Hankel Matrices Acting on the Dirichlet Space

Abstract

The study of the infinite Hankel matrix acting on analytic function spaces dates back to the influential work of Nehari and Widom on the Hardy space $H^2$. Since then, it has been extensively generalized to other settings such as weighted Bergman spaces, Dirichlet type spaces, and Möbius invariant function spaces. Nevertheless, several fundamental operator-theoretic questions, including the boundedness and compactness, remain unresolved in the context of the Dirichlet space. Motivated by this, via Carleson measures, the Widom type condition, and the reproducing kernel thesis, we obtain:

• (i) necessary and sufficient conditions for bounded and compact operators induced by Hankel matrices on the Dirichlet space, thereby answering a folk question in this field (Galanopoulos et al. in Result Math 78(3) Paper No. 106, 2023);

• (ii) necessary and sufficient conditions for bounded and compact operators induced by Cesàro type matrices on the Dirichlet space.

As a beneficial product, we find an intrinsic function-theoretic characterization of functions with positive decreasing Taylor coefficients in the function space ${\mathcal {X}}$ throughly studied by Arcozzi et al. (Lond Math Soc II Ser 83(1):1–18, 2011). In addition, we also show that a random Dirichlet function almost surely induces a compact Hankel type operator on the Dirichlet space.

1 Introduction

Let ${\mathbb {N}}$ be the set of nonnegative integers. Suppose $\varvec{\lambda }=\{\lambda _n\}_{n\in {\mathbb {N}}}$ is a sequence of complex numbers. An infinite Hankel matrix $\varvec{H}_{\varvec{\lambda }}$ induced by the sequence $\varvec{\lambda }$ is given by $\varvec{H}_{\varvec{\lambda }}=(\lambda _{j+k})_{j,k\in {\mathbb {N}}}.$ The Hankel matrix $\varvec{H}_{\varvec{\lambda }}$ is initially defined for all finitely supported sequences in $\ell ^2$. The celebrated Nehari theorem [36] illustrates that $\varvec{H}_{\varvec{\lambda }}$ represents a bounded operator on $\ell ^2$ if and only if there exists a function $\psi$ in $L^\infty$ on the unit circle ${\mathbb {T}}$ such that $\lambda _n$, $n\ge 0$, is the n-th Fourier coefficient of $\psi$.

For $f(z)=\sum _{k=0}^\infty a_kz^k$ in $H({\mathbb {D}})$, the class of functions analytic on the open unit disk ${\mathbb {D}}$, the Hankel matrix $\varvec{H}_{\varvec{\lambda }}$ acts on the function f via

$$\begin{aligned} {\mathcal {H}}_{\varvec{\lambda }}(f)(z):=\sum _{n=0}^\infty \left( \sum _{k=0}^\infty \lambda _{n+k}a_k\right) z^n, \quad z\in {\mathbb {D}}, \end{aligned}$$

whenever the right-hand side makes sense and defines an analytic function on ${\mathbb {D}}$. For a space $X\subseteq H({\mathbb {D}})$ equipped with a norm $\Vert \cdot \Vert _X$, the Hankel type operator ${\mathcal {H}}_{\varvec{\lambda }}$ is bounded on X if ${\mathcal {H}}_{\varvec{\lambda }}(f)$ is well defined in $H({\mathbb {D}})$ for any $f\in X$ and there exists a positive constant C independent of g such that $\Vert {\mathcal {H}}_{\varvec{\lambda }}(g)\Vert _X \le C \Vert g\Vert _X$ for all g in X.

By the Fourier transform, the Hankel matrix $\varvec{H}_{\varvec{\lambda }}$ represents a bounded operator on $\ell ^2$ if and only if ${\mathcal {H}}_{\varvec{\lambda }}$ is bounded on the Hardy space $H^2$. See [38] for details. When ${\varvec{\lambda }}$ is the moment sequence $\{\mu _n\}_{n\in {\mathbb {N}}}$ of a finite positive Borel measure $\mu$ on [0, 1), where $\mu _n=\int _{[0, 1)}t^n d\mu (t)$, the related Hankel matrix is denoted by $\varvec{H}_\mu$ in the literature. Write ${\mathcal {H}}_\mu$ for the corresponding Hankel type operator. Widom [45] proved that ${\mathcal {H}}_\mu$ is bounded on $H^2$ if and only if $\mu _j=O(\frac{1}{j+1})$. It is worth noting that $\{\frac{1}{j+1}\}_{j\in {\mathbb {N}}}$ is the moment sequence of the Lebesgure measure on [0, 1), which corresponds to the classical Hilbert matrix $\varvec{H}=((j+k+1)^{-1})_{j,k\in {\mathbb {N}}}$. See [12, 17,18,19, 29,30,31, 33,34,35, 48] for developments of the Hilbert matrix acting on analytic function spaces.

Since then, operator questions regarding the Hankel matrix have been investigated in various other function spaces of analytic functions on the open unit disk, including Hardy spaces, weighted Bergman spaces, Dirichlet type spaces, and Möbius invariant function spaces. Diamantopoulos’ [16] work illustrates that Widom’s condition remains true for the Dirichlet type space ${\mathcal {D}}_\alpha$ with $0<\alpha <2$. He also noted that the boundedness of ${\mathcal {H}}_\mu$ on the classical Dirichlet space ${\mathcal {D}}$ and the Bergman space $A^2$ coincides. Under a milder condition, using Carleson measures for ${\mathcal {D}}$, Galanopoulos and Peláez [21] completely characterized the boundedness of ${\mathcal {H}}_\mu$ on the Dirichlet space. Chatzifountas, Girela and Peláez studied the boundedness and compactness of ${\mathcal {H}}_\mu$ between distinct Hardy spaces in [14]. Girela and Merchán [25] obtained a method to give complete descriptions of the boundedness and compactness of ${\mathcal {H}}_\mu$ on some Hardy spaces and Möbius invariant function spaces. Bao et al. [6] considered bounded ${\mathcal {H}}_\mu$ on analytic functions spaces in terms of so-called Hankel measures.

In their recent work, Galanopoulos et al. [23] demonstrated that a 1-logarithmic 1-Carleson measure ensures the boundedness of ${\mathcal {H}}_\mu$ on the Dirichlet space. However, they also pointed out that the characterization of the finite positive Borel measure $\mu$ on [0, 1) for which the operator ${\mathcal {H}}_\mu$ is bounded on the Dirichlet space remains unresolved. Via Carleson measures, Widom type conditions and the reproducing kernel thesis, the current work answers this question.

Recall that the Dirichlet space ${\mathcal {D}}$ is a Hilbert space of analytic functions on ${\mathbb {D}}$ equipped with the Dirichlet inner product

$$\begin{aligned} \langle f, g\rangle _{{\mathcal {D}}}=f(0)\overline{g(0)}+\int _{\mathbb {D}}f'(z) \overline{g'(z)} dA(z). \end{aligned}$$

(1.1)

Here $dA(z)=\pi ^{-1}dxdy$ for $z=x+iy$ is the normalized Lebesgue measure on ${\mathbb {D}}$. A finite positive Borel measure $\nu$ on ${\mathbb {D}}$ is said to be a Carleson measure for the Dirichlet space if there is a positive constant C such that

$$\begin{aligned} \left( \int _{\mathbb {D}}|f(z)|^2 d\nu (z)\right) ^{\frac{1}{2}}\le C \Vert f\Vert _{\mathcal {D}}\end{aligned}$$

(1.2)

for all $f\in {\mathcal {D}}$. Denote by $CM({\mathcal {D}})$ the set of Carleson measures for ${\mathcal {D}}$. The smallest such C in (1.2) is denoted by $\Vert \nu \Vert _{CM({\mathcal {D}})}$, the Carleson measure norm of $\nu$. If the identity map $I_d: {\mathcal {D}}\rightarrow L^2({\mathbb {D}}, d\nu )$ is compact, then we say that $\nu$ is a vanishing Carleson measure for ${\mathcal {D}}$. See [2, 5, 20, 21, 44, 47] for this definition of Carleson measures for ${\mathcal {D}}$ and related investigations.

In this paper, we first obtain the following result.

Theorem 1.1

Suppose $\varvec{\lambda }=\{\lambda _n\}_{n\in {\mathbb {N}}}$ is a sequence of complex numbers. Then the Hankel type operator ${\mathcal {H}}_{\varvec{\lambda }}$ is bounded on the Dirichlet space ${\mathcal {D}}$ if and only if $h_{\overline{{\varvec{\lambda }}}}$ is analytic on ${\mathbb {D}}$ and the measure $|h_{\overline{{{\varvec{\lambda }}}}}'(z)|^2dA(z)$ is a Carleson measure for the Dirichlet space ${\mathcal {D}}$, where $h_{\overline{{{\varvec{\lambda }}}}}(z)=\sum _{n=0}^\infty \overline{\lambda _n}z^n$.

In the case of ${\mathcal {H}}_\mu$, by Theorem 1.1, the operator ${\mathcal {H}}_\mu$ is bounded on ${\mathcal {D}}$ if and only if $|h'_\mu (z)|^2dA(z)$ is a Carleson measure for ${\mathcal {D}}$, where

$$\begin{aligned} h_\mu (z)=\int _{[0, 1)}\frac{1}{1-tz}d\mu (t). \end{aligned}$$

Note that in [44] Stegenga’s beautiful characterization of $CM({\mathcal {D}})$ is related to the logarithmic capacity of a finite union of intervals of the unit circle. Based on certain integrals involve the Carleson box and the heightened box, Arcozzi et al. [2, Theorem 1] gave a complete description of $CM({\mathcal {D}})$. There still lacks a direct relation between moment sequences and the boundedness of ${\mathcal {H}}_\mu$ on the Dirichlet space. We will establish these relations through the use of Widom type conditions and the reproducing kernel thesis.

Corresponding to the Dirichlet inner product (1.1), the reproducing kernel of the Dirichlet space ${\mathcal {D}}$ at a point $w\in {\mathbb {D}}$ is

$$\begin{aligned} K_w(z)=1+ \log \frac{1}{1-z{\overline{w}}} \end{aligned}$$

and its normalized reproducing kernel

$$\begin{aligned} k_w(z)=\frac{K_w(z)}{\sqrt{K_w(w)}}. \end{aligned}$$

Now, we state our characterization of the bounded Hankel type operator ${\mathcal {H}}_\mu$ on the Dirichlet space as follows.

Theorem 1.2

Suppose $\mu$ is a finite positive Borel measure on [0, 1). Then the following conditions are equivalent.

• (i) The Hankel type operator ${\mathcal {H}}_\mu$ is bounded on ${\mathcal {D}}$.

