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COMMUTANTS OF TOEPLITZ OPERATORS WITH MONOMIAL SYMBOLS 5 1 AKAKITIKARADZE 0 2 Abstract. In this note we describe the commutant of the multipli- n cation operator by a monomial in the Toeplitz algebra of a complete a J strongly pseudoconvexReinhardt domain. 0 1 Throughoutgiven an n-tuple of nonnegative integers α = (i ,··· ,i ) and 1 n A] z = (z1,··· ,zn) ∈ Cn, we will denote monomial z1i1···znin by zα. Also, Z+ F will denote the set of all nonnegative integers. . Recall that a boundeddomain Ω ⊂ Cn is said to be a complete Reinhardt h t domain if (z1,··· ,zn) ∈ Ω implies (a1z1,··· ,anzn) ∈ Ω for any complex a numbers a ∈ D¯,1 ≤ i ≤ n, where D = {z ∈ C : |z| < 1} denotes the unit m i disk. [ As usual, the Bergman space of square integrable holomorphic functions 1 on a bounded domain Ω ⊂ Cn is denoted by A2(Ω). Recall that given a v bounded measurable function f ∈ L∞(Ω), one defines the corresponding 3 0 ToeplitzoperatorTf :A2(Ω) → A2(Ω)withsymbolf asTf(φ) = P(fφ),φ ∈ 3 A2(Ω),whereP : L2(Ω) → A2(Ω)istheorthogonalprojection. Ifthesymbol 2 f is holomorphic, then T is the multiplication operator on A2(Ω) with 0 f . symbol f. Let T(Ω) denote the C∗-algebra generated by {Tg : g ∈ L∞(Ω)}. 1 We will refer to this algebra as the Toeplitz C∗-algebra of Ω. 0 5 GivenaToplitzoperatorTf,itisofgreatinteresttostudythecommutant 1 of T in the Toeplitz C∗-algebra of Ω. f v: To this end, in the case of the unit disk Ω = D in C, Z. Cuckovic [[Cu], Xi Theorem 1.4] showed that the commutant of Tzk,k ∈N in the Toeplitz C∗- algebra of D consists of the Toeplitz operators with bounded holomorphic r a symbols. Subsequently T. Le [[Le], Theorem 1.1] has generalized Cuckovic’s result to the case of of the unit ball Ω = B ⊂ Cn and a monomial f = n zm1···zmn such that m > 0 for all 1 ≤ i ≤ n. 1 n i In this note we extend Le’s result to the case of strongly pseudoconvex completeReinhardtdomains. (Theorem0.2.) Moreover, ourproofissimpler and computation free. As in [Cu], [Le], the following result plays the crucial role in the proof of Theorem 0.2. Theorem 0.1. Let Ω ⊂ Cn be a bounded complete Reinhardt domain, and let f = z1m1z2m2···znmn be a monomial such that mi > 0 for all 1 ≤ i ≤ n. If S : A2(Ω) → A2(Ω) is a compact operator that commutes with T , then f S = 0. 1 2 AKAKITIKARADZE We will need the following trivial Lemma0.1. Letφ ∈L∞(Ω).Thenthereexistsǫ > 0suchthat |φ|m+1dµ ≥ Ω ǫ |φ|mdµ, for all m ∈ N. R Ω R Proof. Without loss of generality we may assume that µ(φ−1(0)) = 0. Put Ω = |φ|−1((0,ǫ)). Choose ǫ > 0 so that µ(Ω ) ≤ 1µ(Ω). Then for all m ∈ N ǫ ǫ 2 |φ|m+1dµ ≥ |φ|m+1dµ ≥ ǫ |φ|mdµ. Z Z Z Ω Ω\Ωǫ Ω\Ωǫ But |φ|mdµ ≤ ǫmµ(Ω )≤ |φ|mdµ. ǫ Z Z Ωǫ Ω\Ωǫ Therefore 1 |φ|m+1dµ ≥ ǫ |φ|mdµ. Z 2 Z Ω Ω (cid:3) Proof. (Theorem 0.1.) Let us write f = zτ,τ = (m ,··· ,m ). Since Ω is a 1 n complete Reinhardt domain, it is well-known that monomials {zγ,γ ∈ Zn} + form an orthogonal