Some Problems On Fusion Frames And Weyl Type Theorems

Frame theory and spectrum theory are two significant subjects in the study of operator theory,which have both theoretical value and wide applications.Firstly,we investigate some problems of fusion frames in this paper.We know that frames for Hilbert spaces were introduced by Duffin and Schaeffer in 1952 to study some problems in nonharmonic Fourier series and has attracted many math-ematician's attention since 1986.Frame has some properties similar to a basis,however,in contrast to the situation for a basis,the coefficient might not be unique for frame representations.This property makes them more convenient than bases.Inspired by the study of frame vectors for unitary systems,we will investigate fusion frames with special structure,which are generalizations of frames.Secondly,in recent decades,the Weyl theorem and its generalizations related to the distribution of eigenvalue in quantum mechanics,have been an impressive key branch in spectrum theory.Since the analysis of the structure of the spectrum of a bounded linear operator enables people to obtain a deep understanding of this operator,we will focus on two new spectral properties which are new variants of Weyl's theorem.The main works are as follows.Firstly,the concept of fusion frame generators is introduced in this paper.Due to the close rela-tivity of fusion frame generators and complete wandering subspaces,we give the definition of the local commutants for a set of operators at a subspace and investigate some properties of the local commu-tants.By means of local commutants,we present a structure characterization of all complete wandering subspaces for a unital semigroup.In particular,some results of complete wandering subspaces for a special class of unitary systems with the structure similar to the wavelet systems are obtained.Secondly,we define the generalized local commutants,which are generalizations of local commu-tants.It is proved that for a unitary system with a complete wandering subspace,the set of all Parseval fusion frame generators can be completely characterized in terms of operators in the generalized lo-cal commutant of unitary system.We then give the dilation theorems for Parseval fusion frames and Parseval fusion frame generators for unitary groups.Thirdly,we get some results of a Gabor type unitary system which has a complete wandering subspace.We show that Parseval fusion frame generator for Gabor type unitary systems can be dilated to a complete wandering subspace,which is different from the general dilation property.Fourthly,motivated by K-frames and fusion frames,we introduce the concepts of K-fusion frames and K-fusion frame generators.In terms of the operator K,the synthesis operator and the quotient operator some characterizations for K-fusion frames are given.We investigate the condition for tight K-fusion frames to be tight fusion frames.Moreover,we obtain a parametrization theorem of the K-fusion frame generators for unitary systems.Fifthly,we define a new spectrum property(WE),which extends generalized Weyl theorem.By means of the spectrum induced in view of the operators of consistency in the Fredholm and index,we establish the sufficient and necessary conditions for a bounded linear operator for which property(WE)and its perturbation hold.Then we explore conditions for operators so that their direct sums satisfies property(WE).Moreover,we discuss the relation between property(WE)and hypercyclic operators.Finally,we investigate another spectrum property(h),which is related to Weyl theorem.We define a new spectrum by the property of generalized Kato type operators.In terms of this spectrum,we give the sufficient and necessary conditions for a bounded linear operator to has property(h).Meanwhile,the sufficient and necessary conditions for which the property(h)of functional calculus are obtained.Moreover,we study the stability of property(h).