• (ii) The reproducing kernel thesis holds; that is,

  $$\begin{aligned} \sup _{t\in [0,1)}\Vert {\mathcal {H}}_\mu (k_t)\Vert _{{\mathcal {D}}}<\infty , \end{aligned}$$

  where $k_t$ is the normalized reproducing kernel of ${\mathcal {D}}$ at t in [0, 1).

• (iii) The Widom type condition is true; that is,

  $$\begin{aligned} \sum ^\infty _{n=m}n \mu _n^2=O\left( \frac{1}{\log (m+2)}\right) . \end{aligned}$$

To prove Theorem 1.2, we need Cesàro type matrices. As benefit products, the necessary and sufficient condition for the boundedness of Cesàro type matrix acting on the Dirichlet space is obtained.

Let ${{\varvec{\eta }}}=\{\eta _n\}_{n\in {\mathbb {N}}}$ be a sequence of complex numbers. Recall that the Cesàro type matrix $\varvec{C}_{\varvec{\eta }}$ is the following infinite lower triangular matrix:

$$\begin{aligned} \varvec{C}_{\varvec{\eta }}= \begin{pmatrix} \eta _0 & 0 & 0& 0 & \cdots \\ \eta _1 & \eta _1 & 0 & 0 & \cdots \\ \eta _2 & \eta _2 & \eta _2 & 0 & \cdots \\ \vdots & \vdots & \vdots & \ddots & \ddots \end{pmatrix}. \end{aligned}$$

In the same way as operators induced by Hankel matrices, the matrix $\varvec{C}_{\varvec{\eta }}$ induces a Cesàro type operator ${\mathcal {C}}_{\varvec{\eta }}$ as follows:

$$\begin{aligned} {\mathcal {C}}_{\varvec{\eta }}(f)(z)=\sum _{n=0}^\infty \left( \eta _n \sum _{k=0}^n a_k\right) z^n, \quad z\in {\mathbb {D}}, \end{aligned}$$

(1.3)

for $f(z)=\sum _{n=0}^\infty a_nz^n$ in $H({\mathbb {D}})$, whenever the right-hand side of (1.3) makes sense and defines an analytic function on ${\mathbb {D}}$.

If $\eta _n=1/(n+1)$ for every n, then the Cesàro type matrix $\varvec{C}_{\varvec{\eta }}$ is the classical Cesàro matrix $\varvec{C}$. Danikas and Siskakis [15] showed that the operator induced by $\varvec{C}$ is bounded from $H^\infty$ to BMOA. Siskakis [42, 43] studied the action of $\varvec{C}$ on Hardy spaces.

Given a finite positive Borel measure $\mu$ on [0, 1), if

$$\begin{aligned} \eta _n=\mu _n=\int _{[0, 1)}t^n d\mu (t) \end{aligned}$$

for each n, then we write ${\mathcal {C}}_{\varvec{\eta }}$ as ${\mathcal {C}}_\mu$. The operator ${\mathcal {C}}_\mu$ is said to be the Cesàro-like operator which was introduced in [22, 27] recently. We refer to [7, 9, 23, 24] for more results about ${\mathcal {C}}_\mu$ on some spaces of analytic functions. From Theorem 5 in [23], if $\mu$ is a 1-logarithmic 1-Carleson measure, then ${\mathcal {C}}_\mu$ is bounded on ${\mathcal {D}}$. Conversely, if ${\mathcal {C}}_\mu$ is bounded on ${\mathcal {D}}$, then $\mu$ is a $\frac{1}{2}$-logarithmic 1-Carleson measure. In his recent work [10], Blasco studied the boundedness of Cesàro-like operators induced by complex Borel measures on ${\mathbb {D}}$ acting on some weighted Dirichlet spaces.

Our subsequent finding gives necessary and sufficient conditions for the boundedness of every Cesàro type operator on the Dirichlet space ${\mathcal {D}}$.

Theorem 1.3

Suppose ${{\varvec{\eta }}}=\{\eta _n\}_{n=0}^\infty$ is a sequence of complex numbers. Then the following conditions are equivalent.

• (i) The Cesàro type operator ${\mathcal {C}}_{\varvec{\eta }}$ is bounded on ${\mathcal {D}}$.

• (ii) The reproducing kernel thesis holds; that is,

  $$\begin{aligned} \sup _{t\in [0,1)}\Vert {\mathcal {C}}_{\varvec{\eta }}k_t\Vert _{{\mathcal {D}}}<\infty , \end{aligned}$$

  where $k_t$ is the normalized reproducing kernel of ${\mathcal {D}}$ at t in [0, 1).

• (iii) The Widom type condition is true; that is,

  $$\begin{aligned} \sum ^\infty _{n=m}n |\eta _n|^2=O\left( \frac{1}{\log (m+2)}\right) . \end{aligned}$$

Following our previous work on the boundedness of Hankel type operators and Cesàro type operators on the Dirichlet space, we also establish their compactness counterparts. These results are included in Sect. 4.

Let $(X_n)_{n\in {\mathbb {N}}}$ be a sequence of independent identically distributed (i.i.d.) real random variables on a probability space $(\Omega ,{\mathcal {A}},{\mathbb {P}})$. For a sequence $\{\alpha _n\}$ of complex numbers, one formally defines a random power series

$$\begin{aligned} \varphi _\omega (z)=\sum _{n=0}^\infty X_n(\omega )\alpha _n z^n. \end{aligned}$$

By Nehari’s theorem, almost surely the random analytic function $\varphi _\omega$ induces a bounded Hankel type operator on the Hardy space $H^2$ if and only if the symbol function $\varphi _\omega$ is a ${\textit{BMOA}}$ function almost surely. The famous unsolved Anderson’s question [1] is to seek the necessary and sufficient condition to characterize random ${\textit{BMOA}}$ functions. One can consult [28, 37] for recent progress and more details in this subject.

According to Theorem 1.1, the symbol function $h_{\overline{{{\varvec{\lambda }}}}}$ of a bounded Hankel type operator ${\mathcal {H}}_{\varvec{\lambda }}$ on ${\mathcal {D}}$ must be in the Dirichlet space. However, our next corollary illustrates that, under milder conditions, a random Dirichlet function almost surely induced a compact Hankel type operator, and hence almost surely induced a bounded Hankel type operator.

Theorem 1.4

Suppose $\varvec{\lambda }=\{\lambda _n\}_{n\in {\mathbb {N}}}$ is a sequence of complex numbers. Let $(X_n)_{n\in {\mathbb {N}}}$ be a sequence of i.i.d. real random variables satisfying that ${\mathbb {E}}[X_n]=0$ and ${\mathbb {E}}[X_n^4]$ is finite. If $h_{{\varvec{\lambda }}}\in {\mathcal {D}}$, then almost surely the Hankel type operator ${\mathcal {H}}_{\varvec{\omega }}$ is compact on ${\mathcal {D}}$, where $\varvec{\omega }=\left\{ X_n \overline{\lambda _n}\right\} _{n\in {\mathbb {N}}}$.

Denote by ${\mathbb {Z}}$ the set of integers. Let $E\subseteq {\mathbb {Z}}$. As defined by Rudin [40], the integrable function f on the unite circle ${\mathbb {T}}$ is called an E-function if ${\hat{f}}(n)$, the n-th Fourier coefficient of f, is equal to zero for all $n\in {\mathbb {Z}}\setminus E$. Suppose $p>2$, the set $E\subseteq {\mathbb {Z}}$ is called a Rudin’s $\Lambda (p)$ set if there is a positive constant C such that $\Vert f\Vert _{L^p({\mathbb {T}})}\le C\Vert f\Vert _{L^2({\mathbb {T}})}$ for all trigonometric polynomials whose coefficients are equal to 0 on ${\mathbb {Z}}{\setminus } E$. The Rudin’s $\Lambda (p)$ set is a natural generalization of the classical lacunary sequence (the Hadamard set) which plays important roles in many aspects of analysis (see [11, 26, 40] for more details). Recall that an increasing sequence $\{n_k\}_{k\in {\mathbb {N}}}$ of positive integers is said to be lacunary if there exists q in $(1, \infty )$ such that $n_{k+1}/n_k>q$ for all k. It is also known that there exist Rudin’s $\Lambda (p)$ sets rather than Hadamard sets.

Our next result presents a characterization of bounded and compact Hankel type operators on ${\mathcal {D}}$ when non-zero Fourier coefficients of the symbol function are located in a Rudin’s $\Lambda (p)$ set.

Corollary 1.5

Suppose $p>2$, the set E is a Rudin’s $\Lambda (p)$ set and $\varvec{\lambda }=\{\lambda _{n}: n\in E\}$ is a sequence of complex numbers. Then the following statements are equivalent.

• (i) The Hankel type operator ${\mathcal {H}}_{\varvec{\lambda }}$ is compact on ${\mathcal {D}}$.

• (ii) The Hankel type operator ${\mathcal {H}}_{\varvec{\lambda }}$ is bounded on ${\mathcal {D}}$.

• (iii) The symbol function $h_{\overline{{{\varvec{\lambda }}}}}(z)=\sum _{n\in E} \overline{\lambda _n}z^n$ belongs to ${\mathcal {D}}$; that is,

  $$\begin{aligned} \sum _{n\in E} (n+1) |\lambda _{n}|^2<\infty . \end{aligned}$$

We end this introduction by some remarks on the boundedness of Hankel matrix on the Dirichlet space and the following function space ${\mathcal {X}}$ related to the Dirichlet space thoroughly studied by Arcozzi et al. [4]. Precisely, ${\mathcal {X}}$ is an analytic function space on ${\mathbb {D}}$ [4, p. 2] consisting of functions f such that

$$\begin{aligned} \Vert f\Vert ^2_{{\mathcal {X}}}=|f(0)|^2+\Vert |f'|^2dA\Vert _{CM({\mathcal {D}})}<\infty . \end{aligned}$$

Denote by ${\mathcal {X}}_0$ the norm closure in ${\mathcal {X}}$ of the space of polynomials. Equivalently, $f\in {\mathcal {X}}_0$ if and only if $|f'|^2dA$ is a vanishing Carleson measure for ${\mathcal {D}}$. We refer to [4, 39, 46, 47] for more details and applications of ${\mathcal {X}}$ and ${\mathcal {X}}_0$.

Now, Theorem 1.1 reads as that the Hankel type operator ${\mathcal {H}}_{\varvec{\lambda }}$ is bounded on the Dirichlet space ${\mathcal {D}}$ if and only if $h_{\overline{{\varvec{\lambda }}}}$ is in ${\mathcal {X}}$. Such a result may not be surprising to the operator theory on the function spaces. However, there is still a lack of a full function-theoretic description of functions on ${\mathcal {X}}$ (one can consult [4, p. 16] for more details). Hence, our Widom type conditions in Theorems 1.2 and 3.1 seem to be the first intrinsic function-theoretic characterization of functions with positive decreasing Taylor coefficients in the function space ${\mathcal {X}}$. For convenience, we include a separate corollary as follows.

