Toeplitz Operators On The Spaces Of Analytic Functions With Symmetric Measures

In this paper, let D, (?) and (?)D denote the open unit disk, the closed unit disk andthe unit circle in the complex plane C, respectively. Let H be a complex separableinfinite dimensional Hilbert space, and let B(H) denote the algebra of all boundedlinear operators on H. We denote by K(H) the ideal of compact operators in (?).A'(T) and A'-e(T) denote the commutant {A∈B(H) : AT＝TA} and the essentialcommutant {A∈B(H) : AT -TA∈K(H)} of T, respectively.The theory of Toeplitz operators is not only a bridge between function theoryand operator theory, but also has many important applications in control, quantummechanics, probability and statistics, etc. Hence the theory of Toeplitz operators hasbeing the hot problem in the fields of operator theory and analysis, which has attracted the attentions of many scholars.In recent decades, the enthusiasms of the mathematicians for the research of Toeplitz operators on function spaces have been escalated continuously, and many important achievements have been obtained. There are many similarities as well as dif-fereces between the theory of Toeplitz operators on Hardy space H²((?)D) and Bergman space A²(D, dA). In this paper, we consider Toeplitz operators on the spaces of analytic functions with symmetric measures. A finite positive Borel measureμon (?) is said to be a symmetric measure if there is a finite positive Borel measure v on the closed unit interval [0, 1] such that for any continuous function f(z) on (?). In this case, the measure v is called the radialcomponent ofμ. In order thatμis a normalized measure on (?), we supposeÏ…([0,1]) = 1.Throughout this paper,μalways denotes a normalized symmetric measure on (?) andv always denotes the radial component ofμ. For 1≤p <∞, let L^p((?), dμ) denote thespace of measurable functions f on (?) such thatA^p((?), dμ) is the set of those functions in L^p((?), dμ) that are analytic on D. L^∞((?), dμ)denote the set of the essentially bounded measurable functions on (?) respect todμ. As usual, the space of bounded analytic functions on D is denoted by H^∞(D).By the results of Frankfurt, we know that if 1∈supp v then A^p((?), dμ) is theclosed subspace of L^p((?),dμ), the set of polynomials is dense in A²((?), dμ), and K_z(ω) =(?)∈D, is the reproducing kernel of A²((?),dμ). Throughout this paper, wealways suppose that 1∈supp v in order that A²((?),dμ) is a reproducing kernelHilbert space.Now we begin with the definition of Toeplitz operators on A²((?), dμ).Definition 0.1 Let P be the orthogonal projection from L²((?),dμ) onto A²((?),dμ).Forφ∈L^∞((?),dμ), the Toeplitz operator T_φon A²((?),dμ) is defined byIfφ∈H^∞(D), then the Toeplitz operator T_φis called the analytic Toeplitz operatorwith symbolφ.Toeplitz operators can also be defined for unbounded symbols.Whenφ∈L¹((?), dμ), we define a linear operator T_φby T_φf = P(φf) for f∈A²((?). dμ).T_φmay be unbounded on A²((?),dμ), but it is always densely defined. It is easy to seethat H^∞(D), which is dense in A²((?). dμ), is contained in the domain of T_φ.In this paper we will focus on Toeplitz operators on A²((?). dμ).Since Brown and Halmos characterized the algebraic properties of Toeplitz operators on H²((?)D) in 1963, the algebraic properties of Toeplitz operators on variousfunction spaces have received the very big attentions. In chapter 2, we investigate mainly the algebraic properties of Toeplitz operators on A²((?),dμ). We first introduce some fundamental properties of Toeplitz operators, and prove that there are no compact Toeplitz