Research On Some Problems In Operator Theory

The topological structure of the group G(A)of invertible elements in a unital Banach algebra A has attracted topologists f'rom the very beginning of homotopy theory.One landmark among them is Kuiper's theorem.In[116],N.H.Kuiper showed that G(L(H))is contractible,where G(L(H))denotes the group of invertible operators acting on an infinite dimensional complex Hilbert space H.Naturally,the problem of the connected-ness of G(A)should be first considered.It is well known that the invertible group of a von Neumann algebra is path connected in the norm topology.On the other hand,the algebra H? of bounded analytic functions on the unit disk,denoted by G(H?),is not connected.Actually,the topological structure of G(H?)is rather complicated,see[30]for more inf'ormation.However,for a certain invertible function f=eh,which cannot be connected to the constant f'unction 1 via a norm continuous path within G(H?)[14],D.R.Pitts showed that the Toeplitz operator with symbol f could be connected to the identity via a norm continuous path of invertible elements of a nest algebra[31].The main motivation is due to a strong analogy between H,and the "simplest" nest algebra,i.e.,all lower triangular operators with respect to a fixed orthonormal basis {en}n=1? for a complex separable Hilbert space H.In the second chapter,we will show that the result of D.R.Pitts holds in general.We prove that there exists an orthonormal basis F for classical Hardy space H2,such that each invertible analytic Toeplitz operator could be connected to the identity via a norm continuous path of invertible elements of the lower triangular operators with respect to F.The main results are as follows:Proposition 1.Let ?={rn}n=1? be a sequence of real numbers in[0,1)which has a cluster point r in[0,1).Define(?)Then F?={fn:n ? N} is an orthonormal basis for H2.Moreover,for T??A(S),T? admits a matrix representation of the form(?)with respect to F?.Corollary 1.Let ?={rn}n-1? be a sequence of real numbers in[0,1)which has a cluster point r in[0,1).Let F? be the orthonormal basis for H2 described in Proposition 2.2.2.Define P0={0},Pn=span{fi:1?i?n},n ?N.We write the nest T?={Pn,H2:0?n<+?}.Then the mapping ??:??T? is an isometric embedding from H? into T(P??).Notice that T(P??)is the algebra of lower triangular operators with respect to F?.Theorem 1.Suppose that R is an invertible coanalytic Toeplitz operator,then R is connected to I in T(P)-1.Theorem 2.Suppose that ?,??G(H?).Then T? is connected to T? in T(P?)-1 where T(P?)is exactly the algebra of the lower triangular operators with respect to F.In chapter three.Let H be a separable infinite dimensional complex Hilbert space,and L(H)be the algebra of bounded linear operators on H.For T?L(H),let a(T)be the spectrum of T.We say T is quasinilpotent if ?(T)={0}.Denote by LatT the lattice of invariant subspaces and {T}' the algebra of operators commuting with T.If M ?(?) LatA,then M is called a hyperinvariant subspace of T.The power set A(T)of quasinilpotent operator T was introduced by R.G.Douglas and R.Yang(see[62]and[63])in 2016.Definition 1.Suppose that T ? L(H)is quasinilpotent and x?H\{0}.Let(?)Abbreviate k(T,x)to kx unless otherwise specified.Set A(T)={kx:x?0},and call it the power set of T.Since(?),so k?x=kx for all ??0 and x?0.And hence ?(T)(?)[0,1].If T=0,then A(T)={1}.In[62]and[63],the power set of nilpotent operator and the classical Volterra operator are calculated.The power set provide a method to consider hyperinvariant subspace problem for T ?L(H)with an isolated spectral point.In particular,the hyperinvariant subspace problem of quasinilpotent operators can be discussed by means of power set.It was obtained that for each 0???1,the set M?={x?H:x=0 or kx ??