Corollary 1.6

Suppose ${{\varvec{\lambda }}}=\{\lambda _n\}^\infty _{n=0}$ is a decreasing sequence of positive real numbers and $h_{{{\varvec{\lambda }}}}(z)=\sum _{n=0}^\infty \lambda _n z^n$ for $z\in {\mathbb {D}}$. Then $h_{{{\varvec{\lambda }}}}\in {\mathcal {X}}$ if and only if

$$\begin{aligned} \sum ^\infty _{n=m}n \lambda _n^2=O\left( \frac{1}{\log (m+2)}\right) . \end{aligned}$$

For an analytic function f with non-zero Fourier coefficients belonging to a Rudin’s $\Lambda (p)$ set, by Corollary 1.5, it follows that $f\in {\mathcal {X}}$ if and only if $f\in {\mathcal {D}}$. This is a slice generation of Arcozzi, Rochberg, Sawyer, Wick’s work [4] related to the equivalence of Parts (3.b) and (3.c) in Theorem 4. More details of discussions about the function space ${\mathcal {X}}$ are given in Sect. 6.

Throughout this paper, the symbol $A\thickapprox B$ means that $A\lesssim B\lesssim A$. We say that $A\lesssim B$ if there exists a positive constant C such that $A\le CB$.

2 Bounded Hankel Type Operators ${\mathcal {H}}_{\varvec{\lambda }}$ on ${\mathcal {D}}$

In this section we will prove Theorem 1.1. We also consider the Möbius invariance of the norm of a bounded Hankel type operator on the Dirichlet space.

Given a function b in $H({\mathbb {D}})$, define a Hankel type bilinear form on the Dirichlet space, initially for f, g in ${\mathcal {P}}$, as

$$\begin{aligned} T_b(f, g):=\langle fg, b\rangle _{{\mathcal {D}}}. \end{aligned}$$

Here ${\mathcal {P}}$ is the space of polynomials. The norm of the bilinear form $T_b$ is

$$\begin{aligned} \Vert T_b\Vert _{{\mathcal {D}}\times {\mathcal {D}}}=\sup \left\{ |T_b(f, g)|: \Vert f\Vert _{\mathcal {D}}=\Vert g\Vert _{\mathcal {D}}=1, \, f, g\in {\mathcal {P}}\right\} . \end{aligned}$$

For a bounded bilinear form $T_b$ on ${\mathcal {D}}$, $T_b$ is said to be compact on ${\mathcal {D}}$ if $T_b(B_{\mathcal {D}}\times B_{\mathcal {D}})$ is precompact in ${\mathbb {C}}$, where $B_{\mathcal {D}}=\{f\in {\mathcal {D}}: \Vert f\Vert _{\mathcal {D}}\le 1\}$. In other words, $T_b$ is compact on ${\mathcal {D}}$ if and only if for all bounded sequences $\{(f_n, g_n)\}\subseteq {\mathcal {D}}\times {\mathcal {D}}$, the sequence $\{T_b(f_n, g_n)\}$ has a convergent subsequence. See [8] for more descriptions on compact bilinear operators.

With respect to the normalized basis $\{(n+1)^{-1/2}z^n\}$ of the Dirichlet space ${\mathcal {D}}$, the matrix representation of $T_b$ is

$$\begin{aligned} \left( \frac{j+k}{\sqrt{j+1}\sqrt{k+1}}\overline{b_{j+k}}\right) , \quad j, k=0, 1, \ldots , \end{aligned}$$

where $b_{j+k}$ is the $(j+k)$-th Taylor coefficient of b. Arcozzi, i et al. characterized the boundedness and compactness of the bilinear form on the Dirichlet space as follows.

Theorem A

[3, Theorem 1.1] Let $b\in H({\mathbb {D}})$. Then the following assertions hold.

• (i) $T_b$ extends to a bounded bilinear form on ${\mathcal {D}}$ if and only if $b\in {\mathcal {X}}$.

• (ii) $T_b$ extends to a compact bilinear form on ${\mathcal {D}}$ if and only if $b\in {\mathcal {X}}_0$.

Proof of Theorem 1.1

Given a sequence of complex number ${{\varvec{\lambda }}}=\{\lambda _n\}^\infty _{n=0}$, suppose $h_{\overline{{{\varvec{\lambda }}}}}$ is analytic on ${\mathbb {D}}$ and the measure $|h_{\overline{{{\varvec{\lambda }}}}}'(z)|^2dA(z)$ is a Carleson measure for ${\mathcal {D}}$. Then $h_{\overline{{\varvec{\lambda }}}}\in {\mathcal {X}}$. Hence, $h_{{\varvec{\lambda }}}\in {\mathcal {D}}$. Let $f(z)=\sum ^\infty _{k=0} c_k z^k$ belong to ${\mathcal {D}}$. Using the fact that ${\mathcal {D}}$ is a subset of the Hardy space $H^2$, we get

$$\begin{aligned} \sup _{j\in {\mathbb {N}}}\left| \sum _{k=0}^\infty \lambda _{j+k}c_k\right| <\infty . \end{aligned}$$

Write $a_j=\sum _{k=0}^\infty \lambda _{j+k}c_k$. Then $\{a_j\}_{j\in {\mathbb {N}}}$ is a bounded sequence of complex numbers and hence ${\mathcal {H}}_{{\varvec{\lambda }}}(f)(z)=\sum _{j=0}^\infty a_jz^j$ is analytic on ${\mathbb {D}}$. Note that

$$\begin{aligned} \Vert {\mathcal {H}}_{{\varvec{\lambda }}}f\Vert _{\mathcal {D}}^2= |a_0|^2+\sum _{j=0}^{\infty }(j+1)|a_{j+1}|^2. \end{aligned}$$

Let m be a positive integer. For any polynomial $g(z)=\sum ^m_{j=0}b_j z^j$ with $\Vert g\Vert _{\mathcal {D}}\le 1$, we have

$$\begin{aligned} \left| a_0 \overline{b_0}+ \sum _{j=0}^{m-1} (j+1)a_{j+1}\overline{b_{j+1}}\right|&\le |\lambda _0| |c_0| |b_0|+|b_0|\left| \sum ^\infty _{k=1}\lambda _{k}c_{k}\right| \\&\quad +\left| \sum ^{m-1}_{j=0} \overline{b_{j+1}} (j+1)\sum ^\infty _{k=0}\lambda _{j+1+k}c_k\right| \\&= |\lambda _0| |c_0| |b_0|+ |b_0| \left| \int _{\mathbb {D}}\frac{f(z)-f(0)}{z}\overline{h'_{\overline{{\varvec{\lambda }}}} (z)} dA(z)\right| \\&\quad +\left| \int _{\mathbb {D}}f(z)g'_1(z)\ \overline{h'_{\overline{{\varvec{\lambda }}}} (z)} dA(z)\right| \\&\le |\lambda _0| \Vert f\Vert _{{\mathcal {D}}} \Vert g\Vert _{{\mathcal {D}}}+\Vert g\Vert _{{\mathcal {D}}}\\&\quad \times \left( \int _{\mathbb {D}}\left| \frac{f(z)-f(0)}{z}\right| ^2|h'_{\overline{{\varvec{\lambda }}}} (z)|^2 dA(z)\right) ^{\frac{1}{2}}\\&\quad + \left( \int _{\mathbb {D}}|f(z)|^2|h'_{\overline{{\varvec{\lambda }}}} (z)|^2 dA(z)\right) ^{\frac{1}{2}}\\&\quad \times \left( \int _{\mathbb {D}}\left| g_1'(z)\right| ^2 dA(z)\right) ^{\frac{1}{2}}, \end{aligned}$$

where $g_1(z)=\sum ^m_{j=0}\overline{b_j} z^j$. Since $|h'_{\overline{{\varvec{\lambda }}}} (z)|^2 dA(z)$ is a Carleson measure for ${\mathcal {D}}$, there exists a positive constant C such that

$$\begin{aligned} \left| a_0 \overline{b_0}+ \sum _{j=0}^{m-1} (j+1)a_{j+1}\overline{b_{j+1}}\right| \le C\Vert f\Vert _{\mathcal {D}}. \end{aligned}$$

Therefore

$$\begin{aligned} \left( |a_0|^2+\sum _{j=0}^{m-1}(j+1)|a_{j+1}|^2\right) ^{1/2}\le C\Vert f\Vert _{\mathcal {D}}\end{aligned}$$

and $\Vert {\mathcal {H}}_{{\varvec{\lambda }}}f\Vert _{\mathcal {D}}\le C\Vert f\Vert _{\mathcal {D}}.$

On the other hand, suppose the operator ${\mathcal {H}}_{{\varvec{\lambda }}}$ is bounded on ${\mathcal {D}}$. Then, for the function $I(z)=z$ on ${\mathbb {D}}$, we see that ${\mathcal {H}}_{{\varvec{\lambda }}}(I)(z)=\sum _{n=0}^\infty \lambda _{n+1} z^n$ belongs to ${\mathcal {D}}$. This implies that $h_{\overline{{{\varvec{\lambda }}}}}$ is analytic on ${\mathbb {D}}$ and

$$\begin{aligned} \int _{{\mathbb {D}}}|h_{\overline{{{\varvec{\lambda }}}}}'(z)|^2dA(z)<\infty . \end{aligned}$$

The boundedness of the operator ${\mathcal {H}}_{{\varvec{\lambda }}}$ on ${\mathcal {D}}$ yields

$$\begin{aligned} |\langle {\mathcal {H}}_{{\varvec{\lambda }}} f,g\rangle _{{\mathcal {D}}}|\lesssim \Vert f\Vert _{\mathcal {D}}\Vert g\Vert _{\mathcal {D}}\end{aligned}$$

for all f and g in ${\mathcal {D}}$. Moreover, for any polynomials f and g,

$$\begin{aligned} \int _{\mathbb {D}}f(z)g'(z)\, \overline{h'_{\overline{{\varvec{\lambda }}}} (z)} \ dA(z)=\int _{\mathbb {D}}({\mathcal {H}}_{{\varvec{\lambda }}} f)'(z) g'({\overline{z}}) \, dA(z). \end{aligned}$$

(2.1)

Consequently,

$$\begin{aligned} \left| f(0)g(0)\overline{h_{\overline{{\varvec{\lambda }}}} (0)}\right| +\left| \int _{\mathbb {D}}(f(z)g(z))'\overline{h'_{\overline{{\varvec{\lambda }}}} (z)} dA(z)\right| \lesssim \Vert f\Vert _{\mathcal {D}}\Vert g\Vert _{\mathcal {D}}. \end{aligned}$$

Therefore, $T_{h_{\overline{{\varvec{\lambda }}}}}$ is a bounded bilinear form on ${\mathcal {D}}$. It follows from Theorem A that $h_{\overline{{\varvec{\lambda }}}}\in {\mathcal {X}}$ and hence $|h_{\overline{{{\varvec{\lambda }}}}}'(z)|^2dA(z)$ is a Carleson measure for ${\mathcal {D}}$. This completes the whole proof of Theorem 1.1. $\square$

Denote by $\text {Aut}({\mathbb {D}})$ the Möbius group which consists of all one-to-one analytic functions that map ${\mathbb {D}}$ onto itself. Let ${\widetilde{{\mathcal {D}}}}$ be the space of functions f in ${\mathcal {D}}$ with $f(0)=0$. For a sequence $\varvec{\lambda }=\{\lambda _n\}_{n\in {\mathbb {N}}}$ of complex numbers, consider

$$\begin{aligned} {\widetilde{{\mathcal {H}}}}_{\varvec{\lambda }}(f)(z)=\sum _{n=1}^\infty \left( \sum _{k=1}^\infty \lambda _{n+k}a_k\right) z^n, \quad z\in {\mathbb {D}}, \end{aligned}$$

where $f(z)=\sum _{n=1}^\infty a_n z^n \in {\widetilde{{\mathcal {D}}}}$. For the bounded ${\widetilde{{\mathcal {H}}}}_{\varvec{\lambda }}$ on ${\widetilde{{\mathcal {D}}}}$, the norm of ${\widetilde{{\mathcal {H}}}}_{\varvec{\lambda }}$ is Möbius invariant in the following sense.

