Expectations And Frames With Specific Upper And Lower Bounds

In this article, expectations and frames with specific upper and lower bounds are discussed. First, the set Md(A, B) of all d by d matrices C such that A≤Eφ (C)≤B for all unit vectors φ∈Cd is introduced and some properties of Md(A,B)are given; secondly, frames of the Hilbert space Cd with specific upper bound B and lower bound A are discussed and the set Fd(A,B) of all such frames is defined, the relationship between Fd(A,B) and Md(A,B) are obtained, some properties of Fd(A,B)are explored by using some results on frames; expectation operators are introduced and discussed, a reconstruction formula is established; lastly, perturbation properties of mapping Md(·,·) are given. This paper is divided into four chapters:In Chapter1, the background and existent research situation of frame are given. Additionally, corresponding knowledge of frame theory are given.In Chapter2, first, we define Eφ(C)=<φ|C|φ>, called the expectation of a matrix C with respect to the unit vector|φ>∈S(Cd), some properties of Eφ(C) are obtained; and then denote by Md (A, B) the set of all d by d matrices C such that A≤Eφ(C)≤B for all unit vectors φ∈Cd; finally, some properties of the set Md(A,B) are research.In Chapter3, denote by Fd(A, B) the set of all frames for Cd with lower bound A and upper bound B, and then the relationship between them are obtained; additionally, some properties of Fd(A,B) are discussed by using some results on frames.In Chapter4, expectation operators are defined first, some results of them are obtained, and a reconstruction formula is established; finally, the perturbation properties of the mapping Md(·,·) are explored.