Research On The Spectral Properties And Perfect Positive Mapping Of Self-imitation Measures

The concept of a spectral self-affine measure was first proposed by Jorgenden and pedersen.In 1998,they obtained the first example of a spectral measure with the fractal characteristics.Then some mathematicians conjectured that a compatible pair generated a spectral measure.The conjecture has attracted a lot of attention in the past two decades.In this thesis,we also focuse on compatible pair and spectral measure.On the one hand,if a non-singular integer matrix is given,we study con-ditions under which there exists a digit set D(D(?)Zn)such that μM,D is a spectral measure.On the other hand,some spectral Cantor measures in one-dimensional space are considered.Some techniques of operator theory are used in many important re-sults of spectral measures,and the technique of operator theory becomes a necessary tool in the study of spectral measures.On the operator theory,the interpolation,the representation and the extension of completely positive maps are studied.This thesis is divided into four chapters:In chapter 1,we review the basic results of fractal,spectral measure and operators,and give the basic concepts and properties needed in this thesis.In chapter 2,we firstly get the characterizations of M and D whenμM,D is a spectral measure for the case D(?)Z,|D| = 3,i.e.,μM,is a spectral measure if and only if D is a complete residue system(mod 3)and 3 |M,if and only if there exists an integer digit set S such that(M-1D,S)is a compatible pair,and further some spectrums are obtained.Secondly,for the case D(?)Z,|D| =4,we give all the spec-trum measures obtained by the conditions of compatible pair and obtain a necessary condition of a spectral measure.Thirdly,in the n-dimensional Euclidean space Rn,when a non-singular integer matrix M ∈Mn(Z)is given,,some conditions under which there exist integer digit sets D,S such that(M-1D,S)becomes a compatible pair are described.Also,we obtain the characterizations of the complete residue systems(mod M or mod M*)by the decomposition of the matrix M,especially obtain the characterizations of(M[0,1)n)∩ Zn and(M*[0,1)n)∩ Zn.Lastly,for a non-singular integer matrix M given,we get a sufficient condition under which there exists an in-teger digit set such that(M-1D,S)is a compatible pair,i.e.,if the integer digit set D satisfies |D| | |M|,then there exists an integer digit set S such that(M-1D,S)is a compatible pair.As a conclusion,a sufficient condition under which μM,D is a spectral measure has been obtained.In chapter 3,the interpolation of completely positive maps is studied in the finite case and in the infinite case respectively.In the finite case,we have obtained that the structure characterizations and the sufficient and necessary conditions that there exist completely positive maps of preserving trace,trace compression and preserving iden-tity,respectively,between two known self adjoint operators.In the infinite case,we have generalized the results in the finite case and we obtain the characterizations and conditions that there exist completely positive maps of preserving trace,trace com-pression and preserving unit,respectively,between two known self adjoint(positive)operators.In chapter 4,we mainly study the structure representations and the extensions of the positive maps among the three*-algebras:the set of all the bounded lin-ear operators B(H),the set of all the compact operators K(H),and the set of al-1 the trace class operators T(H).Firstly,we get the equivalence of the following spaces:CP(K(H),T(K))≈ NCP(B(K),T(H))≈ T(K(?)H)+,also obtain thatφD∈G NCP(B(H),T(K))if and only if there exists Vi∈ B(H,K)such that φ(X)=(?)and(?),where s≤∞,and φ∈CP(K(H),T((K))if and only if there exists Vi E B(H,K)such that φ(X)=(?)and(?),where s≤∞.Secondly,we give the characterization of φ∈ CP(K(H),T(1C)).Lastly,we consider the extension properties from CP(K(H),K(K))into NCP(B(H),B(K))and from CP(T(H),K(K))into CP(K(H),K(K)),and some sufficient and necessary conditions of φ ∈CP(T(H),K(K)).