arXiv:1308.2214v1 [math.FA] 9 Aug 2013
TOEPLITZNESS OF COMPOSITION OPERATORS IN SEVERAL VARIABLES ˇ ˇ CKOVI ˇ ´ AND TRIEU LE ZELJKO CU C Abstract. Motivated by the work of Nazarov and Shapiro on the unit disk, we study asymptotic Toeplitzness of composition operators on the Hardy space of the unit sphere in Cn . We extend some of their results but we also show that new phenomena appear in higher dimensions.
1. Introduction Let Bn denote the unit ball and Sn the unit sphere in Cn . We denote by σ the surface area measure on Sn , so normalized that σ(Sn ) = 1. We write L∞ for L∞ (Sn , dσ) and L2 for L2 (Sn , dσ). The Hardy space H 2 consists of all analytic functions h on Bn which satisfy Z |h(rζ)|2 dσ(ζ) < ∞. khk2 = sup 0 m0 . Therefore, limm→∞ kΦm (K)k = 0. Now for f a bounded function on Sn , we have Φm (Tf + K) = Φm (Tf ) + Φm (K) = Tf + Φm (K) −→ Tf in the norm topology as m → ∞. This shows that Tf + K is UAT with asymptotic symbol f . In dimension one, Theorem 1.1 shows that the converse of Lemma 3.1 holds. On the other hand, Theorem 1.1 fails when n ≥ 2. We shall show that there exist composition operators that are UAT but cannot be written in the form “Toeplitz + compact”. We first show that composition operators cannot be written in the form “Toeplitz + compact” except in trivial cases. This generalizes Theorem 1.2 to all dimensions. Theorem 3.2. Let ϕ be an analytic selfmap of Bn such that Cϕ is bounded on H 2 . If Cϕ can be written in the form “Toeplitz + compact”, then either Cϕ is compact or it is the identity operator. Proof. Our proof here works also for the one-dimensional case and it is different from Nazarov–Shapiro’s approach (see the proof of Theorem 1.1 in [9]). Suppose Cϕ is not the identity and Cϕ = Tf + K for some compact operator K and some bounded function f . By Lemma 3.1, Cϕ is UAT with asymptotic symbol f on the unit sphere. This then implies that Cϕ is also MSAT with asymptotic symbol f . From Corollary 2.3 we know that Cϕ , being a non-identity bounded composition operator, is MSAT with asymptotic symbol zero. Therefore f = 0 a.e. and hence Cϕ = K. This completes the proof of the theorem. We now provide an example which shows that the converse of Lemma 3.1 (and hence Theorem 1.1) does not hold in higher dimensions. Example 3.3. For z = (z1 , . . . , zn ) in Bn , we define ϕ(z) = (ϕ1 (z), . . . , ϕn (z)) = (0, z1 , 0, . . . , 0). Then ϕ is a linear operator that maps Bn into itself. It follows from [3, Lemma 8.1] that Cϕ is bounded on H 2 and Cϕ∗ = Cψ , where ψ is a linear map given by ψ(z) = (ψ1 (z), . . . , ψn (z)) = (z2 , 0, . . . , 0).
