Hyponormality of Toeplitz operators with polynomial symbols on the weighted Bergman space

Transcription

1 Arab. J. Math :87 94 DOI /s Arabian Journal of Mathematics Ambeswar Phukon Hyponormality of Toeplitz operators with polynomial symbols on the weighted Bergman space Received: 11 September 2016 / Accepted: 30 April 2017 / Published online: 16 May 2017 The Authors This article is an open access publication Abstract This paper gives the complete proof of the Conjecture given by Hazarika and this author jointly which deals with a necessary and sufficient condition for the hyponormality of Toeplitz operator, T ϕ on the weighted Bergman space with certain polynomial symbols under some assumptions about the Fourier coefficients of the symbol ϕ. Mathematics Subject Classification 47B35 47B20 1 Introduction Let D be the open unit disc in the complex plane C. For 1 <α<, letl 2 D, da α denote the Hilbert space consisting of functions on D which are square integrable with respect to the measure da α z = α + 11 z 2 α daz, where da denotes the normalized Lebesgue area measure on D. The inner product on L 2 D, da α is defined as f, g α = D f zgzda αz, f, g L 2 D, da α. The weighted Bergman space A 2 α D is then defined as the closed subspace of L2 D, da α consisting of analytic functions on D. Forany nonnegative integer n, ife n z = Γn+α+2 Γn+1Γ α+2 zn,z D with the usual Gamma function Γs, then the set {e n } forms an orthonormal basis for A 2 α D [12,13]. The reproducing kernel in A2 α D is defined as K z α 1 w =, for z,w D. Forϕ L D, da 1 zw 2+α α, the multiplication operator M ϕ on A 2 α D is defined by M ϕ f = ϕ. f. The orthogonal projection P α of L 2 D, da α onto A 2 α D is given by P α f z = f, K z α α = f w D da 1 z w 2+α α w, for f L 2 D, da α. For ϕ L D, the Toeplitz operator T ϕ with symbol ϕ is defined on A 2 α D by T ϕ f = P α ϕ. f.we,thus,havet ϕ f z = ϕw f w D da 1 z w 2+α α w, f A 2 α D and w D. The Hankel operator on A 2 α D is defined by H ϕ f = I P α ϕ. f. A. Phukon B Kokrajhar Government College, BTC, Kokrajhar, Assam 78370, India aphukon38@gmail.com

2 88 Arab. J. Math :87 94 A bounded linear operator T on a Hilbert space is said to be hyponormal if its self-commutator [T, T ]:= T T TT is positive semi-definite. The hyponormality of Toeplitz operators T ϕ on the Hardy space H 2 T where T = D was first characterised by Cowen [2] which was simplified by Nakazi and Takahashi [9]. For ϕ L T, we write Eϕ ={k H : k 1 and ϕ k ϕ H T}. Then T ϕ is hyponormal if and only if Eϕ is nonempty. The solution was given on the basis of a dilation theorem of Sarason [11]. As for the Bergman space, no such dilation theorem exists [3]; so, the characterization for the hyponormality of Toeplitz operators on the Bergman space is still remained an open question. For ϕ = f +ḡ where f, g are bounded analytic functions, Sadraoui [10] gave a necessary and sufficient condition for the hyponormality of Toeplitz operators on the Bergman space which is equivalently true on the weighted Bergman space too. His theorem is stated below: Theorem 1.1 [10] Let f, g be bounded and analytic in L 2 D, da. The following statements are equivalent: i T f +ḡ is hyponormal; ii Hḡ H ḡ H f H f ; iii Hḡ = CHf, where C is of norm less than or equal to one. Recently, Hwang and Lee [5], Hwang,Lee and Park [6], Lu andliu [7], Luand Shi [8]and Hazarika along with this author in [4] gave some necessary and sufficient conditions for the hyponormality of Toeplitz operators on weighted Bergman space with the class of functions ϕ = f +ḡ where f, g L 2 D, da α.in [4], the authors gave a Conjecture in which it was made an attempt to give a complete criteria for the hyponormality of T ϕ for this class of functions in the weighted Bergman space. The Conjecture is: Conjecture 1.2 Let ϕz = gz+ f z,where fz = a m z m +a N z N,gz = a m z m +a N z N 1 m < N. If a m ā N = a m ā N and α is sufficiently large, then T ϕ on A 2 α D is hyponormal { J=m α j a m 2 a m 2 N m+1 j=0 N j a N 2 a N 2 if a N a N N 2 a N 2 a N 2 m 2 a m 2 a m 2 if a N a N In this paper, we give the proof of this Conjecture for all α> 1. Since, the hyponormality of operators is translation invariant, we may assume that f 0 = g0 = 0. We have the following properties of Toeplitz operators: If f, g L D, da α,then i T f +g = T f + T g ii T f = T f iii T f T g = T fg if f or g is analytic. Also, for the projection P α of L 2 D, da α onto A 2 α D, wehave P α z t z s = { Γs+1Γ s t+α+2 Γs+α+2Γ s t+1 zs t if s t 0 if s < t where s and t are nonnegative integers. Again, for γ k = z k α,wehave γk 2 = zk 2 α = zk, z k α = z k z k da α z = α + 1 z 2k 1 z 2 α daz = α D t k 1 t α dt = α + 1Bk + 1,α+ 1 = D Γk + 1Γ α + 2 Γα+ k The proof of the Conjecture To prove the Conjecture we need some specific lemmas. Lemma 2.1 [1,8] Fix m 1. Then for α 1,

