Compact Product ofHankel and Toeplitz Operators on the Hardy Space

Cheng Chu

ABSTRACT. In this paper, we study the product of a Hankel oper ator and a Toeplitz operator on the Hardy space. We give necessary and sufficient conditions for when such a product HfTg is compact.

l. Introduction

Let D be the open unit disk in the complex plane. Let L2 denote the Lebesgue space of square integrable functions on the unit circle 3D. The Hardy space H2 is the subspace of L2 of analytic functions on D. Let P be the orthogonal projection from L2 to H2. For / £ L°°, the space of essentially bounded Lebesgue measur able functions on 3D, the Toeplitz operator Tf and the Hankel operator Hf with symbol f are defined on H2 by

Tfh = P(fh), and

Hfh = PU(fh),

for h G H2. Here, U is the unitary operator on L2 defined by Uh(z) = zh(z), where f{z) = f(z). Clearly,

Hf = Hp,

where f*(z) = f(z). Hankel operators are often defined in an alternative way (see for example [13], [17]). It is easy to see that the two definitions are unitarily equivalent.

Let us first look at the compactness of Toeplitz and Hankel operators in dividually. The only compact Toeplitz operator is the zero operator (see, e.g., [17, p. 194]). For the Hankel operator, we have the following theorem, usually referred to as Hartmans Criterion.

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Contemporary MathematicsVolume 638, 2015http://dx.doi.org/10.1090/conm/638/12804

Asymptotic Bohr radius for the polynomialsin one complex variable

Cheng Chu

Abstract. We consider the Bohr radius Rn for the class of complex polyno-mials in one variable of degree at most n. It was conjectured by R. Fournier

in 2008 that Rn = 13+ π2

3n2 + o( 1n2 ). We prove this conjecture is true.

1. Introduction

Let D be the open unit disk in the complex plane C and H∞ be the Banachspace of bounded analytic functions on D with the norm

||f ||∞ = supz∈D

|f(z)|.

Also let Pn denote the subspace of H∞ consisting of all the complex polynomialsof degree at most n. The Bohr radius R for H∞ is defined as

R = sup{r ∈ (0, 1) :

∞∑k=0

|ak|rk � ||f ||∞, for all f(z) =

∞∑k=0

akzk ∈ H∞}.

Bohr’s famous power series theorem [1] shows that R = 13 .

In 2005, Guadarrama [4] considered the Bohr type radius for the class Pn

defined by

(1.1) Rn = sup{r ∈ (0, 1) :

n∑k=0

|ak|rk � ||p||∞, for all p(z) =

n∑k=0

akzk ∈ Pn},

and gave the estimate

C1

3n/2< Rn − 1

3< C2

log n

n,

for some positive constants C1 and C2. Later in 2008, Fournier obtained an explicitformula for Rn by using the notion of bounded preserving functions. He proved thefollowing theorem [2]

2010 Mathematics Subject Classification. Primary 41-XX.Partially supported by National Science Foundation Grant DMS 1300280.

c©2015 American Mathematical Society

39

PROCEEDINGS OF THEAMERICAN MATHEMATICAL SOCIETYVolume 144, Number 6, June 2016, Pages 2533–2537http://dx.doi.org/10.1090/proc/12900

Article electronically published on October 20, 2015

A NOTE ON THE SPECTRAL AREA

OF TOEPLITZ OPERATORS

CHENG CHU AND DMITRY KHAVINSON

(Communicated by Pamela Gorkin)

Abstract. In this note, we show that for hyponormal Toeplitz operators,there exists a lower bound for the area of the spectrum. This extends theknown estimate for the spectral area of Toeplitz operators with an analyticsymbol.

1. Introduction

Let D be the open unit disk in the complex plane. Let L2 denote the Lebesguespace of square integrable functions on the unit circle ∂D. The Hardy space H2 isthe subspace of L2 of analytic functions on D. Let P be the orthogonal projectionfrom L2 to H2. For ϕ ∈ L∞, the space of bounded Lebesgue measurable functionson ∂D, the Toeplitz operator Tϕ and the Hankel operator Hϕ with symbol ϕ aredefined on H2 by

Tϕh = P (ϕh),

and

(1.1) Hϕh = U(I − P )(ϕh),

for h ∈ H2. Here U is the unitary operator on L2 defined by

Uh(z) = z̄h(z̄).

