Orthogonal Lie Algebra Of Operators And Related Topics

Let H be a complex,separable Hilbert space,and B(H)the algebra of all bounded linear operators on H.Skew-symmetric matrices are a special class of matrices in matrix theory and play important roles in various mathematic branches.In particular,the skew-symmetric ma-trices constitute an orthogonal Lie algebra,which is a class of finite-dimensional classical Lie algebras in Lie theory.The orthogonal Lie algebra of operators OC is a analogue of orthogonal Lie algebra in infinite dimensions,which is defined by OC={X ? B(H):CXC=-X*},where C is a conjugation on H,namely,C is a conjugate-linear,isometric involution.It is know that OC as a symplectic type Cartan factor is closely related to the classification of bounded symmetric fields in Banach Spaces and it plays an important role in the structure of JB*-triples.An operator T in OC is said to be C-skew symmetric,and T can be written as a skew symmetric matrix relative to an orthonormal basis of H.Recently,there has been growing interest in skew symmetric operators,and abundant results have been obtained.In this paper,we study the algebraic structure of OC by using the related results of skew-symmetric operators,including Lie ideals,Lie derivations,quadratic ideals and special operators in OC.In the 1970s,P.de la Harpe proved that each of nontrivial ideal of OC is composed of compact operators and contains all finite rank operators in OC.The second chapter completely characterizes Lie ideals of OC.Theorem 1.L is a Lie ideal of OC if and only if there exists an associative ideal I of B(H)such that L=I?OC.As applications,we establish the dual relations among Schatten p-classes in OC.These results are skew-symmetric versions of the classical Schatten p-classes theory.Fur-thermore,we prove that the approximate point spectra,the approximate defect spectra,the left spectra and the right spectra of Lie derivations of OC all coincide.The third chapter characterizes quadratic ideals of OC.We first establish the ap-proximate range inclusion theorem in OC by quadratic operators.It follows from the theorem that we describe quadratic ideals of OC,which are complete and are closed in the weak operator topology.Theorem 2.Let I be a quadratic ideal of OC.Then the following are equivalent:(?)I is complete and is closed in WOT;(?)I is complete and is closed in SOT;(?)I=TOCT for some T ? SC with a closed range;(?)I=IM for some closed subspace M of H.In view of the important role of quadratic operators in quadratic ideals of OC,we completely determine the spectra of quadratic operators and its various parts.The previous results show that although OC is not associative algebra,its structure is very similar to B(H).This inspired us to study whether some of the classical results in B(H)are still valid for OC.The aim of fourth chapter is to discuss several special operators in OC.First,we prove that invertible operators in Oc constitute an arc-wise connected,open subset and characterize norm limits of invertible operators.Theorem 3.Invertible operators in OC constitute an arc-wise connected,open subset.Theorem 4.Let T ? OC.Then T can be approximated in norm by invertible operators in OC if and only if at least one of the following holds:(?)Ran T is not closed;(?)dim ker T=?;(?)dim ker T<? is even.Secondly,we establish the Weyl-von Neumann-Berg Theorem in OC.Theorem 5.If T1,T2,…,Tm are commuting normal operators in OC and ?>0 then there are simultaneously diagonalizable normal operators D1,…,Dm in OC and compact operators K1,…,Km such that for 1?j?m,?Kj?<? and Dj=Tj+Kj.Thirdly,we prove that irreducible operators are dense in OC.Theorem 6.If T ? OC p ?(1,?),then,given ?>0,there exists K ? Bp(H)with?K?p<? such that T+K ? OC is irreducible.In addition,we describe the means of unitary operators in OC and determine the structure of the set formed by Fredholm operators in OC.The fifth chapter characterizes complex symmetric Toeplitz operators by the method that we develop on the basis of skew-symmetric operators.An operator T ? B(H)is called complex symmetric operator,if there exists a conjugation C such that CTC=T*.In this paper,we completely describe when Toeplitz operators with certain trigonometric symbols on the Hardy space of the unit disk are complex symmetric.