Some Issues On Higher-dimensional Numerical Ranges And Nest Algebras

This paper is devoted to the investigation of maps preserving higher-dimensional numerical ranges on Hilbert spaces, Jordan homomorphisms of nest algebras, decom-posability of finite rank operators in Lie ideals of nest algebras and non-selfadjoint subalgebras of B(H) which have the double commutant property. It consists of four chapters.In Chapter 1, we introduce the background, review the developments and achievements until now. At the same time, we give preliminaries and the main results of this paper.In Chapter 2, we characterize nonlinear maps preserving the higher-dimensional numerical ranges. The results of this chapter are as follows.Theorem A Let H and K be two complex Hilbert space with the dimension greater than k. Suppose that φ:B(H)�'B(K) is a surjective map. Then φ satisfies that Wk(AB-BA*)= Wk(φ(A)φ(B)-φ(B)φ(A)*) for all A,B∈B(H)if and only if there is a real number γ∈{-1,1} and a unitary operator U ∈ B(H,K) such that φ(A)=γUAU* for all A∈(H).Theorem B Let H and K be two complex Hilbert space with the dimension greater than k. Suppose that φ:B(H)�'B(K) is a surjective map. Then φ satisfies Wk(φ(A)φ(B)+φ(B)φ(A))= Wk(AB+BA) for all A B∈B(H) if and only if there exists a unitary operator U:H�' K and a scalar η∈{1,-1} such that φ(A)=ηUAU* for all A∈B(H) or there exists a conjugate unitary operator U:H�'K and a scalar η∈{1,-1} such that φ(A)=ηUA*U* for all A∈B(H).Theorem C Let H and K be two complex Hilbert space with the dimension greater than k. Suppose that φ:B(H)�'B(K) is a surjective map. Then φ satisfies Wk(φ(A)φ(B)+φ(B)φ(A)*)= Wk(AB+BA*) for all A, B∈B(H) if and only if there is a unitary operator U:H�'K and a scalar η∈{1,-1} such that φ(A)=ηUAU* for all A∈B(H).Theorem D Let H be a complex separable Hilbert space of dimension greater than k and φ:B(H)�'B(H) be a multiplicative map. Suppose that k≥2. Then φ satisfies Wk(φ(A))=Wk(A) for all A ∈ B(H) if and only if there is a unitary operator U ∈ B(H) such that φ(A)=UAU*.In Chapter 3, we study Jordan homomorphisms of nest algebras and decom-posability of finite rank operators in Lie ideals of nest algebras. Making a use of known results, we prove a surjective Jordan homomorphism of nest algebras is ei-ther a homomorphism or an anti-homomorphism, and get a necessary and sufficient condition for decomposability of closed Lie ideal of nest algebras. The results of this chapter are as follows.Theorem E Let N1 and N2 be nest, AlgN1 and AlgN2 be nest algebra. Suppose that φ:AlgN1�'AlgN2 is a surjective Jordan homomorphism and there exists P ∈ N1 such that φ(P)＝0,I.Then φ is either a homomorphism or an anti-homomorphism.Theorem F Let N be a nest on a separable complex Hilbert space H. Then each closed Lie ideal in AlgN is decomposable if and only if N satisfies one of the following conditions:(1) N has no finite-dimensional atoms;(2) N has only a finite-dimensional atom and such an atom is 1-dimensional.In Chapter 4, we first introduce connected space and decompose the block- closed bimodule over a maximal abelian selfadjoint subalgebras of B(H)into a direct sum of some connected subspaces.Then we discuss non-selfadjoint subalgebras of B(H)which habe the double commutant property.The results of this chapter are as follows.Theorem G Let M∈B(H)be a masa,R=(?)iREi,Fi,·i∈Ω be a block-closed bimodule over M.Suppose that S=CI+R.Then S=S" if and only if one of the following conditions is true:(1)∑iEi=I and R=(?)(?)iREi,Fi;(2)∑iEi≠I≠∑iFi.Theorem H Let M(?)B(H)be a masa,D be a subspace of M,R=(?)iREi,Fi,i∈ Q be a block-closed bimodule over M.Suppose that S=D+R satisfies S=S" Then the following are equivalent.(1)∑iEi=I；(2)∑iFi=I.