Proposition 2.1

Let $\varvec{\lambda }=\{\lambda _n\}_{n\in {\mathbb {N}}}$ be a sequence of complex numbers. Suppose the Hankel type operator ${\widetilde{{\mathcal {H}}}}_{\varvec{\lambda }}$ is bounded on ${\widetilde{{\mathcal {D}}}}$. Then there exist positive constants $C_1$ and $C_2$ depending only on $\varvec{\lambda }$ such that

$$\begin{aligned} C_1 \Vert {\widetilde{{\mathcal {H}}}}_{\varvec{\lambda }}\Vert \le \Vert {\widetilde{{\mathcal {H}}}}_{\varvec{\lambda _\phi }}\Vert \le C_2 \Vert {\widetilde{{\mathcal {H}}}}_{\varvec{\lambda }}\Vert \end{aligned}$$

for all $\phi \in \text {Aut}({\mathbb {D}})$, where $\varvec{\lambda _\phi }=\left\{ \overline{\lambda _{\phi , n}}\right\} _{n\in {\mathbb {N}}}$ with

$$\begin{aligned} h_{\overline{{\varvec{\lambda }}}}\circ \phi (z)= \sum _{n=0}^\infty \lambda _{\phi , n} z^n. \end{aligned}$$

Proof

By Theorem 1.1, the boundedness of ${\widetilde{{\mathcal {H}}}}_{\varvec{\lambda }}$ on ${\widetilde{{\mathcal {D}}}}$ yields

$$\begin{aligned} \Vert |h'_{\overline{{\varvec{\lambda }}}}|^2dA\Vert _{CM({\widetilde{{\mathcal {D}}}})}<\infty . \end{aligned}$$

From the proof of Theorem 1.1, there are positive constants $C_1$ and $C_2$ depending only on $\varvec{\lambda }$ such that

$$\begin{aligned} C_1\Vert |h'_{\overline{{\varvec{\lambda }}}}|^2dA\Vert _{CM({\widetilde{{\mathcal {D}}}})}\le \Vert {\widetilde{{\mathcal {H}}}}_{\varvec{\lambda }}\Vert \le C_2 \Vert |h'_{\overline{{\varvec{\lambda }}}}|^2dA\Vert _{CM({\widetilde{{\mathcal {D}}}})}. \end{aligned}$$

(2.2)

Let $\phi \in \text {Aut}({\mathbb {D}})$. For $g\in {\widetilde{{\mathcal {D}}}}$, by the change of variables, we see

$$\begin{aligned} \int _{\mathbb {D}}|g(z)|^2 |(h_{\overline{{\varvec{\lambda }}}}\circ \phi )'(z)|^2dA(z)= \int _{\mathbb {D}}|g(\phi ^{-1}(w))|^2 |h'_{\overline{{\varvec{\lambda }}}}(w)|^2 dA(w). \end{aligned}$$

Also,

$$\begin{aligned} \int _{\mathbb {D}}|g'(z)|^2 dA(z)= \int _{\mathbb {D}}|(g\circ \phi ^{-1})'(w)|^2 dA(w). \end{aligned}$$

Thus there is $C>0$ such that

$$\begin{aligned} \int _{\mathbb {D}}|g(z)|^2 |(h_{\overline{{\varvec{\lambda }}}}\circ \phi )'(z)|^2dA(z)\le C \int _{\mathbb {D}}|g'(z)|^2 dA(z) \end{aligned}$$

for all $g\in {\widetilde{{\mathcal {D}}}}$ if and only if

$$\begin{aligned} \int _{\mathbb {D}}|k(w)|^2 |h'_{\overline{{\varvec{\lambda }}}}(w)|^2 dA(w)\le C \int _{\mathbb {D}}|k'(z)|^2 dA(z) \end{aligned}$$

for all $k\in {\widetilde{{\mathcal {D}}}}$. This means

$$\begin{aligned} \Vert |h'_{\overline{{\varvec{\lambda }}}}|^2dA\Vert _{CM({\widetilde{{\mathcal {D}}}})}=\Vert |(h_{\overline{{\varvec{\lambda }}}}\circ \phi )'|^2 dA\Vert _{CM({\widetilde{{\mathcal {D}}}})}. \end{aligned}$$

Combining this with (2.2), we get the desired result. $\square$

Remark 2.2

By a personal communication, Professor Blasco informed us that Proposition 2.1 can be stated as follows. Suppose $\phi \in \text {Aut}({\mathbb {D}})$ and $\varvec{\lambda }=\{\lambda _n\}_{n\in {\mathbb {N}}}$ is a sequence of complex numbers. Then ${\widetilde{{\mathcal {H}}}}_{\varvec{\lambda }}: {\widetilde{{\mathcal {D}}}}\rightarrow {\widetilde{{\mathcal {D}}}}$ is bounded if and only if ${\widetilde{{\mathcal {H}}}}_{\varvec{\lambda _\phi }}: {\widetilde{{\mathcal {D}}}}\rightarrow {\widetilde{{\mathcal {D}}}}$ is bounded and $\Vert {\widetilde{{\mathcal {H}}}}_{\varvec{\lambda _\phi }}\Vert = \Vert {\widetilde{{\mathcal {H}}}}_{\varvec{\lambda }}\Vert .$

3 Bounded Hankel Type Operators ${\mathcal {H}}_\mu$ and Cesàro Type Operators ${\mathcal {C}}_{\varvec{\eta }}$ on ${\mathcal {D}}$

This section is devoted to an intrinsic description of the boundedness of ${\mathcal {H}}_\mu$ on ${\mathcal {D}}$. Indeed, we can characterize bounded Hankel type operators ${\mathcal {H}}_{{\varvec{\lambda }}}$ on ${\mathcal {D}}$ induced by a decreasing sequence of positive numbers. The proof is based on our characterization of bounded Cesàro type operators on ${\mathcal {D}}$.

The following Hilbert’s double theorem for the Dirichlet space plays a crucial role in our proofs. By using Schur’s test, one can show that inequality (see the proof of Theorem 2 in [41] for more details).

Theorem B

[41, p. 814] Let $f(z)=\sum _{k=0}^\infty a_k z^k$ in ${\mathcal {D}}$. Then there exists a positive constant C independent of f such that

$$\begin{aligned} \sum ^\infty _{m=1}\sum _{n=1}^\infty \frac{|a_n| |a_m|}{\log (n+m+1)}\le C\sum ^\infty _{n=1} n|a_n|^2. \end{aligned}$$

We first prove Theorem 1.3.

Proof of Theorem 1.3

Since $\Vert k_w\Vert _{\mathcal {D}}=1$ for any $w\in {\mathbb {D}}$, we obtain the implication $(\textrm{i})\Rightarrow (\textrm{ii})$.

Next, for each $t\in [0,1)$, recall that the normalized reproducing kernel $k_t$ for the Dirichlet space is

$$\begin{aligned} k_t(z)=\left( 1+\log \frac{1}{1-t^2}\right) ^{-\frac{1}{2}}\left( 1+\sum _{n=1}^\infty \frac{1}{n} t^n z^n\right) . \end{aligned}$$

By direct computations,

$$\begin{aligned} \Vert {\mathcal {C}}_{\varvec{\eta }}k_t\Vert ^2_{{\mathcal {D}}} = \left( 1+\log \frac{1}{1-t^2}\right) ^{-1}\left( |\eta _0|^2+\sum ^\infty _{n=0}(n+1)|\eta _{n+1}|^2\left( 1+\sum ^{n+1}_{k=1}\frac{t^{k}}{k}\right) ^2\right) . \end{aligned}$$

Thus for any positive integer m and $t\in [0,1)$,

$$\begin{aligned} \Vert {\mathcal {C}}_{\varvec{\eta }}k_t\Vert ^2_{{\mathcal {D}}}&\gtrsim \left( \log \frac{e}{1-t}\right) ^{-1}\sum ^\infty _{n=m}(n+1)|\eta _{n+1}|^2\left( \sum ^m_{k=1}\frac{t^{k}}{k}\right) ^2\\&\gtrsim \left( \log \frac{e}{1-t}\right) ^{-1} t^{2m}(\log (m+1))^2\sum ^\infty _{n=m}(n+1)|\eta _{n+1}|^2. \end{aligned}$$

In particular, let $t=\frac{m}{m+1}\in (0,1)$. Then

$$\begin{aligned} \frac{1}{\log (m+1)} \Vert {\mathcal {C}}_{\varvec{\eta }}k_t\Vert ^2_{{\mathcal {D}}} \gtrsim \sum ^\infty _{n=m} (n+1)|\eta _{n+1}|^2. \end{aligned}$$

By (ii), we have $\Vert {\mathcal {C}}_{\varvec{\eta }}k_t\Vert ^2_{{\mathcal {D}}}\lesssim 1$ and hence

$$\begin{aligned} \sum ^\infty _{n=m}n |\eta _n|^2=O\left( \frac{1}{\log (m+2)}\right) , \end{aligned}$$

which gives $(\textrm{ii})\Rightarrow (\textrm{iii})$.