} is a hyperinvariant subspace for T if M? is closed set.They found that if A(T)contains at least two points and M? is closed,then T has a nontrivial hyperinvariant subspace.The following question was raised by R.G.Douglas and R.Yang in 2018[63]:is there a nontrivial quasinilpotent operator T such that A(T)is a single point?This question was answered by Y.X.Liang and R.Yang in[65]for the Volterra integral operator Vh on the classical Hardy-Hilbert space.Naturally,we consider the f'ollowing:what other quasinilpotent operators with sin-gleton power set have nontrivial hyperinvariant subspaces besides Volterra integral opera-tor?If this problem can be answered,then the hyperinvariant subspace of quasinilpotentoperators will be solved.In chapter three,we give a class of weighted shifts with singleton power set {1},and construct a weighted shift with power set[0,1].Theorem 3.Let Aen=?nen+1(n?0)be an injective unilateral weighted shift,ifA is strongly strictly cyclic quasinilpotent operator,then A(A)={1}.Corollary 2.Let Aen=?nen+1(n?0)be an injective unilateral weighted shift with weight sequence that is monotone decreasing and in lp for some 1?p<?,then?A(A)={1}.Proposition 2.For a natural number k,let nk=2k and=4-k.Suppose that {ej(k)}j=1nk is an orthonormal basis of Hk and (?) and Ak=akJnk.Putting (?).Then A(A)=[0,1].In chapter four,we will discuss truncated Toeplitz operators on model space over the upper half plane.Model spaces are Hilbert spaces of the form where 6 is an inner function,H2 is the classical Hardy space on the open unit disk D,and ?denotes the orthogonal complement in H2.On a functional analysis level,model spaces are the orthogonal complements of the nontrivial invariant subspaces of the unilateral shift Sf=zf on H2.These subspaces were characterized as ?H2 by Beurling in his famous 1949 paper.In other words,the spaces(?H2)?are the invariant subspaces of the backward shift operator S*f=f-f(0)/z on H2.In 2007,D.Sarason[77]showed the properties of functions in model space in detail,and he gave the definition of truncated Toeplitz operators.Then he proved that all trun-cated Toeplitz operators are complex symmetric operators,its algebraic characterization and the concrete forms of zero rank,one rank and finite rank operators are obtained.On this basis,we know that the classical Hardy space defined on the unit disk D can be transformed into Hardy space on the upper half plane C+ by an unitary operator.So it is natural to transform general model space into model space over the upper half plane,and corresponding truncated Toeplitz operator can also be defined.Let ?+ be an inner function,the corresponding model space over the upper half plane K?+ is defined to be K?+=H2(C+)(?)?+H2(C+).Let ??L2(R)and ?H?(C+)be inner function.We define truncated Toeplitz operator by:A?f=P?+(?f),for all f?K?+,where P?+ is projection from L2(R)to K?+.In chapter four,we describe algebra properties of truncated Toeplitz operators on model space over the upper half plane.We extend the results obtained by D.Sarason in 2007.These results seem trivial,but the calculations are complicated.The main results are as follows:Theorem 4.The operators in J(K?+)are C+-symmetric.Theorem 5.(i)For ??C+,then(?)(?)For ??C+\{i},we get(?)(?)(?)For ??C+\{i} we obtain that(?)(?)Theorem 6.If ??L2(R),then A?=0 if and only if ??(z+i)(?+H2(C+)+(?)Theorem 7.The bounded operator A+?J(K?+)if and only if there are functions?+,?+ in K?+ such that A+-S?+A+S?+*=(1-?(z))?+(?)hi+hi(?)(1+?(z))?+,where (?)Theorem 8.The following assertions are true:(?)Let ??C+,then the operators h?(?) and (?)h? are in J(K?+).(?)(?)(?)If ?+ has an ADC+ at the point?of R,then h?(?)h? is the truncated Toeplitz operator with symbol(?)(iv)The only rank-one operators in J(K?+)are the nonzero scalar multiples of the operators in(i)and(?).