3 Arab. J. Math : i H z m z k ξ = { ξ m ξ k if 0 k < m ξ m ξ k γ 2 k γ 2 k m ξ k m if m k; ii the functions {H z m z k } are orthogonal in L2 D, da α ; iii H z m H z m z k ξ = ω 2 mk ξ k ; k = 0, 1, 2,...,where iv H z m z k α = ω mk γ k. ωmk 2 = γk+m 2 γk 2 γk+m 2 γk 2 γ 2 k γ 2 k m if 0 k < m if m k; Lemma 2.2 [5] For 0 m N and let K i ={k i A 2 α D : k iz = c Nk+i z Nk+i } where, i = 0, 1, 2,...,N 1,wehave Nk+i! 2 ΓNk+i m+α+2γ α+2 P α z m k i z 2 α = c ΓNk+i+α+2 2 ΓNk+i m+1 Nk+i 2 if m i k=1 Nk+i! 2 ΓNk+i m+α+2γ α+2 c ΓNk+i+α+2 2 ΓNk+i m+1 Nk+i 2 if m > i Lemma 2.3 Let k 1 and 1 m < N. Let Then, lim k ξk ζk = m2 N 2 Proof By simplifying, ξk = ζk = ξk ζk = k + m! Γk + m + α + 2 k! 2 Γk m + α + 2 Γk + α Γk m + 1 k + N! Γk + N + α + 2 k! 2 Γk N + α + 2 Γk + α Γk N + 1 m k+ j j=0 k j m j=0 k+α+2+ j k+α+2 j N k+ j j=0 k j j=0 k+α+2+ j N k+α+2 j = m k+ j m k+α+2 j j=0 k j j=0 k+α+2+ j j= m k+α+2+ j N k+ j N k+α+2 j j=0 k j j=0 k+α+2+ j j= N k+α+2+ j If Q n k is a polynomial of degree n in k,then m k + j m k + α + 2 j j=0 k j j=0 k + α j is a polynomial Q 2m 2 k. Now, m k + j = k m + m jk + Lk m m j where, L = m 1 m;and m k + α + 2 j = k + α + 2 m = j k + α Lk + α + 2 m 2 k m + mα + 2k + j α k m 2 +

4 90 Arab. J. Math :87 94 Similarly, = k m + j k + m 1α + 2k m Lk m 2 + mα + 2 m j=0 j k + α k j = k m j m 1α + 2 jk + L 1 k m where, L 1 = m 2.m 1 And, k + α j = k m + j=0 + mα j k j α m 1α + 2 j j + L k m 2 + j + L 1 k m 2 + If we denote the coefficients of k n s of the polynomial Q n k by C n s, then we have C 2m = 0, C 2 = 0. C 2m 2 = α α j m 1α + 2 j + 2L + = α + 2 m + 2 = m 2 α + 1 j m 1α + 2 j + 2m j 2L 1 + m 2 j j + j mα + 2 j j mα Thus there exists a polynomial Q 2m 3 k such that ξk = m 2 α + 1k 2m 2 + Q 2m 3 k. Similarly, ζk = N 2 α + 1k 2N 2 + Q 2N 3 k. Therefore, ξk lim k ζk = lim m 2 α + 1k 2m 2 + Q 2m 3 k k N 2 α + 1k 2N 2 + Q 2N 3 k. j= N j= m 2 j k + α j m2 = k + α j N 2. Lemma 2.4 Let f z = a m z m + a N z N and gz = a m z m + a N z N, with 1 m < N. Let α> 1and a m ā N = a m ā N. Then for i = j, we have H f k iz, H f k jz α = Hḡki z, Hḡk j z α. Proof H f k iz, H f k jz α = ā m H z m k i z +ā N H z N k i z, ā m H z m k j z +ā N H z N k j z α = a m 2 H z m k i z, H z m k j z α + a N 2 H z N k i z, H z N k j z α + a m ā N H z N k i z, H z m k j z α +ā m a N H z m k i z, H z N k j z α = a m ā N H z N k i z, H z m k j z α +ā m a N H z m k i z, H z N k j z α. j