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We claim that Φ(Cϕ ) = 0. For j 6= 2, Cϕ Tzj = Tϕj Cϕ = 0 since ϕj = 0 for such j. Also, (Tz 2 Cϕ )∗ = Cϕ∗ Tz2 = Cψ Tz2 = Tψ2 Cψ = 0. Hence Tz 2 Cϕ = 0. It follows that Φ(Cϕ ) = Tz1 Cϕ Tz1 + · · · + Tzn Cϕ Tzn = 0, which implies Φm (Cϕ ) = 0 for all m ≥ 1. Thus, Cϕ is UAT with asymptotic symbol zero. On the other hand, since (ϕ ◦ ψ)(z) = (0, z2 , 0, . . . 0), we conclude that for any non-negative integer s, Cϕ∗ Cϕ (z2s ) = Cψ Cϕ (z2s ) = Cϕ◦ψ (z2s ) = z2s . This shows that the restriction of Cϕ on the infinite dimensional subspace spanned by {1, z2 , z22 , z23 , . . .} is an isometric operator. As a consequence, Cϕ is not compact on H 2 . Theorem 3.2 now implies that Cϕ is not of the form “Toeplitz + compact” either. Theorem 1.2 shows that on the Hardy space of the unit disk, a composition operator Cϕ is UAT if and only if it is either a compact operator or the identity. Example 3.3 shows that in dimensions n ≥ 2, there exists a noncompact, non-identity composition operator which is UAT. It turns out that there are many more such composition operators. In the rest of the section, we study uniform asymptotic Toeplitzness of composition operators induced by linear selfmaps of Bn . We begin with a proposition which gives a lower bound for the norm of the product Tf Cϕ when ϕ satisfies certain conditions. This estimate will later help us show that certain composition operators are not UAT. Proposition 3.4. Let ϕ be an analytic selfmap of Bn such that Cϕ is bounded. Suppose there are points ζ, η ∈ Sn so that hϕ(z), ηi = hz, ζi for a.e. z ∈ Sn . Let f be a bounded function on Sn which is continuous at ζ. Then we have kTf Cϕ k ≥ |f (ζ)|. s Proof. For an integer s ≥ 1, put gs (z) = 1 + hz, ηi and hs = Cϕ gs . Then for a.e. z ∈ Sn , s s hs (z) = gs (ϕ(z)) = 1 + hϕ(z), ηi = 1 + hz, ζi . Because of the rotation-invariance of the surface measure on Sn , we see that khs k = kgs k. Now, we have kTf Cϕ k ≥
|hTf hs , hs i| |hTf Cϕ gs , hs i| |hf hs , hs i| = = kgs kkhs k kgs kkhs k khs k2 R f (z)|1 + hz, ζi|2s dσ(z) = SnR . 2s Sn |1 + hz, ζi| dσ(z)
We claim that the limit as s → ∞ of the quantity inside the absolute value is f (ζ). From this the conclusion of the proposition follows.
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To prove the claim we consider R 2s R |f (z) − f (ζ)| · |1 + hz, ζi|2s dσ(z) SnR f (z)|1 + hz, ζi| dσ(z) R − f (ζ) ≤ Sn 2s dσ(z) 2s |1 + hz, ζi| Sn Sn |1 + hz, ζi| dσ(z) R R 2s U + Sn \U |f (z) − f (ζ)| · |1 + hz, ζi| dσ(z) R = 2s Sn |1 + hz, ζi| dσ(z) R 2s S \U |1 + hz, ζi| dσ(z) ≤ sup |f (z) − f (ζ)| + 2kf k∞ Rn , (3.1) 2s z∈U Sn |1 + hz, ζi| dσ(z)
where U is any open neighborhood of ζ in Sn . By the continuity of f at ζ, the first term in (3.1) can be made arbitrarily small by choosing an appropriate U . For such a U , we may choose another open neighborhood W of ζ with W ⊆ U such that sup |1 + hz, ζi| : z ∈ Sn \U < inf |1 + hz, ζi| : z ∈ W .
This shows that the second term in (3.1) converges to 0 as s → ∞. The claim then follows.
Using Proposition 3.4, we give a sufficient condition under which Cϕ fails to be UAT. Proposition 3.5. Let ϕ be a non-identity analytic selfmap of Bn such that Cϕ is bounded. Suppose that ϕ is continuous on Bn and there is a point ζ ∈ Sn and a unimodular complex number λ so that hϕ(z), ζi = λhz, ζi for all z ∈ Sn . Then Cϕ is not UAT. Proof. Since ϕ is a non-identity map, Corollary 2.3 shows that Cϕ is MSAT with asymptotic symbol zero. To prove that Cϕ is not UAT, it suffices to show that Cϕ is not UAT with asymptotic symbol zero. Let f (z) = hϕ(z), zi for z ∈ Sn . By the hypothesis, the function f is continuous on Sn and f (ζ) = hϕ(ζ), ζi = λhζ, ζi = λ. For any positive integer m, formula (2.1) gives Φm (Cϕ ) = Tf m Cϕ . Since ϕ satisfies the hypothesis of Proposition 3.4 with η = λζ and f m is continuous at ζ, we may apply Proposition 3.4 to conclude that kΦm (Cϕ )k = kTf m Cϕ k ≥ |f m (ζ)| = 1. This implies that Cϕ is not UAT with asymptotic symbol zero, which is what we wished to prove. Our last result in the paper provides necessary and sufficient conditions for a class of composition operators to be UAT. Theorem 3.6. Let ϕ(z) = Az where A : Cn → Cn is a non-identity linear map with kAk ≤ 1. Then Cϕ is UAT if and only if all eigenvalues of A lie inside the open unit disk.