$(\textrm{iii})\Rightarrow (\textrm{i})$. The Widom type condition gives

$$\begin{aligned} |\eta _n|^2\lesssim \frac{1}{(n+1)\log (n+2)} \end{aligned}$$

for all $n\in {\mathbb {N}}$. For $f(z)=\sum ^\infty _{k=0}a_k z^k$ in ${\mathcal {D}}$, we get

$$\begin{aligned} \left| \eta _n\sum ^n_{k=0}a_k\right|&\le |\eta _n|\left( \sum ^n_{k=0}\frac{1}{k+1}\right) ^{\frac{1}{2}}\left( \sum ^n_{k=0}(k+1)|a_k|^2\right) ^{\frac{1}{2}}\\&\le |\eta _n|\left( \log (n+2)\right) ^{\frac{1}{2}} \Vert f\Vert _{{\mathcal {D}}}\\&\lesssim \Vert f\Vert _{{\mathcal {D}}} \end{aligned}$$

for all nonnegative integers n. Thus ${\mathcal {C}}_{\varvec{\eta }}(f)$ is well defined in $H({\mathbb {D}})$.

Because of (iii),

$$\begin{aligned} \sum ^\infty _{n=0}(n+1)|\eta _{n+1}|^2<\infty . \end{aligned}$$

(3.1)

Let $g(z)=\sum ^\infty _{k=0}a_k z^k$ be in the Dirichlet space ${\mathcal {D}}$. Then

$$\begin{aligned} \Vert {\mathcal {C}}_{\varvec{\eta }}g\Vert ^2_{{\mathcal {D}}}&\le |\eta _0 a_0|^2 +2 \sum ^\infty _{n=0}(n+1)|\eta _{n+1}|^2 |a_0|^2\\&\quad + 2\sum ^\infty _{n=0}(n+1)|\eta _{n+1}|^2\left( \sum ^{n+1}_{k=1} |a_k| \right) ^2. \end{aligned}$$

Using (3.1), (iii), and Theorem B, we deduce

$$\begin{aligned} \Vert {\mathcal {C}}_{\varvec{\eta }}g\Vert ^2_{{\mathcal {D}}}&\lesssim \Vert g\Vert _{\mathcal {D}}^2+ \small {\sum ^\infty _{n=0}(n+1)|\eta _{n+1}|^2 \left( \sum ^{\infty }_{k=1} |a_k| \chi _{\{k\le n+1\}}(k) \right) \left( \sum ^{\infty }_{j=1} |a_j|\chi _{\{j\le n+1\}}(j) \right) }\\&\thickapprox \Vert g\Vert _{\mathcal {D}}^2 +\sum ^\infty _{k=1} \sum ^\infty _{j=1} |a_k| |a_j|\left( \sum ^\infty _{n=\max \{k-1,j-1\}}(n+1)|\eta _{n+1}|^2\right) \\&\lesssim \Vert g\Vert _{\mathcal {D}}^2 + \sum ^\infty _{k=1} \sum ^\infty _{j=1} \frac{|a_k| |a_j|}{\log (k+j+1)}\\&\lesssim \Vert g\Vert ^2_{{\mathcal {D}}}, \end{aligned}$$

where $\chi _{\{k\le n+1\}}(k)$ is the characteristic function on ${\{k\le n+1\}}\cap {\mathbb {N}}^+$. Here ${\mathbb {N}}^+$ is the set of positive integers. Hence ${\mathcal {C}}_{\varvec{\eta }}$ is bounded on ${\mathcal {D}}$. The proof of Theorem 1.3 is complete. $\square$

The proof of Theorem 1.2 follows from the next more general result.

Theorem 3.1

Suppose ${{\varvec{\lambda }}}=\{\lambda _n\}^\infty _{n=0}$ is a decreasing sequence of positive real numbers. Then the following conditions are equivalent.

• (i) The Hankel type operator ${\mathcal {H}}_{{\varvec{\lambda }}}$ is bounded on ${\mathcal {D}}$.

• (ii) The reproducing kernel thesis holds; that is,

  $$\begin{aligned} \sup _{t\in [0,1)}\Vert {\mathcal {H}}_{{\varvec{\lambda }}} k_t\Vert _{{\mathcal {D}}}<\infty , \end{aligned}$$

  where $k_t$ is the normalized reproducing kernel of ${\mathcal {D}}$ at t in [0, 1).

• (iii) The Widom type condition is true; that is,

  $$\begin{aligned} \sum ^\infty _{n=m}n \lambda _n^2=O\left( \frac{1}{\log (m+2)}\right) . \end{aligned}$$

Proof

If ${\mathcal {H}}_{{\varvec{\lambda }}}$ is bounded on ${\mathcal {D}}$, it is clear that (ii) holds.

$(\textrm{ii})\Rightarrow (\textrm{iii})$. Since $\{\lambda _n\}$ is a decreasing sequence of positive numbers,

$$\begin{aligned} \Vert {\mathcal {H}}_{{\varvec{\lambda }}} k_t\Vert _{{\mathcal {D}}}^2&\ge \left( 1+\log \frac{1}{1-t^2}\right) ^{-1}\sum ^\infty _{n=1} (n+1) \left( \sum ^\infty _{k=1}\lambda _{n+k+1}\frac{t^{k}}{k}\right) ^2 \\&\ge \left( 1+\log \frac{1}{1-t^2}\right) ^{-1}\sum ^\infty _{n=m} (n+1) \left( \sum ^n_{k=1}\lambda _{n+k+1}\frac{t^{k}}{k}\right) ^2 \\&\ge \left( 1+\log \frac{1}{1-t^2}\right) ^{-1}\sum ^\infty _{n=m} (n+1) \lambda _{2n+1}^2 \left( \sum ^m_{k=1}\frac{t^{k}}{k}\right) ^2 \\&\gtrsim \left( \log \frac{e}{1-t}\right) ^{-1} t^{2m}(\log (m+1))^2 \sum ^\infty _{n=m}(n+1)\lambda _{2n+1}^2 \end{aligned}$$

for all $t\in [0, 1)$ and all positive integers m.

For any fixed integer m, taking $t=\frac{m}{m+1}$ in the above, we get

$$\begin{aligned} \sum ^\infty _{n=m}(2n+1)\lambda _{2n+1}^2\lesssim \frac{1}{\log (m+1)} \end{aligned}$$

and

$$\begin{aligned} \sum ^\infty _{n=m}(2n+2)\lambda _{2n+2}^2\lesssim \frac{1}{\log (m+1)}. \end{aligned}$$

Then the desired result holds.

$(\textrm{iii})\Rightarrow (\textrm{i})$. It follows from (iii) that

$$\begin{aligned} \lambda _k^2 \lesssim \frac{1}{k \log (k+1)} \end{aligned}$$

(3.2)

for all positive integers k. Let $f(z)=\sum ^\infty _{k=0}a_k z^k$ belong to ${\mathcal {D}}$. By (3.2) and the monotonicity of $\{\lambda _k\}$,

$$\begin{aligned} \left| \sum ^\infty _{k=0}\lambda _{n+k}a_k\right|&\le \lambda _n |a_0| +\sum ^\infty _{k=1}\lambda _{n+k}|a_k| \\&\lesssim \Vert f\Vert _{{\mathcal {D}}}+\Vert f\Vert _{{\mathcal {D}}}\left( \sum ^\infty _{k=1}\frac{\lambda ^2_{n+k}}{k+1}\right) ^{\frac{1}{2}}\\&\lesssim \Vert f\Vert _{{\mathcal {D}}}+ \Vert f\Vert _{{\mathcal {D}}}\left( \sum ^\infty _{k=1}\frac{1}{(k+1)^2\log (k+1)}\right) ^{\frac{1}{2}}\\&\lesssim \Vert f\Vert _{{\mathcal {D}}} \end{aligned}$$

for all nonnegative integers n. Then ${\mathcal {H}}_{{\varvec{\lambda }}}(f)$ is analytic on the unit disk ${\mathbb {D}}$ for any f in ${\mathcal {D}}$.

Observe that by condition (iii), we have

$$\begin{aligned} \sum ^\infty _{n=0}(n+1)\lambda _{n+1}^2<\infty . \end{aligned}$$

(3.3)

Then the Hölder inequality gives $\sum ^\infty _{k=0} \lambda _k |a_k|\lesssim \Vert f\Vert _{\mathcal {D}}.$ Consequently,

$$\begin{aligned} \Vert {\mathcal {H}}_{{\varvec{\lambda }}} f\Vert ^2_{{\mathcal {D}}}&\le \left( \sum ^\infty _{k=0} \lambda _k |a_k|\right) ^2 + \sum ^\infty _{n=0}(n+1)\left( \sum ^\infty _{k=0}\lambda _{n+k+1} |a_k| \right) ^2 \nonumber \\&\lesssim \Vert f\Vert _{\mathcal {D}}^2+ \sum ^\infty _{n=0}(n+1)\left( \sum ^{n+1}_{k=0} \lambda _{n+k+1} |a_k| \right) ^2 \nonumber \\&\quad +\sum ^\infty _{n=0}(n+1)\left( \sum ^\infty _{k=n+1} \lambda _{n+k+1} |a_k| \right) ^2. \end{aligned}$$

(3.4)

By Theorem 1.3 and the monotonicity of the sequence ${\varvec{\lambda }}$, it is true that

$$\begin{aligned} \sum ^\infty _{n=0}(n+1)\left( \sum ^{n+1}_{k=0} \lambda _{n+k+1} |a_k| \right) ^2&\le \sum ^\infty _{n=0}(n+1) \lambda _{n+1}^2 \left( \sum ^{n+1}_{k=0} |a_k| \right) ^2 \nonumber \\&\le \Vert {\mathcal {C}}_{{\varvec{\lambda }}} f_2\Vert ^2_{\mathcal {D}}\lesssim \Vert f\Vert ^2_{\mathcal {D}}, \end{aligned}$$

(3.5)

where $f_2(z)=\sum ^\infty _{k=0}|a_k| z^k$ with the same Dirichlet norm of f.

Note that the monotonicity of the sequence ${\varvec{\lambda }}$ again, the Hölder inequality, and formula (3.3). Then

$$\begin{aligned} \sum ^\infty _{n=0}(n+1)\left( \sum ^\infty _{k=n+1} \lambda _{n+k+1} |a_k| \right) ^2&\le \sum ^\infty _{n=0}(n+1)\left( \sum ^\infty _{k=n+1} \lambda _{k} |a_k| \right) ^2 \nonumber \\&\lesssim \Vert f\Vert ^2_{\mathcal {D}}\sum ^\infty _{n=0}(n+1) \left( \sum ^\infty _{k=n+1} \frac{\lambda ^2_k}{k}\right) \nonumber \\&\thickapprox \Vert f\Vert ^2_{\mathcal {D}}\sum ^\infty _{k=1}\frac{\lambda ^2_k}{k} \sum ^{k-1}_{n=0}(n+1) \nonumber \\&\thickapprox \Vert f\Vert ^2_{\mathcal {D}}\sum ^\infty _{k=1} k \lambda ^2_k \lesssim \Vert f\Vert ^2_{\mathcal {D}}. \end{aligned}$$

(3.6)

Joining (3.4), (3.5), and (3.6), we get the boundedness of ${\mathcal {H}}_{{\varvec{\lambda }}}$ on ${\mathcal {D}}$. The proof is complete. $\square$

4 Compact Hankel and Cesàro Type Operators on ${\mathcal {D}}$

In this section, we give corresponding results about the compactness of Hankel and Cesàro type operators on ${\mathcal {D}}$.