basis of A2(Ω). Assume that there exits a nonzero compact operator S : A2(Ω) → A2(Ω) that commutes with Tzτ. Thus S(gzmτ) = S(g)zmτ, for all g ∈ A2(Ω),m ∈ N. Let α,β ∈ Zn be such + that hS(zα),zβi 6= 0. Write S(z ) = c zγ,c ∈ C. Thus c 6= 0. It A2(Ω) α γ γ γ β follows that for any m ∈ N P kS(zα+mτ)k = kzmτS(zα)k ≥ |c |kzβ+mτk . A2(Ω) A2(Ω) β A2(Ω) Let β′ ∈ Nn,k ∈ N be such that zβzβ′ = zkτ. Such β′,k exist because m > 0, for all 1 ≤ i ≤ m. Let ǫ′ > 0 be such that ǫ′||gzβ′|| ≤ ||g|| i A2(Ω) A2(Ω) for all g ∈ A2(Ω). Then for ǫ = |c |ǫ′ > 0, we have β kS(zα+mτ)k ≥ ǫkz(m+k)τk ,m ≥ 0. A2(Ω) A2(Ω) By Lemma 0.1, there exists δ′ > 0 such that for all m ≥ 0 kz(m+k)τk ≥ δ′kzmτk . A2(Ω) A2(Ω) Put δ = ǫδ′. Combining the above inequalities we get that for all m ≥ 0 kS(zα+mτ)k ≥ δkzmτk . A2(Ω) A2(Ω) On the other hand since ||zα+mτ||A2(Ω) ≤ ||zα||L∞(Ω)||zmτ||A2(Ω), we get that kS(zα+mτ)kA2(Ω) ≥ δ ,m ∈ N. kzα+mτkA2(Ω) kzαkL∞(Ω) COMMUTANTS OF MONOMIALS 3 However, thesequence{ zα+mτ },m ∈ Nconvergesto0weakly. Thus kzα+mτkA2(Ω) compactness of S implies that kS(zα+mτ)k A2(Ω) lim = 0, m→∞ kzα+mτkA2(Ω) a contradiction. (cid:3) In the proof of Theorem 0.2 we will use the Hankel operators. Recall that given a function φ∈ L∞(Ω), the Hankel operator H :A2(Ω)→ L2(Ω) with φ symbol φ is defined by H (g) = φg−P(φg),g ∈ A2(Ω). φ The proof of the following uses Theorem 0.1 and is essentially the same as in [[Cu], page 282]. Theorem 0.2. Let Ω ⊂ Cn be a bounded smooth strongly pseudoconvex complete Reinhardt domain, and let f = zm1···zmn,m > 0 be a monomial. 1 n i If S is an element of the Toeplitz C∗-algebra of Ω which commutes with T , f then S is a multiplication operator by a bounded holomorphic function on Ω. Proof. Recall that for any g ∈ L∞(Ω) and a holomorphic ψ ∈ A∞(Ω) we have [T ,T ] = H ∗H . On the other hand H is a compact operator for all g ψ ψ¯ g z¯i 1 ≤ i ≤ n, as easily follows from [Pe]. Thus, [T ,T ] is a compact operator g zi for all 1 ≤ i ≤ n,g ∈ L∞(Ω). This implies that the commutator [s,T ] is zi compact for any s ∈ T(Ω),1 ≤ i ≤ n. If S ∈ T(Ω) commutes with T , then f so do compact operators [S,T ],1 ≤ i ≤ n. Therefore, by Theorem 0.2 we zi have [S,T ] = 0,1 ≤ i ≤ n. This implies that S = T for some bounded zi g holomorphic g by [[SSU], proof of Theorem 1.4]. (cid:3) References [Cu] Z. Cuckovic, Commutants of Toeplitz operators on the Bergman space, Pacific J. Math. 162 (1994), no. 2, 277–285. [Le] T.Le,The commutants of certain Toeplitz operators on weighted Bergman spaces, J. Math. Anal. Appl.348 (2008), no. 1, 1–11. [Pe] M. Peloso, Hankel operators on weighted Bergman spaces on strongly pseudoconvex domains, Illinois J. Math. 38 (1994), no. 2, 223–249. [SSU] N.Salinas, A. Sheu,H. Upmeier, Toeplitz operators on pseudoconvex domains and foliation C∗-algebras, Ann.Math. Vol. 130, No. 3 (1989), 531–565. Department of Mathematics, Universiy of Toledo, Toledo, Ohio, USA E-mail address: [email protected]