Commuting Toeplitz and Hankel Operators on Weighted Bergman Space

Transcription

1 Int. J. Contemp. Math. Sciences, Vol. 7, 2012, no. 41, Commuting Toeplitz and Hankel Operators on Weighted Bergman Space Jun Yang 1 Department of Mathematics Shanghai Maritime University Shanghai, , China yangjundlut@yahoo.com.cn Yan Ren College of Automation Shenyang Aerospace University Shenyang , China Abstract In this paper,we give some results concerning Hankel operators and characterize when the Hankel operator and Toeplitz operator commute on weighted Bergman space. Mathematics Subject Classification: 47B35, 47B47 Keywords: Toeplitz operator, Hankel operator, Mellin transform, weighted Bergman space 1 Introduction and Preliminaries Let da denote Lebesgue area measure on the unit disk D, normalized so that the measure of D equals 1. For α > 1, we denote by da α the measure da α (z = (α + 1(1 z 2 α da(z. For 1 p < +, the space L p (D, da α is a Banach space. The weighted Bergman space A 2 α is the closed subspace of analytic functions in the Hilbert space L 2 (D, da α. For each z D, the 1 Supported by the National Natural Science Foundation of China (No , Innovation Program of Shanghai Municipal Education Commission (No.13YZ090 and Science & Technology Program of Shanghai Maritime University (No

2 2036 Jun Yang and Yan Ren application:l z : A 2 α C is continuous and can be represented as L z (f = f(z = f, K z (α α, where K (α z (w = 1 (1 w z = 2+α n=0 Γ(n α n!γ(2 + α (w zn, z, w D. This means that, if P α is the orthogonal projection from L 2 (D, da α onto A 2 α, then P α can be defined by (P α f(w = f, K w (α 1 α = f(z D (1 wz da α(z. 2+α For a function f L (D, da α (z, we define the Toeplitz operator T f : A 2 α A 2 α with symbol f by T f (h = P (fh. It is well known thatt f (h(z = D f(wh(wk(α z (wda α (w, z D. Let U : L 2 (D, da α (z L 2 (D, da α (z be the unitary operator defined by Uf(z = f(z = f( z, where f belongs to L 2 (D, da α (z. Let g be in L 2 (D, da α (z, we define a bounded linear operator M g on L 2 (D, da α (z as: M g (f = gf. Then we can define the small Hankel operator H g : A 2 α(d A 2 α(d ash g = P UM g. In 1964, Brown and Halmos [1] showed the necessary and sufficient conditions for two bounded Toeplitz operators T φ and T ψ commute on Hardy space. On Bergman space of the unit disk, Axler and Cuckovic [2] characterized commuting Toeplitz operators with harmonic symbols. Stroethoff [3] extended their results to essentially commuting Toeplitz operators. With the help of Mellin transform, Cuckovic and Rao [4] studied Toeplitz operators with monomial symbols. Louhichi, Strouse and Zakariasy [5] gave necessary and sufficient conditions for the product of two Toeplitz operators to be a Toeplitz operator and some other results. In this paper, we investigate the commutativity of Toeplitz operators and Hankel operators on the weighted Bergman space of the unit disk. 2 Some Basic Results An operator that will arise in our study of Toeplitz operators is the Mellin transform, defined for any function φ L 1 ([0, 1], rdr, by the formula φ α (z is the Mellin transform : φ α (z = 1 φ(r(1 0 r2 α r z 1 dr. which is a bounded holomorphic function in the half plane z : Rez 2}.