5 Arab. J. Math : Since, H z m k i z, H z m k j z α = H z m c Nk+i z Nk+i, H z m c Nk+ j z Nk+ j = c Nk+i c Nk+ j H z m z Nk+i, H z m z Nk+ j α α = 0 by Lemma 2.1. And, H z N k i z, H z N k j z α = 0. Similarly, Hḡki z, Hḡk j z α = a m ā N H z N k i z, H z m k j z α +ā m a N H z m k i z, H z N k j z α. Since, a m ā N = a m ā N, the proof is complete. Now we are ready to prove Conjecture 1.2. Proof Let K i ={k i A 2 α D : k iz = c Nk+i z Nk+i },wherei = 0, 1, 2,...,N 1.. By Theorem 1.1, T ϕ is hyponormal if and only if That is, if and only if That is, if and only if H f H f H f Hḡ H ḡ k i z, H f H f k iz, H f k iz α + i = j;i, j 0 k i z, k i z Hḡ α i = j;i, j 0 k i z 0. 1 α k i z, Hḡ k i z 0. α H f k iz, H f k jz α Hḡki z, Hḡki z α Hḡk i z, Hḡk j z α 0. 2 Thus, using Lemma 2.4 in 2, we have that T ϕ is hyponormal if and only if H f k iz, H f k iz α Hḡki z, Hḡki z α 0. 3 We have, M f k iz, M f k iz α = ā m z m +ā N z N k i z, ā m z m +ā N z N k i z α = a m 2 z m k i z, z m k i z α + a N 2 z N k i z, z N k i z α +ā m a N zm k i z, z N k i z α + a m ā N z N k i z, z m k i z α. Similarly, Mḡki z, Mḡki z α = a m 2 z m k i z, z m k i z α + a N 2 z N k i z, z N k i z α +ā m a N z m k i z, z N k i z α + a m ā N z N k i z, z m k i z α.

6 92 Arab. J. Math :87 94 Therefore, using the assumption a m ā N = a m ā N,wehave Also, we have M f k iz, M f k iz α = a m 2 a m 2 + a N 2 a N 2 = a m 2 a m 2 Mḡki z, Mḡki z α c Nk+i 2 z m z Nk+i, z m z Nk+i α + a N 2 a N 2 c Nk+i 2 z N z Nk+i, z N z Nk+i α Nk + i + m!γα+ 2 ΓNk + i + m + α + 2 c Nk+i 2 Nk i!γα+ 2 ΓNk i + α + 2 c Nk+i 2. 4 Tḡk i z, Tḡk i z α = ā m T z m k i z +ā N T z N k i z, ā m T z m k i z +ā N T z N k i z α = a m 2 T z m k i z, T z m k i z α + a N 2 T z N k i z, T z N k i z α +ā m a N T z m k i z, T z N k i z α +a m ā N T z N k i z, T z m k i z α. And, T f k iz, T f k iz α = a m 2 T z m k i z, T z m k i z α + a N 2 T z N k i z, T z N k i z α +ā m a N T z m k i z, T z N k i z α + a m ā N T z N k i z, T z m k i z α. Therefore, using the assumption a m ā N = a m ā N and Lemma 2.2, wehave Tḡki z, Tḡki z α T f k iz, T f k iz α = a m 2 a m 2 P α z m k i z 2 α + a N 2 a N 2 P α z N k i z 2 α = a m 2 a m 2 P α z m k i z 2 α + + a N 2 a N 2 P α z N k i z 2 α = a m 2 a m 2 k=1 + a m 2 a m 2 i=m + a N 2 a N 2 k=1 i=m P α z m k i z 2 α Nk + i! 2 ΓNk + i m + α + 2Γ α + 2 ΓNk + i + α c Nk+i 2 ΓNk + i m + 1 Nk + i! 2 ΓNk + i m + α + 2Γ α + 2 ΓNk + i + α c Nk+i 2 ΓNk + i m + 1 Nk + i! 2 ΓNk + i N + α + 2Γ α + 2 ΓNk + i + α c Nk+i 2. 5 ΓNk + i N + 1