The result below is the compact version of Theorem 1.1.

Theorem 4.1

Suppose $\varvec{\lambda }=\{\lambda _n\}_{n\in {\mathbb {N}}}$ is a sequence of complex numbers. Then the Hankel type operator ${\mathcal {H}}_{\varvec{\lambda }}$ is compact on the Dirichlet space ${\mathcal {D}}$ if and only if $h_{\overline{{{\varvec{\lambda }}}}}$ is analytic on ${\mathbb {D}}$ and the measure $|h_{\overline{{{\varvec{\lambda }}}}}'(z)|^2dA(z)$ is a vanishing Carleson measure for the Dirichlet space ${\mathcal {D}}$.

Proof

Suppose $h_{\overline{{\varvec{\lambda }}}}\in {\mathcal {X}}_0$. Then the identity map

$$\begin{aligned} I_d: {\mathcal {D}}\rightarrow L^2({\mathbb {D}}, |h'_{\overline{{\varvec{\lambda }}}}|^2dA) \end{aligned}$$

is compact. Let $\{f_k\}_{k=1}^\infty$ be a bounded sequence in ${\mathcal {D}}$ such that $\{f_k\}_{k=1}^\infty$ tends to 0 uniformly in compact subsets of ${\mathbb {D}}$ as $k\rightarrow \infty$. From the proof of Theorem 1.1, we obtain

$$\begin{aligned} |\langle {\mathcal {H}}_{{\varvec{\lambda }}} f_k,g\rangle _{{\mathcal {D}}}|&\lesssim |\lambda _0| |f_k(0)| |g(0)|+\Vert g\Vert _{{\mathcal {D}}}\left( \int _{\mathbb {D}}\left| \frac{f_k(z)-f_k(0)}{z}\right| ^2|h'_{\overline{{\varvec{\lambda }}}} (z)|^2 dA(z)\right) ^{\frac{1}{2}}\\&\quad + \Vert g\Vert _{\mathcal {D}}\left( \int _{\mathbb {D}}|f_k(z)|^2|h'_{\overline{{\varvec{\lambda }}}} (z)|^2 dA(z)\right) ^{\frac{1}{2}} \end{aligned}$$

for all $g\in {\mathcal {D}}$. Hence, for any $\varepsilon >0$, there is an integer $k_0$ such that

$$\begin{aligned} |\langle {\mathcal {H}}_{{\varvec{\lambda }}} f_k,g\rangle _{{\mathcal {D}}}|\lesssim \Vert g\Vert _{\mathcal {D}}\, \varepsilon \end{aligned}$$

for all $k>k_0$ and all $g\in {\mathcal {D}}$. Then $\Vert {\mathcal {H}}_{{\varvec{\lambda }}} f_k\Vert _{\mathcal {D}}\rightarrow 0$ as $k\rightarrow \infty$. Thus ${\mathcal {H}}_{{\varvec{\lambda }}}$ is compact on ${\mathcal {D}}$.

Conversely, suppose ${\mathcal {H}}_{{\varvec{\lambda }}}$ is compact on ${\mathcal {D}}$. For a bounded sequence $\{(f_n, g_n)\}_{n=1}^\infty \subseteq {\mathcal {D}}\times {\mathcal {D}}$, both $\{f_n\}$ and $\{g_n\}$ are bounded sequences in ${\mathcal {D}}$. Then both $\{{\mathcal {H}}_{{\varvec{\lambda }}}(f_n)\}$ and $\{{\mathcal {H}}_{{\varvec{\lambda }}}(g_n)\}$ have convergent subsequences. Because of (2.1), $T_{h_{\overline{{\varvec{\lambda }}}}}$ extends to a compact bilinear form on ${\mathcal {D}}$. Then Theorem A yields $h_{\overline{{\varvec{\lambda }}}}\in {\mathcal {X}}_0$. The proof is complete. $\square$

For the compactness of ${\mathcal {C}}_{\varvec{\eta }}$ on ${\mathcal {D}}$, we also have the following conclusion.

Theorem 4.2

Suppose ${{\varvec{\eta }}}=\{\eta _n\}_{n=0}^\infty$ is a sequence of complex numbers. Then the following conditions are equivalent.

• (i) The Cesàro type operator ${\mathcal {C}}_{\varvec{\eta }}$ is compact on ${\mathcal {D}}$.

• (ii) The reproducing kernel thesis holds; that is,

  $$\begin{aligned} \lim _{t\rightarrow 1^{-}}\Vert {\mathcal {C}}_{\varvec{\eta }}k_t\Vert _{{\mathcal {D}}}=0, \end{aligned}$$

  where $k_t$ is the normalized reproducing kernel of ${\mathcal {D}}$ at t in [0, 1).

• (iii) The Widom type condition is true; that is,

  $$\begin{aligned} \sum ^\infty _{n=m}n |\eta _n|^2=o\left( \frac{1}{\log (m+2)}\right) . \end{aligned}$$

Proof

$(\textrm{i})\Rightarrow (\textrm{ii})$. Note that $\Vert k_t\Vert _{\mathcal {D}}=1$ for each $t\in [0, 1)$ and $k_t$ tends to zero uniformly in compact subsets of ${\mathbb {D}}$ as $t\rightarrow 1^{-}$. Then (ii) holds.

$(\textrm{ii})\Rightarrow (\textrm{iii})$. Checking the proof of Theorem 1.3, we see

$$\begin{aligned} \log (m+1) \sum ^\infty _{n=m}(n+1)|\eta _{n+1}|^2 \lesssim \Vert {\mathcal {C}}_{\varvec{\eta }}k_{\frac{m}{m+1}}\Vert ^2_{{\mathcal {D}}} \end{aligned}$$

(4.1)

for all positive integers m. Taking $m \rightarrow \infty$ in (4.1), we obtain (iii).

$(\textrm{iii})\Rightarrow (\textrm{i})$. Let m be a positive integer. For $f(z)=\sum _{n=0}^\infty a_nz^n$ in ${\mathcal {D}}$, consider

$$\begin{aligned} {\mathcal {C}}_{\varvec{\eta }}^{(m)} (f)(z)=\sum _{n=0}^m \left( \eta _n \sum _{k=0}^n a_k\right) z^n, \quad z\in {\mathbb {D}}. \end{aligned}$$

Then ${\mathcal {C}}_{\varvec{\eta }}^{(m)}$ is a finite rank operator. Thus, ${\mathcal {C}}_{\varvec{\eta }}^{(m)}$ is compact on ${\mathcal {D}}$. Because of (iii), for every $\epsilon >0$, there exists a positive integer N such that

$$\begin{aligned} \sum ^\infty _{n=m} (n+1)|\eta _{n+1}|^2< \frac{\epsilon }{\log (m+2)} \end{aligned}$$

for $m>N$. Using the proof of Theorem 1.3, we have

$$\begin{aligned} \sum ^\infty _{n=m} (n+1) |\eta _{n+1}|^2 \left( \sum ^{n+1}_{k=1} |a_k| \right) ^2&\lesssim \sum ^\infty _{k=1} \sum ^\infty _{j=1} |a_k| |a_j|\sum ^\infty _{n=\max \{k-1,j-1, m\}}(n+1)|\eta _{n+1}|^2\\&\lesssim \epsilon \sum ^\infty _{k=1} \sum ^\infty _{j=1} \frac{|a_k| |a_j|}{\log (k+j+m)}\\&\lesssim \epsilon \Vert f\Vert _{\mathcal {D}}^2 \end{aligned}$$

for all $m>N$. Thus, for $m>N$,

$$\begin{aligned} \Vert ({\mathcal {C}}_{\varvec{\eta }}-{\mathcal {C}}_{\varvec{\eta }}^{(m)}) (f)\Vert ^2_{\mathcal {D}}&\lesssim |a_0|^2\sum ^\infty _{n=m} (n+1)|\eta _{n+1}|^2+\sum ^\infty _{n=m} (n+1) |\eta _{n+1}|^2 \left( \sum ^{n+1}_{k=1} |a_k| \right) ^2\\&\lesssim \epsilon \Vert f\Vert _{\mathcal {D}}^2. \end{aligned}$$

In other words, $\Vert {\mathcal {C}}_{\varvec{\eta }}-{\mathcal {C}}_{\varvec{\eta }}^{(m)}\Vert \rightarrow 0$ as $m\rightarrow \infty$. Hence ${\mathcal {C}}_{\varvec{\eta }}$ is compact on ${\mathcal {D}}$. The proof is finished. $\square$

The compact result corresponding to Theorem 3.1 also holds.

Theorem 4.3

Suppose ${{\varvec{\lambda }}}=\{\lambda _n\}^\infty _{n=0}$ is a decreasing sequence of positive real numbers. Then the following conditions are equivalent.

• (i) The Hankel type operator ${\mathcal {H}}_{{\varvec{\lambda }}}$ is compact on ${\mathcal {D}}$.

• (ii) The reproducing kernel thesis holds; that is,

  $$\begin{aligned} \lim _{t\rightarrow 1^-}\Vert {\mathcal {H}}_{{\varvec{\lambda }}} k_t\Vert _{{\mathcal {D}}}=0, \end{aligned}$$

  where $k_t$ is the normalized reproducing kernel of ${\mathcal {D}}$ at t in [0, 1).

• (iii) The Widom type condition is true; that is,

  $$\begin{aligned} \sum ^\infty _{n=m}n \lambda _n^2=o\left( \frac{1}{\log (m+2)}\right) . \end{aligned}$$

Proof

By the proof of Theorem 3.1, the arguments here are similar to that of Theorem 4.2. We omit it. $\square$

5 Random Hankel Type Operators on ${\mathcal {D}}$ and a Result Related to Rudin’s $\Lambda (p)$ Sets

In this section, we prove Theorem 1.4 and Corollary 1.5. Many known real random variables sequences $(X_n)_{n\in {\mathbb {N}}}$ satisfy the conditions in Theorem 1.4. First of all, all bounded mean zero random variables are included in Thereom 1.4. The typical example is the Bernoulli random variables sequence $(X_n)_{n\in {\mathbb {N}}}$. In other words, $(X_n)_{n\in {\mathbb {N}}}$ is independent such that ${\mathbb {P}}(X_n=1)={\mathbb {P}}(X_n=-1)=1/2$. In addition, for the classical independent normal distribution sequence $X_n=N(0,1)$, we have ${\mathbb {E}}[X_n]=0$ and ${\mathbb {E}}[X_n^4]=3$ (cf. [32, p. 2]). Therefore, Theorem 1.4 can be applied to the corresponding random Gaussian analytic Dirichlet functions.