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ˇ ˇ CKOVI ˇ ´ AND TRIEU LE ZELJKO CU C
Proof. Since kAk ≤ 1, all eigenvalues of A lie inside the closed unit disk. We first show that if A has an eigenvalue λ with |λ| = 1, then Cϕ is not UAT. Let ζ ∈ Sn be an eigenvector of A corresponding to λ. We claim that A∗ ζ = λζ. In fact, we have |(A∗ − λ)ζ|2 = |A∗ ζ|2 − 2ℜhA∗ ζ, λζi + |λζ|2 = |A∗ ζ|2 − 2ℜhζ, λAζi + |ζ|2 = |A∗ ζ|2 − 2ℜhζ, λλζi + |ζ|2 = |A∗ ζ|2 − 1 ≤ 0. This forces A∗ ζ = λζ as claimed. As a result, for z ∈ Sn , we have hϕ(z), ζi = hAz, ζi = hz, A∗ ζi = hz, λζi = λhz, ζi. We then apply Proposition 3.5 to conclude that Cϕ is not UAT. We now show that if all eigenvalues of A lie inside the open unit disk then Cϕ is UAT. Put f (z) = hϕ(z), zi = hAz, zi for z ∈ Sn . Since |Az| ≤ 1 and Az is not a unimodular multiple of z, we see that |f (z)| < 1 for z ∈ Sn . Since f is continuous and Sn is compact, we have kf kL∞ (Sn ) < 1. For any integer m ≥ 1, formula (2.1) gives kΦm (Cϕ )k = kTf m Cϕ k ≤ kTf m kkCϕ k ≤ (kf kL∞ (Sn ) )m kCϕ k. Since kf kL∞ (Sn ) < 1, we conclude that limm→∞ kΦm (Cϕ )k = 0. Therefore, Cϕ is UAT with asymptotic symbol zero. Acknowledgements. We wish to thank the referee for useful suggestions that improved the presentation of the paper. References [1] Jos´e Barr´ıa and P. R. Halmos, Asymptotic Toeplitz operators, Trans. Amer. Math. Soc. 273 (1982), no. 2, 621–630. MR 667164 (84j:47040) [2] Arlen Brown and Paul R. Halmos, Algebraic properties of Toeplitz operators, J. Reine Angew. Math. 213 (1963/1964), 89–102. MR 0160136 (28 #3350) [3] Carl C. Cowen and Barbara D. MacCluer, Composition operators on spaces of analytic functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995. MR 1397026 (97i:47056) ˇ ˇ ckovi´c and Mehdi Nikpour, On the Toeplitzness of the adjoint of composition [4] Zeljko Cuˇ operators, J. Math. Anal. Appl., to appear. [5] A. M. Davie and N. P. Jewell, Toeplitz operators in several complex variables, J. Functional Analysis 26 (1977), no. 4, 356–368. MR 0461195 (57 #1180) [6] Ronald G. Douglas, Banach algebra techniques in operator theory, Academic Press, New York, 1972, Pure and Applied Mathematics, Vol. 49. MR 0361893 (50 #14335) [7] Avraham Feintuch, On asymptotic Toeplitz and Hankel operators, The Gohberg anniversary collection, Vol. II (Calgary, AB, 1988), Oper. Theory Adv. Appl., vol. 41, Birkh¨ auser, Basel, 1989, pp. 241–254. MR 1038338 (91f:47040) [8] Valentin Matache, S-Toeplitz composition operators, Complex and harmonic analysis, DEStech Publ., Inc., Lancaster, PA, 2007, pp. 189–204. MR 2387289 (2009c:47029) [9] Fedor Nazarov and Joel H. Shapiro, On the Toeplitzness of composition operators, Complex Var. Elliptic Equ. 52 (2007), no. 2-3, 193–210. MR 2297770 (2008a:47043)
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[10] Walter Rudin, Function theory in the unit ball of Cn , Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], vol. 241, Springer-Verlag, New York, 1980. MR 601594 (82i:32002) [11] Joel H. Shapiro, Composition operators and classical function theory, Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993. MR 1237406 (94k:47049) [12] , Every composition operator is (mean) asymptotically Toeplitz, J. Math. Anal. Appl. 333 (2007), no. 1, 523–529. MR 2323505 (2008d:47050) Department of Mathematics and Statistics, Mail Stop 942, University of Toledo, Toledo, OH 43606 E-mail address:
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