3 Commuting Toeplitz and Hankel operators 2037 Let φ L 1 (D, da α be a radial function, i.e. suppose that:φ(z = φ( z, z D. In fact, if we define the function φ r on [0, 1] by φ r (s = φ(s, then a direct calculation shows that: T φ (z k, z l 0 for k l α = 2(α + 1 φ α (2k + 2 for k = l. So that if k N : 1 T φ (z k Γ(k α = 2(1 + α φr 2k+1 (1 r 2 α drz k k!γ(2 + α 0 Γ(k α = 2(1 + α k!γ(2 + α φ α,r(2k + 2z k. Thus T φ is a diagonal operator on A 2 α with coeffcient sequence Γ(k α (2(1 + α k!γ(2 + α φ α,r(2k + 2 k=0. This makes it relatively simple to work with the product of two operators with such radial symbols. Now, we define the radialization of a function f L 1 (D, da α by : rad(f(z = 1 2π f(e it zdt. It is clear that a function f is a radial if and 2π 0 only if rad(f = f. Let R be the space of square integrable radial functions on D. As before, we identify these functions with the associated functions on [0, 1] that are square integrable with respect to rdr measure. By using that trigonometric polynomials are dense in L 2 (D, da α and that, for k 1 k 2, e ik1θ R is orthogonal to e ik2θ R we see that: L 2 (D, da α = k Z e ikθ R α. Definition 2.1 Let φ be a function in L 1 (D, da α which is of the form e ikθ f where f is a radial function. Then we say that φ is a quasihomogeneous function of quasihomogeneous degree k. and A direct calculation gives the following lemma which we shall use often. Lemma 2.2 Let k, p Z + and let φ be an integrable radial function, then, T e φ(z k Γ(k + p α = 2(α + 1 ipθ (k + p!γ(2 + α φ α(2k + p + 2z k+p, 0 if k < p T e φ(z k = ipθ 2(α + 1 Γ(k p+2+α φ (k p!γ(2+α α(2k p + 2z k p if k p.

4 2038 Jun Yang and Yan Ren and Lemma 2.3 Let k, p Z + and let φ be an integrable radial function, then, H e ipθ φ(z k = 0 0 if k > p H e φ(z k = ipθ 2(α + 1 Γ( k+p+α+2 φ ( k+p!γ(2+α α(p + 2z k+p if k p. Proof. For k, p Z +, we have H e φ(z k = P (U(e ipθ φ(rz k = P ( ipθ e ipθ φ(rz k = P (e ipθ φ(rz k = Γ(n + α + 2 n!γ(α + 2 eipθ φ(rz k, z n α z n n 0 = n 0 Γ(n + α + 2 n!γ(α + 2 If 0 k p, that H e ipθ φ(z k 1 2π 0 0 φ(rr k+n+1 e i( k+p nθ (1 + α(1 r 2 α dθ π drzn. 1 2π Γ( k + p + α + 2 = φ(rr k+p k+1 (1 + α(1 r 2 α dθ ( k + p!γ(α π drzn Γ( k + p + α = 2(1 + α φ(rr p+1 (1 r 2 α drz k+p ( k + p!γ(α Γ( k + p + α + 2 = 2(1 + α ( k + p!γ(α + 2 φ α(p + 2z k+p. Similarly, H e ipθ φ(z k = 0, fork > p. Thus it is easy to get H e ipθ φ(z k = 0. 3 Commutativity of Toeplitz and Hankel operators Theorem 3.1 Let p 1, p 2 Z + and φ 1 (r, φ 2 (r be two bounded radial functions. Then H e ip 1 θ φ 1 (rh e ip 2 θ φ 2 (r = H e ip 2 θ φ 2 (rh e ip 1 θ φ 1 (r if and only if one of the following conditions holds: (ap 1 = p 2 ; (b φ 1 (p = 0 or φ 2 (p = 0.