operators with harmonic symbols other than the zero operator, and also consider the spectrums of Toeplitz operators with real-valued harmonic symbols.For a linear operator T (not necessarily bounded) on A²((?),dμ) whose domain contains H^∞(D), the Berezin transform of T is the function BT(z) on D defined by BT(z) = (Tk_z,k_z),where k_z is the normalized reproducing kernel K_z/||K_z||₂.The Berezin transform Bφ(z) of a functionφ∈L¹ ((?),dμ) is defined to be the Berezin transform of the Toeplitz operator T_φ.If f∈L¹((?),dμ),then the radialization of f.denoted by f^#,is the function on D defined byWe characterize the relationship between Berezin transform and harmonic functions,and obtain the following proposition.Proposition 0.2 Ifφ∈L¹((?),dμ),thenφis harmonic on D if and only if B_φ=φandφ^#∈C((?)).We apply the above proposition to obtaining the following theorem.Theorem 0.3 Suppose that f and g are bounded harmonic functions on D. For the Toeplitz operators T_f,T_g on A²((?),dμ),then the following results hold:(1) T_fT_g=T_fg if and only if either g is analytic on D, or f is analytic on D.(2) T_fT_g=T_gT_f if and only if either f and g are analytic on D,or (?) and (?) are analytic on D,or there exist constants a, b∈C,not both 0,such that af+bg is constant on D.We also consider when the product of two Toeplitz operators with harmonic symbols is still a Toeplitz operator with a harmonic symbol. In particular, if supp v= {0.1},then on A²((?),dμ) we obtain the sufficient and necessary conditions that the product of two Toeplitz operators with harmonic symbols is still a Toeplitz operator with a harmonic symbol. Hence, on A²((?),dμ),we solve the zero product problem of two Toeplitz operators with harmonic symbols, and prove that there are no nontrivial isometric Toeplitz operators with harmonic symbols and nontrival idempotent Toeplitz operators with harmonic symbols.Finally, we discuss the Toeplitz operators with general symbols and prove that the commutator idea of (?)(H^∞(D)+C((?))) is K(A²((?),dμ)),where (?)(H^∞(D)+C((?))) is the norm closed subalgebra generated by {T_φ(?)φ∈H^∞(D)+C((?))}.In chapter 3,we study the reducing subspaces and commutants of analytic Toeplitz operators on A²((?),dμ).It is interest to characterize the commutant of an operator, for such a characterization should help us to understand the structure of the operator. But, except for normal operators, there exist only a few operators whose commutants are clear. In fact, it is very difficult to characterize the commutants of a special class of subnormal operators-analytic Toeplitz operators, too. In 1973, Deddens and Wong characterized the commutants of analytic Toeplitz operators on H²((?)D) in terms of the commutants of isometric operators and raised six questions which have stimulated much of the further work on the problem. Subsequently, continuing the work of Deddens and Wong, a series of results were obtained. Now we know that there are many differences between the reducing subspaces of analytic Toeplitz operators on H²((?)D) and that on A²(D,dA). On H²((?)D),the Toeplitz operator induced by a finite Blaschke product of order greater than one has finitely many minimal reducing subspaces. However, on A²(D, dA), the Toeplitz operator induced by a Blaschke product of order two has only two nontrivial reducing subspaces.Firstly, we characterize the reducing subspaces of T_? by the results of Stessin and Zhu, and obtain the following theorem.Let M_i=∨{z^kn+i,k=0,1.2,…},i=0,1,…,n-1.Theorem 0.4 (1) Suppose that supp v = {0,1}.