Recall that the spectrum of a linear operator T , denoted as sp(T ), is the set ofcomplex numbers λ such that T − λI is not invertible; here I denotes the identityoperator. Let [T ∗, T ] denote the operator T ∗T −TT ∗, called the self-commutator ofT . An operator T is called hyponormal if [T ∗, T ] is positive. Hyponormal operatorssatisfy the celebrated Putnam inequality [11]

Theorem 1.1. If T is a hyponormal operator, then

‖[T ∗, T ]‖ � Area(sp(T ))

π.

Notice that a Toeplitz operator with analytic symbol f is hyponormal, and itis well known that sp(Tf ) = f(D). The lower bounds of the area of sp(Tf ) wereobtained in [9] (see [2], [1], [13], and [14] for generalizations to uniform algebras andfurther discussions). Together with Putnam’s inequality, such lower bounds wereused to prove the isoperimetric inequality (see [4], [5], and the references there).Recently, there has been revived interest in the topic in the context of analyticToeplitz operators on the Bergman space (cf. [3], [10], and [7]). Together with

Received by the editors March 23, 2015 and, in revised form, July 13, 2015.2010 Mathematics Subject Classification. Primary 30J99,47B35.

c©2015 American Mathematical Society

2533

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Complex Anal. Oper. Theory (2018) 12:849–857https://doi.org/10.1007/s11785-017-0740-y

Complex Analysisand Operator Theory

Normal Truncated Toeplitz Operators

Cheng Chu1

Received: 20 August 2017 / Accepted: 20 October 2017 / Published online: 29 October 2017© Springer International Publishing AG 2017

Abstract The characterization of normal truncated Toepltiz operators is first givenby Chalendar and Timotin. We give an elementary proof of their result without usingthe algebraic properties of truncated Toeplitz operators.

Keywords Truncated Toeplitz operator · Normal operator

Mathematics Subject Classification Primary 47

1 Introduction

LetD be the open unit disk in the complex plane. Let L2 denote the Lebesgue space ofsquare integrable functions on the unit circle ∂D. The Hardy space H2 is the subspaceof analytic functions on D whose Taylor coefficients are square summable. Then itcan also be identified with the subspace of L2 of functions whose negative Fouriercoefficients vanish. Let P and P⊥ be the orthogonal projections from L2 to H2 and[H2]⊥, respectively. Here [H2]⊥ is the orthogonal complement of H2 in L2. Forf ∈ L∞, the space of essentially bounded Lebesgue measurable functions on ∂D, theToeplitz operator T f with symbol f ∈ L∞ is defined by

T f h = P( f h),

for h ∈ H2.

Communicated by Nikolai Vasilevski.

B Cheng Chucheng.chu@vanderbilt.edu

1 Department of Mathematics, Vanderbilt University, Nashville, TN, USA

Complex Analysis and Operator Theoryhttps://doi.org/10.1007/s11785-018-0814-5

Complex Analysisand Operator Theory

Bounded Composition Operators and Multipliers of SomeReproducing Kernel Hilbert Spaces on the Bidisk

Cheng Chu1

Received: 20 December 2017 / Accepted: 8 June 2018© Springer International Publishing AG, part of Springer Nature 2018

AbstractWe study the boundedness of composition operators on the bidisk using reproducingkernels. We show that a composition operator is bounded on the Hardy space H2(D2)

if some associated function is a positive kernel. This positivity condition naturallyleads to the study of the sub-Hardy Hilbert spaces of the bidisk, which are analogs ofde Branges–Rovnyak spaces on the unit disk. We discuss multipliers of those spacesand obtain some classes of bounded composition operators on the bidisk.

Keywords Composition operator · Hardy space · Reproducing kernel

Mathematics Subject Classification Primary 47B33; Secondary 47B32

1 Introduction

LetD denote the open unit disk inCwith boundary T. The bidiskD2 and the torus T2

are the subsets ofC2 which are Cartesian products of two copiesD andT, respectively.The Hardy space H2(D) is the closure of the analytic polynomials in L2(T) and theHardy space H2(D2) (or H2) is the closure of the analytic polynomials in L2(T2, dσ)

(or L2(T2)), where dσ is the normalized Haar measure on T2. H∞(D2) is the space

of bounded analytic functions on D2 with norm

|| f ||∞ = sup(z1,z2)∈D2

| f (z1, z2)|.

For a bounded domain � ⊂ Cd (� = D or D2), the composition operator Cϕ

on H2(�) is defined by Cϕ f = f ◦ ϕ, for an analytic self-map ϕ of �. One of the

Communicated by Vladimir Bolotnikov.

B Cheng Chucheng.chu@vanderbilt.edu

1 Department of Mathematics, Vanderbilt University, Nashville, TN, USA