何　莉， 曹廣福

(廣州大學(xué) 數(shù)學(xué)與信息科學(xué)學(xué)院， 廣東 廣州　510006)

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Bergman-Sobolev空間上Toeplitz算子的本性范數(shù)

何莉， 曹廣福

(廣州大學(xué) 數(shù)學(xué)與信息科學(xué)學(xué)院， 廣東 廣州510006)

文章研究了Bergman-Sobolev上Toeplitz算子的某些性質(zhì)，主要通過該類算子的符號(hào)函數(shù)在邊界處的行為計(jì)算了它們的本性范數(shù).

Bergman-Sobolev空間; Toeplitz算子; 本性范數(shù)

0　Introduction

Denote by R the real number set, N the natural number set and N*the positive integer set.

Forβ∈R and 1≤p<+∞, the Sobolev space Lβ,pis the completion of all functionsf∈() for which

Forp=2, the space Lβ,2is a Hilbert space with the inner product

?f∈Lβ,2,g∈Lβ,2.

Here,L2denotes the usual Lebesgue spaceL2(,dA) and the notation·,·L2denotes the standard inner product inL2.

Whenp=+∞, the corresponding Sobolev space is written as

Lβ,∞={f:→

with ‖f‖Lβ,∞=‖βf‖L∞+‖f‖L∞.SinceeachfunctioninLβ,∞can be extended to a continuous function on the closed unit disc} by Sobolev’s embedding theorem (see Theorem 5.4 of Ref.[1]), we will use the same notation between a function in Lβ,∞and its continuous extension onin this paper.

Tuf=P(uf)

In this paper, we calculate the essential norm of Toeplitz operators on Bergman-Sobolev space with positive integer derivative in terms of the boundary value of the corresponding symbol.

1　Essential norm of Toeplitz operators

Lemma 1For eachλ∈

Proof. See Proposition 3.2 of the paper given in footnote*HE L, CAO G F. Toeplitz operators on Bergman-Sobolev space with positive integer derivative[J]. Sci China Math Ser A, 2016, preprint..

Proof. See Lemma 3.3 of the paper given in footnote①.

Proof. See Lemma 3.4 of the paper given in footnote①.

Lemma 4Letu,v∈Lβ,∞andζ∈. Then, limλ→ζ).

Proof. See Lemma 3.5 of the paper given in footnote①.

Theorem 1Letu∈Lβ,∞,β∈N*. Then, ‖Tu‖e=maxζ∈}|u(ζ)|.

Proof. Setρ=maxζ∈|u(ζ)| for simplicity. Choose some pointη∈so thatu(η)=ρ. For anyK∈,

byLemma4withv=1, this indicates ‖Tu‖e≥maxζ∈|u(ζ)|.

②LEE Y J. Compact sums of Toeplitz products and Toeplitz algebra on the Dirichlet space[J]. Tohoku Math J, preprint,2016.

for everyj>j0.

Moreover, sinceuis continuous on, we can choose somer∈(0,1) such that |u(z)|≤ρ+εfor everyr<|z|<1.

asj→∞. Since

for eachj∈N*, it is not difficult to get that

∫|z|≤r|βfj|2dA<ε

for everyj>jβ. Notice that

for eachj∈N*, where

asj→∞, we have

Direct calculation follows that

(1)

by Minkowski inequality. Since

by Cauchy-Schwarz inequality, where

is a positive number, there exists an integerj*≥0 such that

(2)

whenj>j*because ‖kfj‖A2→0 for each integer 0≤k≤β-1 asj→∞ by Lemma 3. Furthermore, for everyj>jβ,

(3)

Then, by combining the inequalities (1), (2) and (3), we have

‖β(ufj)‖L2≤‖

asj>max{jβ,j*}. This implies that

2　Main result

The main result is the calculation of the essential norm of the Toeplitz operators in terms of the boundary value of their corresponding symbols. That is

Theorem 2Letu∈Lβ,∞,β∈N*. Then,

‖Tu‖e=maxζ∈|u(ζ)|.

AcknowledgmentsThe authors would like to thank professor YOUNG J L in Korea for helpful discussions.

[1]ADAMS R A. Sobolev spaces[M]∥A subsidiary of Harcourt Brace Jovanovich, Pure and applied mathematics. New York-London: Academic Press, 1975:65.

[2]COHN W S, VERBITSKY I E. On the trace inequalities for Hardy-Sobolev functions in the unit ball ofn[J]. Indian Univ Math J, 1994, 43(4): 1079-1097.

[3]BRUNA J, ORTEGA J M. Interpolation along manifolds in Hardy-Sobolev spaces[J]. J Geom Anal, 1997, 7(1): 17-45.

[5]CASCANTE C, ORTEGA J M. Carleson measures for weighted Hardy-Sobolev spaces[J]. Nagoya Math J, 2007, 186: 29-68.

[6]TCHOUNDJA E. Carleson measures for the generalized Bergman spaces via aT(1)-type theorem[J]. Ark Mat, 2008, 46(2): 377-406.

[7]CHO H R, ZHU K H. Holomorphic mean Lipschitz spaces and Hardy-Sobolev spaces on the unit ball[J]. Complex Var Elliptic Equ, 2012, 57(9): 995-1024.

[8]CAO G F, HE L. Fredholmness of Multipliers on Hardy-Sobolev spaces[J]. J Math Anal Appl, 2014, 418(1): 1-10.

[9]CAO G F, HE L. Hardy-Sobolev spaces and their multipliers[J]. Sci China Math Ser A, 2014, 57(11): 2361-2368.

[10]HE L, CAO G F. Composition operators on Hardy-sobolev spaces[J]. Indian J Pure Appl Math, 2015, 46(3): 255-267.

[11]HE L, CAO G F. Toeplitz operators with unbounded symbols on Segal-Bargmann space[J]. J Math Res Appl, 2015, 35(3): 237-255.

[12]HONG C K. On the essential maximal numerical range[J]. Acta Sci Math, 1979, 41: 307-315.

【責(zé)任編輯： 周全】

date: 2016-01-05;Revised date: 2016-04-18

s: National Natural Science Foundation of China (11501136); The key discipline construction project of subject groups focus on Mathematics and information science in the construction project of the high-level university of Guangdong Province (4601-2015); Guangzhou University (HL02-1517) and (HL02-2001)

Essential norm of Toeplitz operators on Bergman-Sobolev space

HE Li, CAO Guang-fu

(School of Mathematics and Information Sciences, Guangzhou University, Guangzhou 510006, China)

In this paper, we study some properties of Toeplitz operators on the Bergman-Sobolev space. Mainly, we calculate the essential norm of these operators in terms of the boundary value of their corresponding symbols.

Bergman-Sobolev space; Toeplitz operator; essential norm

O 177.1Document code: A

1671- 4229(2016)04-0018-04

O 177.1

A

Biography: HE Li(1986-), female, Doctor of science. E-mail: helichangsha1986@163.com.