7 Arab. J. Math : Thus by applying 4, 5in3 and simplifying, we have that T ϕ is hyponormal if and only if a m 2 a m 2 i + m!γα+ 2 Γi + m + α + 2 c i 2 + Nk + i + m!γα+ 2 k=1 ΓNk + i + m + α + 2 Nk + i!2 ΓNk + i m + α + 2Γ α + 2 ΓNk + i + α c Nk+i 2 + Nk + i + m!γα+ 2 ΓNk + i m + 1 i=m ΓNk + i + m + α + 2 Nk + i!2 ΓNk + i m + α + 2Γ α + 2 ΓNk + i + α c Nk+i 2 + a N 2 a N 2 N + i!γα+ 2 ΓNk + i m + 1 ΓN + i + α + 2 c i 2 + Nk i!γα+ 2 k=1 ΓNk i + α + 2 Nk + i!2 ΓNk 1 + i + α + 2Γ α + 2 c ΓNk + i + α Nk+i 2 0 ΓNk 1 + i + 1 or equivalently, a m 2 a m 2 k + m!γα+ 2 Γk + m + α + 2 c k 2 + k=m k + m!γα+ 2 Γk + m + α + 2 k!2 Γk m + α + 2Γ α + 2 c Γk + α k 2 + a N 2 a N 2 k + N!Γα+ 2 Γk m + 1 Γk + N + α + 2 c k 2 k + N!Γα+ 2 + Γk + N + α + 2 k!2 Γk N + α + 2Γ α + 2 c Γk + α k Γk N + 1 k=n Let for all k 1 ξk = = k+m! Γk+m+α+2 k!2 Γk m+α+2 Γk+α+2 2 Γk m+1 k+n! Γk+N+α+2 k!2 Γk N+α+2 Γk+α+2 2 Γk N+1 m k+ j j=0 k j m j=0 k+α+2+ j k+α+2 j N k+ j j=0 k j j=0 k+α+2+ j N k+α+2 j. 7 Now, if a N a N then, a N 2 a N 2 a m 2 ξk, a m 2 for all k 1. 8 For all k = 1, 2,...,N 1, we have m! Γm+α+2 N! ΓN+α+2 Hence from 8and9, T ϕ is hyponormal if and only if By Lemma 2.3,weget a N 2 a N 2 a m 2 a m 2 k+m! Γk+m+α+2 k+n! Γk+N+α+2 j=m ξn. 9 α j N m+1 j=0 N j. 10 m2 lim ξk = k N If a N a N,then a N 2 a N 2 a m 2 ξk for all k 1. a m 2 12 Hence in this case T ϕ is hyponormal if and only if a N 2 a N 2 a m 2 a m 2 m2 N Thus the results follow from 10and13.

8 94 Arab. J. Math : Conclusion This theorem may give a clue to the readers to think about the generalised form of the hyponormality of Toeplitz operators with a class of polynomial symbols relaxing some of the restrictions to the Fourier coefficients. Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License creativecommons.org/licenses/by/4.0/, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original authors and the source, provide a link to the Creative Commons license, and indicate if changes were made. References 1. Arazy, J.; Fisher, S.D.; Peetre, J.: Hankel operators on weighted Bergman spaces. Am. J. Math. 110, Cowen, C.C.: Hyponormality of Toeplitz operators. Proc. Am. Math. Soc. 103, Hwang, I.S.: Hyponormal Toeplitz operators on the Bergman space. J. Korean Math. Soc. 42, Hazarika, M.; Phukon, A.: On hyponormality of Toeplitz operators on the weighted Bergman space. Commun. Math. Appl. 3, Hwang, I.S.; Lee, J.: Hyponormal Toeplitz operators on the weighted Bergman spaces. Math. Inequal. Appl. 15, Hwang, I.S.; Lee, J.; Park, S.W.: Hyponormal Toeplitz operators with polynomial symbols on weighted Bergman spaces. J. Inequal. Appl. 2014, Lu, Y.; Liu, C.: Commutativity and hyponormality of Toeplitz operators on the weighted Bergman space. J. Korean Math. Soc. 46, Lu, Y.; Shi, Y.: Hyponormal Toeplitz operators on the weighted Bergman space. Integr. Equ. Oper. Theory 65, Nakazi, T.; Takahashi, K.: Hyponormal Toeplitz operators and extremal problems of Hardy spaces. Trans. Am. Math. Soc. 338, Sadraoui, H.: Hyponormality of Toeplitz operators and composition operators. Thesis, Purdue University Sarason, D.: Generalized interpolation on H.Trans.Am.Math.Soc.127, Zhu, K.: Theory of Bergman Spaces. Springer, New York Zhu, K.: Operator Theory in Function Spaces. Marcel Dekker, INC., New York 2007