| Approximation of operators in infinite dimensional spaces is always an important re-search subject in the operator theory and operator algebras.Relative research results not only make us have a clear understanding of the structures of operators,but also furnish us a multi-tude of skills and techniques for other areas.And these results also promote the development of the operator theory and operator algebras.The information of spectrums of operators,especially the spectral picture of an operator and the fine spectral picture,plays an important role in the study of operator structures and approximation and classification of operators.This also provides very useful and effective tools for the research work.People are now accustomed to studying and describing the prop-erties and features of operators or operator algebras,through the spectrum and the spectral picture of an operator.When dealing with approximation of operators,it is worth trying to approximate an operator from other points and direction.This may also provide a new angle about the past work and achievements,help us understand these results better,or even help us find some new research direction of operator approximation.The research about the numerical range of an operator is an important subject of oper-ator theory.Since Toeplitz and Hausdorff showed the famous Toeplitz-Hausdorff theorem in 1918-1919,numerical ranges of operators have caused people's close attention and been widely applied to other areas and subjects.And numerical ranges of operators play a signif-icant role in theory and practice.In this thesis,by combining the theory of operator approximation and the research about numerical ranges,we use the numerical range of an operator to study the problem of operator approximation.Unless otherwise stated,in this paper we always denote by ? complex separable infinite dimensional Hilbert space,by B(?)the set of all bounded linear operators on ?,and by?(?)the set of all compact operators on ?.For given operator T G B(?),the numerical range of T is defined by (?)The Toeplitz-Hausdorfif theorem showed that W(T)is always a convex set of the com-plex plane.The famous spectrum conclusion theorem proves that (?),where o(T)denotes the spectrum of T and (?)denotes the closure of the numeri-cal range of T.When doing research,the class of operators which have special property or satisfy some condition,plays a very important role in the research of operator theory.And it also benefits the study of the property and structure of single operator.So special classes of operators are alway became our objective when we do research.In this thesis,we do some research on the approximation of the numerical ranges of operators in special operator classes.It is worth emphasizing that when we deal the prob-lem of operator approximation,the perturbation happens in the original class of operators or operator algebras.That is,we must make sure that the operator after perturbation should be still in the original class of operators or operator algebras.In 2003,Bourin studied that approximation of the numerical range of operator in B(?).Bourin showed that the set of operators with closed numerical ranges are norm-dense in?(?).Furthermore,Bourin proved that for given T ? ?(?)and ?>0,there exists an operator K ? ?(?)with ||K||<? such that W(T + K)is closed.So for any operator in given operator class or operator algebra,after a small compact perturbation,it can admit a closed numerical range.But the new operator whether still in the given operator class or operator algebra is unknown,which is not easy to answer.The research about the above question is our main work.In 2015,Zhu Sen studied the numerical ranges of normal operators and transaloid op-erators.He showed that if N ? ?(?)is normal(or transaloid),then for given ?>0,there exists a K ? ?(?)with ||K||<? such that W(N + K)is closed and N + K is normal(transaloid,respectively).For the convenience of statement,we introduce the following definition.Definition 1.An operator class A is said to be strongly approximately numerically closed,if for(?)T?A and?>0,there exists a K ??(?)with ||K||<? such that (?).From the above definition and illustration,we know that ?(?),the class of normal operators and the class of transaloid operators are all strongly approximately numerically closed.It is well-known that weighted shift operators is a concrete and useful operator class in the research of operator theory,which also plays an significant role when studying the properties and structures of operators.Definition 2.(?) be an orthonormal basis of ? and (?),where(?).Define(?),for (?).we call A? B(?)be an weighted shift operator,and ?an?n?? be the weighted sequence of A.If ? = ?,we say that A is a unilateral weighted shift;If ? = ?,we call that A is a bilateral weighted shift.If T ? ?