5 Commuting Toeplitz and Hankel operators 2039 Proof. On one hand, since for k 0. H e ip 1 θ φ 1 (rh e ip 2 θ φ 2 (r(z k = 0 if k > p2 H e ip 1 θ φ 1 (r2(α + 1 Γ( k+p 2+α+2 ( k+p 2!Γ(α+2 φ α,2(p 2 + 2z k+p 2 if p 2 k 0. (i If 0 k p 2 and p 2 p 1 k p 2, then H e ip 1 θ φ 1 (rh e ip 2 θ φ 2 (r(z k = 2(α + 1 Γ( k + p 2 + α + 2 ( k + p 2!Γ(α + 2 φ α,2(p 2 + 2H e ip 1 θ φ 1 (r(z k+p 2 = 2(α + 1 Γ( k + p 2 + α + 2 ( k + p 2!Γ(α + 2 2(α + 1Γ(p 1 p 2 + k + α + 2 (p 1 p 2 + k!γ(α + 2 φ α,2 (p φ α,1 (p 1 + 2(z k p 2+p 1 (ii In other conditions, it is easy to get H e ip 1 θ φ 1 (rh e ip 2 θ φ 2 (r(z k = 0. On the other hands, (i If 0 k p 1 and p 1 p 2 k p 1, then H e ip 2 θ φ 2 (rh e ip 1 θ φ 1 (r(z k = 4(α Γ( k + p 1 + α + 2 ( k + p 1!Γ(α + 2 φ α,1(p φ α,2 (p Γ(p 2 p 1 + k + α + 2 (p 2 p 1 + k!γ(α + 2 (zk p 1+p 2. (ii In other conditions,h e ip 2 θ φ 2 (rh e ip 1 θ φ 1 (r(z k = 0. If p 1 = p 2, it is easy to get H e ip 1 θ φ 1 (rh e ip 2 θ φ 2 (r = H e ip 2 θ φ 2 (rh e ip 1 θ φ 1 (r. If p 1 p 2, without lost of generality, we can suppose p 1 < p 2, then we obtain (1 If p 2 < 2p 1, and p 1 p 1 k p 1, 4(α Γ( k + p 1 + α + 2 Γ(p 1 p 2 + k + α + 2 ( k + p 2!Γ(α + 2 (p 1 p 2 + k!γ(α + 2 φ α,2(p φ α,1 (p 1 + 2(z k p 2+p 1 = 4(α Γ( k + p 1 + α + 2 ( k + p 1!Γ(α + 2 φ α,1 (p 1 + 2(z k p 1+p 2 then we get φ α,1 (p = 0 or φ α,2 (p = 0. Γ(p 2 p 1 + k + α + 2 (p 2 p 1 + k!γ(α + 2 φ α,2(p 2 + 2

6 2040 Jun Yang and Yan Ren (2 If 2p 1 < p 2, and 0 k p 1, 4(α Γ( k + p 1 + α + 2 ( k + p 1!Γ(α + 2 φ α,2(p Γ(p 2 p 1 + k + α + 2 (p 2 p 1 + k!γ(α + 2 φ α,1(p (z k p 1+p 2 = 0, that is to see φ α,2 (p = 0 or φ α,1 (p = 0. (3 If p 2 p 1 k p 1, 4(α Γ( k + p 2 + α + 2 ( k + p 2!Γ(α + 2 φ α,2(p Γ(p 1 p 2 + k + α + 2 (p 1 p 2 + k!γ(α + 2 φ α,1(p (z k p 2+p 1 = 0, then we also get φ α,2 (p = 0 or φ α,1 (p = 0. Conversely the equation is hold on. Analogous the proof above we can also have the Theorem below. Theorem 3.2 Let φ 1 and φ 2 be two integrable radial functions on D such that T φ1 and H φ2 are bounded operators, then T φ1 H φ2 = H φ2 T φ1. References [1] A. Brown, P. R. Halmos, Algebraic properties of Toeplitz operators, J. Reine Angew. Math., 213(1963, [2] S. Axler, Zeljko Cuckovic, Commuting Toeplitz operators with harmonic symbols, Integral Equations Operator Theory, 14(1991, [3] K. Stroethof, Essentially commuting Toeplitz operators with harmonic symbols, Can. J. Math., 45(1993, [4] Zeljko Cuckovic, N. V. Rao, Mellin transform, monomial symbols, and commuting Toeplitz operators, J. Funct. Anal., 154(1998, [5] I. Louchichi, E. Strouse, L. Zakariasy, Products of Toeplitz operators on the Bergman space, Integral Equations Operator Theory, 54(2006, Received: July, 2012