For a complex number sequence $\varvec{\lambda }=\{\lambda _n\}_{n\in {\mathbb {N}}}$ and a sequence of i.i.d. real random variables $(X_n)_{n\in {\mathbb {N}}}$ with ${\mathbb {E}}[X_n]=0$ and ${\mathbb {E}}[X_n^4]<\infty$, if $h_{{\varvec{\lambda }}}\in {\mathcal {D}}$, then $\sum _{n=0}^\infty |\lambda _n|^2<\infty$. Moreover, by i.i.d. and ${\mathbb {E}}[X_n^2]\le \sqrt{{\mathbb {E}}[X_n^4]}$, we have

$$\begin{aligned} {\mathbb {E}}\left[ \sum _{n=0}^\infty |\lambda _n X_n|^2\right] =\sum _{n=0}^\infty |\lambda _n|^2{\mathbb {E}}[|X_n|^2]<\infty \end{aligned}$$

and almost surely

$$\begin{aligned} h_{{\varvec{\lambda }},\omega }(z)=\sum _{n=0}^\infty X_n(\omega )\lambda _n z^n \end{aligned}$$

is analytic on the unit disk.

For our proof, we need the following elementary inequalities which may be well known. However, we cannot locate literature. For the sake of completeness, we have included a brief proof.

Lemma 5.1

Let $(X_n)_{n\in {\mathbb {N}}}$ be a sequence of i.i.d. real random variables with ${\mathbb {E}}[X_n]=0$ and ${\mathbb {E}}[X_n^4]=1$. Then for all finite complex numbers $a_1,\ldots , a_n$, we have

$$\begin{aligned} {\mathbb {E}}\left[ \left| \sum _{i=1}^n a_i X_i\right| ^4\right] \le 2\left[ \sum _{i=1}^n |a_i|^2\right] ^2. \end{aligned}$$

Proof

For any fixed natural number $n\ge 1$, let $A_n=\sum _{i=1}^n a_i X_i.$ It is easy to see

$$\begin{aligned} |A_n|^2=\sum _{i=1}^n |a_i|^2X_i^2+2\sum _{i<j}Re(a_i{\bar{a}}_j)X_iX_j. \end{aligned}$$

Note that $(X_n)_{n\in {\mathbb {N}}}$ is a sequence of i.i.d. real random variables with ${\mathbb {E}}[X_n]=0$ for each n. Then

$$\begin{aligned} {\mathbb {E}}\big [|A_n|^4\big ]&={\mathbb {E}}\left[ \left( \sum _{i=1}^n |a_i|^2X_i^2\right) ^2\right] +4{\mathbb {E}}\left[ \left( \sum _{i<j}Re(a_i{\bar{a}}_j)X_iX_j \right) ^2\right] \\&=\sum _{i,j}|a_i|^2|a_j|^2{\mathbb {E}}[X_i^2X_j^2]+4\sum _{i<j}(Re(a_i{\bar{a}}_j))^2{\mathbb {E}}[X_i^2X_j^2]\\&\le \sum _{i,j}|a_i|^2|a_j|^2{\mathbb {E}}[X_i^2X_j^2]+4\sum _{i<j}|a_i{\bar{a}}_j|^2{\mathbb {E}}[X_i^2X_j^2]. \end{aligned}$$

Observe that $\big ({\mathbb {E}}[X_i^2X_j^2]\big )^2\le {\mathbb {E}}[X_i^4]{\mathbb {E}}[X_j^4]= 1$, hence

$$\begin{aligned} {\mathbb {E}}\left[ \left| \sum _{i=1}^n a_i X_i\right| ^4\right] \le 2\left[ \sum _{i=1}^n|a_i|^2\right] ^2, \end{aligned}$$

and the proof is completed. $\square$

Let $f\in H({\mathbb {D}})$. For $0<p<\infty$ and $0\le r<1$, let

$$\begin{aligned} M_p(f, r)=\left( \frac{1}{2\pi }\int _0^{2\pi }|f(re^{i\theta })|^p d\theta \right) ^{\frac{1}{p}} \end{aligned}$$

and

$$\begin{aligned} M_\infty (f, r)=\max _{\theta \in [0, 2\pi ]}|f(re^{i\theta })|. \end{aligned}$$

Denote by $H^p$ the Hardy space consisting of those functions f in $H({\mathbb {D}})$ with

$$\begin{aligned} \Vert f\Vert _{H^p}=\sup _{0<r<1} M_p(f, r)<\infty . \end{aligned}$$

It is well known (cf. [20, p. 3]) that ${\mathcal {D}}$ is a subset of $H^p$ for any $p>0$.

From a result of Brown and Shields [13, p. 300], if an analytic function f satisfies $M_p(f', r)\in L^2([0, 1], dr)$ with $2<p\le \infty$, then $|f'(z)|^2dA(z)$ is a Carleson measure for ${\mathcal {D}}$. We strengthen this conclusion as follows.

Lemma 5.2

Let $2<p\le \infty$. Suppose $\phi$ is an analytic function on the open unit disk ${\mathbb {D}}$ with $M_p(\phi ', r)\in L^2([0, 1], dr)$. Then $|\phi '(z)|^2dA(z)$ is a vanishing Carleson measure for the Dirichlet space.

Proof

Let $2<p<\infty$. Suppose $\{f_m\}_{m=1}^\infty$ is a sequence in ${\mathcal {D}}$ such that $\sup _m \Vert f_m\Vert _{\mathcal {D}}<\infty$ and functions $f_m$ tend to zero uniformly in compact subsets of ${\mathbb {D}}$ as $m\rightarrow \infty$. Due to $M_p(\phi ', r)\in L^2([0, 1], dr)$, for any $\epsilon >0$, there exists a $\delta$ in (0, 1) such that

$$\begin{aligned} \int _\delta ^1 \left( \int _0^{2\pi } |\phi '(re^{i\theta })|^p d\theta \right) ^{\frac{2}{p}}dr<\epsilon . \end{aligned}$$

Since $p>2$, using the Hölder inequality with indices $\frac{p}{2}$ and $\frac{p}{p-2}$, we get for any $r\in (0,1)$

$$\begin{aligned} \int _0^{2\pi } |f_m(re^{i\theta })|^2 |\phi '(re^{i\theta })|^2 \frac{d\theta }{2\pi }&\le \left( \int _0^{2\pi } |\phi '(re^{i\theta })|^p \frac{d\theta }{2\pi }\right) ^{\frac{2}{p}}\\&\quad \times \left( \int _0^{2\pi } |f_m(re^{i\theta })|^{\frac{2p}{p-2}} \frac{d\theta }{2\pi }\right) ^{\frac{p-2}{p}}\\&\le \left( \int _0^{2\pi } |\phi '(re^{i\theta })|^p \frac{d\theta }{2\pi }\right) ^{\frac{2}{p}} \Vert f_m\Vert _{H^{\frac{2p}{p-2}}}^2. \end{aligned}$$

Therefore, for the above $\delta \in (0,1)$,

$$\begin{aligned} \int _{\{z\in {\mathbb {D}}: |z|>\delta \}} |f_m(z)|^2 |\phi '(z)|^2 dA(z)&= \int _\delta ^1 \left( \frac{1}{\pi }\int _0^{2\pi } |f_m(re^{i\theta })|^2 |\phi '(re^{i\theta })|^2 d\theta \right) dr\\&\le 2 (2\pi )^{-2/p} \Vert f_m\Vert _{H^{\frac{2p}{p-2}}}^2 \\&\quad \times \int _\delta ^1\left( \int _0^{2\pi } |\phi '(re^{i\theta })|^p d\theta \right) ^{\frac{2}{p}}dr\\&\lesssim \epsilon \Vert f_m\Vert _{H^{\frac{2p}{p-2}}}^2 \lesssim \epsilon \Vert f_m\Vert _{{\mathcal {D}}}^2\lesssim \epsilon . \end{aligned}$$

Since functions $f_m$ tend to zero uniformly in $\{z\in {\mathbb {D}}: |z|\le \delta \}$, there is a positive integer N such that

$$\begin{aligned} \int _{\{z\in {\mathbb {D}}: |z|\le \delta \}} |f_m(z) \phi '(z)|^2 dA(z) \le \epsilon \end{aligned}$$

for all $m>N$. Consequently,

$$\begin{aligned} \lim _{m\rightarrow \infty }\int _{\mathbb {D}}|f_m(z) \phi '(z)|^2 dA(z)=0. \end{aligned}$$

Thus $|\phi '(z)|^2dA(z)$ is a vanishing Carleson measure for ${\mathcal {D}}$. The proof of the case of $p=\infty$ is similar. We omit it. $\square$

Note that if $p=2$ and $\phi \in H({\mathbb {D}})$, then $M_p(\phi ', r)\in L^2([0, 1], dr)$ if and only if $\phi \in {\mathcal {D}}$. In general, the measure $|\phi '(z)|^2dA(z)$ with $\phi \in {\mathcal {D}}$ is not a vanishing Carleson measure for the Dirichlet space.

Remark 5.3

For $\phi \in H({\mathbb {D}})$, if $|\phi '(z)|^2dA(z)$ is a vanishing Carleson measure for ${\mathcal {D}}$, then of course $|\phi '(z)|^2dA(z)$ is a Carleson measure for ${\mathcal {D}}$. By a personal communication, Professor Blasco has a condition weaker than that in Lemma 5.2 to show that $|\phi '(z)|^2dA(z)$ is a Carleson measure for ${\mathcal {D}}$.

Now, we are ready to prove Theorem 1.4.

