates are obtained.The main results of this thesis are as follows.(1) We prove the convex in the case that dimH=n<∞or dimH=∞, the convex combinations of orthonormal bases are{x∈H:1/n≤||x||2≤1} and{x∈H:0<||x||≤1}, respectively.(2) The images of the orthonormal basis for Hilbert space under different operators are investigated. Also, we proved the identity for Parseval frames holds by the mean of operator theory.(3) We present a sufficient condition and a necessary condition for a frame operator being diagonal.(4) We give a characterization for effective sequences, and also proved that the only effectiveness preserving linear mapping are unitary operators in B(H).(5) We give a necessary and sufficient condition for a sequence obtained by adding a unit vector into an orthonormal basis to be an effective sequence.(6) We characterize the linear separable and additive separable maps between tensor product spaces, and also study the relationships between strongly separable operators and rank-one preservers.(7) All the forms of additive separable operators between tensor product spaces are given.(8) It is proved that ifφ:B(x)→B(y) is a surjective additive mapping with m(φ(T))＝m(T),(?)T∈B(x), or q(φ(T))＝q(T),(?)T∈B(x), then there exist two linear or two conjugate linear surjective isometries U: x→y and V:y→x such that φ(T)=UTV,(?)VT∈B(x).(9) A necessary and sufficient condition for an allowable generalized quantum gate to be realizable is obtained.(10) Some properties of restricted allowable generalized quantum gates are given and it is proved that the set of its extreme points is just the set of unitary operators.