(?)admits an upper triangular matrix representation with some orthonormal basis,i.e.T is called triangular.Firstly,we study the unilateral(bilateral)weighted shifts and triangular operators,and get the following results.Theorem 3.The class of unilateral(bilateral)weighted shift operators is strongly approxi-mately numerically closed.Theorem 4.The class of triangular operators is strongly approximately numerically closed.Starting with the properties of normal operators,one can define new operator classes according to different properties.An operator T ??(?)is called normaloid if ||T|| r(T).T ??(?)is called to be an hyponormal operator if T*T—TT*? 0.An operator T is transaloid if T—? is normaloid for every ? ?C.T is quasinormal if(T*T)T = T(T*T).An operator T is subnormal if there is a Hilbert space ? and a normal operator N ? B(C)such that ?(?)L,??(?)?and T = N|?.By now there have been a lot results about these new defined operator classes.For the perturbation of the numerical ranges of normaloid operators and hyponormal operators,we have the following theorem.Theorem 5.The class of normaloid operators and the class of hyponormal operators are strongly approximately numerically closed.Notice that both subnormal operators and quasinormal operators are hyponormal.With the same skills and techniques used in the discuss of hyponormal operators,we get the fol-lowing result.Theorem 6.The class of subnormal operators and the class of quasinormal operators are strongly approximately numerically closed.Let ?n(?)denote the set of all nilponet operators of order at most n.That is Nn(?)={T ? ?(?):Tn =0}.About nilpotent operators and quasinilpotent operators we obtain the following theorem.Theorem 7.For each natural number n,Nn(?)and the class of quasinormal operators are strongly approximately numerically closed.In 1978,Cowen and Douglas introduced and studied a new operator relative to com-plex geometry,which later is called Cowen-Douglas operator.As a special operator class,Cowen-Douglas operators contain some triangular operators,weighted shift operators and some conjugation of subnormal operators.It plays a important role in the research of the theory of non-commutative operator approximation.Let ? be a bounded connected open set of C and n be a positive integer.Denote by Bn(?)the set of all operators in ?(?)which satisfying(?)?(?)?p(?)(?)?(T),where?p(T)denotes the point spectrum of ?;(?)dim ker(?—T)= n,for each ? ??;(?)ran(T—?)??,for each ? ??;(?)?{ker(?-T):???}=?The operator in Bn(?)is called to be an Cowen-Douglas operator.For the class of Cowen-Douglas operators,we have the following theorem.Theorem 8.Let ? be a bounded connected open set of C and n be a positive integer.The class of Cowen-Douglas operators are strongly approximately numerically closed.In this thesis,we show that not all of special operator classes are strongly approximately numerically closed by giving a counterexample.Actually,we prove that every pure quasi-normal operator admits a numerical range which is open disk centered at the origin.By this we can easily prove that the class of pure quasinormal operators is not strongly approximately numerically closed.We know that,special operator class may not admit an algebraic structure or topology structure.But the research objectives which we deal with often have some algebraic structure or topology structure.So operator algebras with special structure cause our attention.As a most important part of non-selfadjoint operator algebras,nest algebras play an important role in operator theory and operator algebras,causing most mathematicians' attention.The achievements and research techniques of nest algebras greatly promote the development of operator theory.A nest ? in ? is a chain of closed subspaces of ? containing {0} and ? which is closed under intersection and closed span.The nest algebra associated with ? is defined as T(N)? {T? ?(?):TN(?)?,for every N ?N}.For each ? ??,define N_=? {N'??:?'<N},N+ =?{?'??:?'>?}.The subspaces N(?)?_is called the atoms of ?.If the atoms of ? span ?,then ? is atomic;If there are no atoms,? is called continuous.As different nests,determine different nest algebras and the complication of nest alge-bras.This makes the research work on nest algebras become more difficult in most cases.In this thesis,the approximation of numerical ranges of operators in nest algebras has great deal with the choice of nests.Through our unremitting efforts,we study the different kind of nests,and give unified and concise results.Theorem 9.For any nest N,the nest algebra T(?)associated with ? is strongly approx-imately numerically closed.In 2006,Garcia and Putinar introduced the concept of complex symmetric operators.The research about complex symmetric operators has a close relationship with operator the-ory and function theory,which attracts a lot of mathematicians' and working mathematician'attention.At the end of this thesis,we study the density of non-complex symmetric oper-ators on finite dimensional space.To proceed,we recall some notation and terminology of complex symmetric operators.Definition 10.Let C:??? be a conjugate-linear map.IfC2 = I and... |
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