Proof

If ${\mathbb {E}}[X_n^4]=0$, then $X_n=0$ almost surely. Hence without loss of generality, we can assume that ${\mathbb {E}}[X_n^4]=1$. Recall that

$$\begin{aligned} h_{\overline{\varvec{\lambda }},\omega }(z)=\sum _{n=0}^\infty X_n(\omega )\overline{\lambda _n} z^n. \end{aligned}$$

By Theorem 1.1, we shall show that almost surely

$$\begin{aligned} |h_{\overline{\varvec{\lambda }},\omega }'(z)|^2dA(z) \end{aligned}$$

is a vanishing Carleson measure for ${\mathcal {D}}$. By Lemma 5.2, it is sufficient to show that

$$\begin{aligned} {\mathbb {E}}\left[ \int _0^1 \left( \int _0^{2\pi }|h_{\overline{\varvec{\lambda }},\omega }'(re^{i\theta })|^4\frac{d\theta }{2\pi }\right) ^{1/2} dr\right] <\infty . \end{aligned}$$

It follows from Fubini’s theorem that

$$\begin{aligned}&{\mathbb {E}}\left[ \int _0^1 \left( \int _0^{2\pi }|h_{\overline{\varvec{\lambda }},\omega }'(re^{i\theta })|^4\frac{d\theta }{2\pi }\right) ^{1/2} dr\right] \\&\quad =\int _0^1\int _\Omega \left( \int _0^{2\pi }|h_{\overline{\varvec{\lambda }},\omega }'(re^{i\theta })|^4\frac{d\theta }{2\pi }\right) ^{1/2}d{\mathbb {P}}(w)dr. \end{aligned}$$

Applying the Hölder inequality with respect to the probability measure $d{\mathbb {P}}$, we have

$$\begin{aligned}&\int _0^1\int _\Omega \left( \int _0^{2\pi }|h_{\overline{\varvec{\lambda }},\omega }'(re^{i\theta })|^4\frac{d\theta }{2\pi }\right) ^{1/2}d{\mathbb {P}}(w)dr\\&\quad \le \int _0^1\left( \int _\Omega \int _0^{2\pi }|h_{\overline{\varvec{\lambda }},\omega }'(re^{i\theta })|^4\frac{d\theta }{2\pi }d{\mathbb {P}}(w)\right) ^{1/2}dr. \end{aligned}$$

Consequently,

$$\begin{aligned} \int _\Omega \int _0^{2\pi }|h_{\overline{\varvec{\lambda }},\omega }'(re^{i\theta })|^4\frac{d\theta }{2\pi }d{\mathbb {P}}(w) = \int _0^{2\pi }\int _\Omega |h_{\overline{\varvec{\lambda }},\omega }'(re^{i\theta })|^4d{\mathbb {P}}(w)\frac{d\theta }{2\pi }. \end{aligned}$$

Recall that

$$\begin{aligned} h_{\overline{\varvec{\lambda }},\omega }'(re^{i\theta })=\sum _{n=1}^\infty \overline{\lambda _n} n r^{n-1}e^{i(n-1)\theta }X_n(\omega ) \end{aligned}$$

and Lemma 5.1, we get

$$\begin{aligned} \int _0^1\left( \int _\Omega \left| \sum _{n=1}^\infty \overline{\lambda _n} n r^{n-1}e^{i(n-1)\theta }X_n\right| ^4d{\mathbb {P}}(w)\right) ^\frac{1}{2} dr \le 3\sum _{n=1}^\infty \int _0^1 | \overline{\lambda _n} n r^{n-1}|^2dr. \end{aligned}$$

Therefore, there is a positive constant C such that

$$\begin{aligned} {\mathbb {E}}\left[ \int _0^1\left( \int _0^{2\pi }|h_{{\varvec{\lambda }},\omega }'(re^{i\theta })|^4\frac{d\theta }{2\pi }\right) ^{1/2}dr\right] \le C\sum _{n=1}^\infty n|\lambda _n|^2<\infty . \end{aligned}$$

This completes the whole proof. $\square$

To prove Corollary 1.5, we start with a simple observation.

Lemma 5.4

Suppose $p>2$, E is a Rudin’s $\Lambda (p)$ set and $\varvec{\lambda }=\{\lambda _{n}: n\in E\}$ is a sequence of complex numbers. If $\sum _{n\in E}n|\lambda _n|^2<\infty$, then $|h_{\overline{{\varvec{\lambda }}}}'(z)|^2dA(z)$ is a vanishing Carleson measure for the Dirichlet space, where $h_{\overline{{\varvec{\lambda }}}}(z)=\sum _{n\in E}\overline{\lambda _n} z^n.$

Proof

Let $p>2$. For any $0<r<1$, observe that

$$\begin{aligned} h_{\overline{{\varvec{\lambda }}}}'(re^{i\theta })=e^{-i\theta }\sum _{n\in E}\overline{\lambda _n} n r^{n-1}e^{in \theta }, \end{aligned}$$

we have

$$\begin{aligned} \Vert h_{\overline{{\varvec{\lambda }}}}'(re^{i\theta })\Vert _{L^p([0, 2\pi ], d\theta )}= \left\| \sum _{n\in E}\overline{\lambda _n} n r^{n-1}e^{in \theta }\right\| _{L^p([0, 2\pi ], d\theta )}. \end{aligned}$$

Since E is a Rudin’s $\Lambda (p)$, there exists a positive constant C such that

$$\begin{aligned} \left\| \sum _{n\in E}\overline{\lambda _n} n r^{n-1}e^{in \theta }\right\| ^2_{L^p([0, 2\pi ], d\theta )}&\le C \left\| \sum _{n\in E}\overline{\lambda _n} n r^{n-1}e^{in \theta }\right\| ^2_{L^2([0, 2\pi ], d\theta )}\\&=C \sum _{n\in E}|\lambda _n|^2n^2r^{2(n-1)}. \end{aligned}$$

Recall that $\sum _{n\in E}n|\lambda _n|^2<\infty$, therefore,

$$\begin{aligned} \int _0^1 \left\| \sum _{n\in E}\overline{\lambda _n} n r^{n-1}e^{in \theta }\right\| ^2_{L^p([0, 2\pi ], d\theta )}dr&\le C \int _0^1 \sum _{n\in E}|\lambda _n|^2n^2r^{2(n-1)}dr\\&< +\infty . \end{aligned}$$

The lemma then follows from Lemma 5.2. $\square$

Proof of Corollary 1.5

Suppose ${\mathcal {H}}_{\varvec{\lambda }}$ is bounded on ${\mathcal {D}}$. By Theorem 1.1, $h_{\overline{{{\varvec{\lambda }}}}}\in {\mathcal {D}}$. By the definition of Dirichlet norm, the condition (iii) holds. $(\textrm{i})\Rightarrow (\textrm{ii})$ is clear. The implication $(\textrm{iii})\Rightarrow (\textrm{i})$ follows from Lemma 5.4 and Theorem 4.1. The proof is complete. $\square$

6 Final Remarks

In this section, we give some remarks about some functions in ${\mathcal {X}}$ and the action of Hankel matrices on the Bergman space $A^2$.

Our results, as we stated in Corollary 1.6, in this paper yield a complete characterization of functions in ${\mathcal {X}}$ with decreasing Taylor’s sequences of positive numbers. More precisely, suppose ${{\varvec{\lambda }}}=\{\lambda _n\}^\infty _{n=0}$ is a decreasing sequence of positive real numbers. From Theorems 1.1 and 3.1, the measure $|h'_{{\varvec{\lambda }}}(z)|^2 dA(z)$ is a Carleson measure for ${\mathcal {D}}$ (i.e. $h_{{\varvec{\lambda }}}\in {\mathcal {X}}$) if and only if

$$\begin{aligned} \sum ^\infty _{n=m}n \lambda _n^2=O\left( \frac{1}{\log (m+2)}\right) . \end{aligned}$$

By Theorems 4.1 and 4.3, the measure $|h'_{{\varvec{\lambda }}}(z)|^2 dA(z)$ is a vanishing Carleson measure for ${\mathcal {D}}$ (i.e. $h_{{\varvec{\lambda }}}\in {\mathcal {X}}_0$) if and only if

$$\begin{aligned} \sum ^\infty _{n=m}n \lambda _n^2=o\left( \frac{1}{\log (m+2)}\right) . \end{aligned}$$

Let $p>2$, for an analytic function f with non-zero Fourier coefficients belonging to a Rudin’s $\Lambda (p)$ set E, we write

$$\begin{aligned} f(z)=\sum _{n\in E} a_nz^n, \end{aligned}$$

by Corollary 1.5, the following statements are equivalent:

• (a) $f\in {\mathcal {X}}$;

• (b) $f\in {\mathcal {X}}_0$;

• (c) $\sum _{n\in E} (n+1) |a_n|^2<\infty .$

Suppose $\varvec{\lambda }=\{\lambda _n\}_{n\in {\mathbb {N}}}$ is a sequence of complex number. Let $(X_n)_{n\in {\mathbb {N}}}$ be a sequence of i.i.d. real random variables with ${\mathbb {E}}[X_n]=0$ and ${\mathbb {E}}[X_n^4]<\infty$. If $h_{{\varvec{\lambda }}}\in {\mathcal {D}}$, Theorems 1.4 and 4.1 yield that

$$\begin{aligned} {\mathbb {P}}(h_{{\overline{{{\varvec{\lambda }}}}, \omega }}\in {\mathcal {X}})={\mathbb {P}}(h_{\overline{{{\varvec{\lambda }}}}, \omega }\in {\mathcal {X}}_0)=1, \end{aligned}$$

where $h_{\overline{{{\varvec{\lambda }}}}, \omega }(z)=\sum _{n=0}^\infty X_n(\omega )\overline{\lambda _n} z^n$.

Next we consider Hankel type operators ${\mathcal {H}}_{\varvec{\lambda }}$ on the Bergman space $A^2$ which is also a Hilbert space of analytic functions on ${\mathbb {D}}$ and it is equipped with the inner product

$$\begin{aligned} \langle f, g\rangle _{A^2}=\int _{\mathbb {D}}f(z) \overline{g(z)} dA(z). \end{aligned}$$

From [16, p. 349], $(A^2)^*\cong {\mathcal {D}}$ and ${\mathcal {D}}^*\cong A^2$ under the pairing

$$\begin{aligned} \langle f, g \rangle = \sum ^\infty _{k=0}a_k b_k, \end{aligned}$$

(6.1)

where $f(z)=\sum ^\infty _{k=0}a_kz^k$ and $g(z)=\sum ^\infty _{k=0}b_k z^k$.

For a sequence $\varvec{\lambda }=\{\lambda _n\}_{n\in {\mathbb {N}}}$ of complex numbers, it is easy to see

$$\begin{aligned} \langle {\mathcal {H}}_{{\varvec{\lambda }}} f, g \rangle =\langle f, {\mathcal {H}}_{{\varvec{\lambda }}} g \rangle . \end{aligned}$$

Consequently, ${\mathcal {H}}_{{\varvec{\lambda }}}$ is bounded (resp. compact) on the Dirichlet space ${\mathcal {D}}$ if and only if ${\mathcal {H}}_{{\varvec{\lambda }}}$ is bounded (resp. compact) on $A^2$. If we replace the pairing (6.1) by the Cauchy pairing

$$\begin{aligned} (f, g )= \sum ^\infty _{k=0}a_k\overline{ b_k}. \end{aligned}$$

Then

$$\begin{aligned} ( {\mathcal {H}}_{{\varvec{\lambda }}} f, g )=( f, {\mathcal {H}}_{\overline{{\varvec{\lambda }}}} g ). \end{aligned}$$

Hence we also get that ${\mathcal {H}}_{{\varvec{\lambda }}}$ is bounded (resp. compact) on ${\mathcal {D}}$ if and only if ${\mathcal {H}}_{\overline{{\varvec{\lambda }}}}$ is bounded (resp. compact) on $A^2$.