(i) When n = 2, M₀(?)M₁ are the only nontrivially minimal reducing subspaces of T_z² on A²((?),dμ). Moreover,T_z² has exactly two nontrivially reducing sub- spaces.(ii) When n>2,T_zⁿ on A²((?),dμ) has nontrivially minimal reducing subspaces other than M_i,0≤i≤n-1.In fact,T_zⁿ has infinitely many distinct minimal reducing subspaces in A²((?),dμ).(2) If supp v is not a subset of {0,1}, then M_i,0≤i≤n-1, are the only nontrivially minimal reducing subspaces of T_zⁿ on A²((?),dμ).Moreover,T_zⁿ has exactly 2ⁿ-2 nontrivially reducing subspaces.The above theorem indicates that the reducing lattice of T_zⁿ on A²((?),dμ) is related to the measureμ.Secondly, we characterize the extremal function of M which is a reducing subspace of an analytic Toeplitz operators on A²((?),dμ).For any closed subspaces M ofA²((?),dμ),letWe call that m is the order of zero of M at the origin.Theorem 0.5 Let M≠{0} be a reducing subspaces of T_φon A²((?),dμ),φ∈H^∞(D),φ(0)=0,and let m be the order of zero of M at the origin, then the extremal problemhas a unique solution G with ||G||₂-1 and G^(m)(0)>0.Furthermore,G∈ker T_φ^*The function G in the theorem above is called the extremal function of M.We obtain a sufficient condition that an analytic Toeplitz operator is irreducible by the above theorem.Corollary 0.6 Ifφ∈H^∞(D) and dim ker T_{φ-φ(0)}^*= 1,then T_φis irreducible.When supp v={0.1},we also study the reducing subspaces of the Toeplitz operator induced by a finite Blaschke product of order.the Toeplitz operator induced by a Blaschke product of order two has only two nontrivial reducing subspaces, and the Toeplitz operator induced by a Blaschke product of order greater than two has infinitely nontrivial reducing subspaces.In section 3.2, we extend the results of Deddens and Wong to the space A²((?),dμ). and obtain a new characterization of the commutants of analytic Toeplitz operators. Our main results are the following theorems.Theorem 0.7 Let g(z) = b₀+b₁z^p1+b₂z^p2+…∈H^∞(D),b_k≠0, k=1,2,…,thenA'(T_zⁿ)∩A'_e(T_g)=A'(T_zⁿ)∩A'(T_g)=A'(T_z^s),s=gcd(n,p₁,p₂,…). Here s=gcd(n,p₁,p₂,…) denotes that s is the great common divisor of n,p₁,p₂,….Theorem 0.8 Letφ=zⁿ F and F∈H^∞(D) is an outer function, thenwhere s = max{r: there is f,g∈H^∞(D) such that zⁿ-f(z^r) and F(z) = g(z^r)}.In the case of A²(D,dA),(?)u(?)kovi(?) raised the following question in 1994.If T essentially commutes with Bergman shift,then what is the set of all functions f such that TT_f=T_fT implies that T is an analytic Toeplitz operator?In 2000, Axler,(?)u(?)kovi(?) and Rao raised the analogous question again.If an operator T in the algebra generated by the Toeplitz operators commutes with a nonconstant analytic Toeplitz operator, then is T an analytic Toeplitz operator?The following corollaries are related to the above question. Let (?)(L^∞((?),dμ)) denotes the norm closed subalgebra generated by {T_φ:φ∈L^∞((?),dμ)}.Corollary 0.9 Suppose that S∈(?)(L^∞((?),dμ)) commutes with T_zⁿF,where F∈H^∞(D) is an outer function, then there isψ∈H^∞(D) such that S=T_ψCorollary 0.10 Suppose that the composition operators on A²((?),dμ) induced by M(o|¨)bius transforms are bounded invertible operators, andφ=φ_aⁿF,whereφ_a(z)=(?),a∈D, F∈H^∞(D) is an outer function. If S∈(?)(L^∞((?),dμ))∩A'(T_φ),then S is an analytic Toeplitz operator. Corollary 0.11 Suppose that the composition operators on A²((?),dμ) induced by M(o|¨)bius transforms are bounded invertible operators. If S∈(?)(L^∞((?), dμ))∩A'(T_φ),φ(z)=φ_a(z)φ_b(z),a,b∈D,then S is an analytic Toeplitz operator.In section 3.3, we consider the commutant of T_φ,where the inner factor ofφ∈H^∞(D) is a finite Blaschke product.