# Riesz multiplier convergent spaces of operator valued series and a version
of Orlicz- Pettis theorem

Mahmut Karakuş and Ramazan Kama  Van Yüzüncü Yıl University, Faculty of
Science, Department of Mathematics, 65100 - Van, Turkey<EMAIL_ADDRESS><EMAIL_ADDRESS>Si̇i̇rt University, Faculty of Education, Department of
Mathematics and Physical Sciences Education, 56100 - Si̇i̇rt, Turkey
<EMAIL_ADDRESS>
###### Abstract.

It is not usual to characterize an operator valued series via completeness of
multiplier spaces. In this study, by using a series of bounded linear
operators, we introduce the space $M^{\infty}_{R}\big{(}\sum_{k}T_{k}\big{)}$
of Riesz summability which is a generalization of the Cesàro summability.
Therefore, we give the completeness criteria of these spaces with
$c_{0}(X)$-multiplier convergent operator series. It is a natural consequence
that one can characterize the completeness of a normed space through
$M^{\infty}_{R}\big{(}\sum_{k}T_{k}\big{)}$ which will be assumed that is
complete for every $c_{0}(X)$-multiplier Cauchy operator series. Then, we
characterize the continuity and the (weakly) compactness of the summing
operator $\mathcal{S}$ from the multiplier space
$M^{\infty}_{R}\big{(}\sum_{k}T_{k}\big{)}$ to an arbitrary normed space $Y$
through $c_{0}(X)$-multiplier Cauchy and $\ell_{\infty}(X)$-multiplier
convergent series, respectively. We also prove that if $\sum_{k}T_{k}$ is
$\ell_{\infty}(X)$-multiplier Cauchy, then the multiplier space of weakly
Riesz-convergence associated to the operator valued series
$M^{\infty}_{wR}\big{(}\sum_{k}T_{k}\big{)}$ is subspace of
$M^{\infty}_{R}\big{(}\sum_{k}T_{k}\big{)}$. Among other results, finally, we
obtain a new version of the well-known Orlicz-Pettis theorem by using Riesz-
summability.

###### Key words and phrases:

Riesz summability; operator valued series; multiplier convergent series;
summing operator; Orlicz-Pettis Theorem

###### 2000 Mathematics Subject Classification:

46B15, 40A05, 46B45, 40C05

##  1\. Introduction

Let $\mathbb{N}$ be the set of positive integers, $\mathbb{R}$ and
$\mathbb{C}$ also be the real and complex fields as usual, respectively. By
$\omega$, we denote the space of all real (or complex) valued sequences and
any vector subspace of $\omega$ is also called as a sequence space. A $K$
space is a locally convex sequence space $X$ containing $\phi$ on which
coordinate functionals $\pi_{k}(x)=x_{k}$ are continuous for every
$k\in\mathbb{N}$, the set of positive integers, where $\phi$ is the space of
finitely non-zero sequences spanned by the set $\\{e^{k}:k\in\mathbb{N}\\}$.
$e^{k}$ is the sequence whose only non-zero term is 1 in the $k^{th}$ place
for all $k\in\mathbb{N}$, and $e$ is also the sequence with $e=(1,1,...)$. A
complete linear metric (or complete normed) $K$ space is called an $FK$ (or a
$BK$) space. Let $X\supset\phi$ be a $BK$ space and $x=(x_{k})\in X$. Then, by
$x^{[n]}=\sum_{k=1}^{n}x_{k}e^{k}$ for all $n\in\mathbb{N}$, we denote the
$n^{th}$ section of $x$. It is said that $X\supset\phi$ is an $AK$ space if
$\big{\|}x^{[n]}-x\big{\|}_{X}\to 0$, as $n\to\infty$, for each $x=(x_{k})\in
X$. The spaces $\ell_{\infty}$, $c$ and $c_{0}$ of bounded, convergent and
null sequences are $BK$ spaces, respectively, with the sup norm
$\|x\|_{\infty}=\sup_{k\in\mathbb{N}}|x_{k}|$.

Let $X$, $Y$ be any two sequence spaces and $A=(a_{nk})$ be an infinite matrix
of complex numbers $a_{nk}$, where $k,n\in\mathbb{N}$. Then, we say that $A$
defines a matrix transformation from $X$ into $Y$ and we denote it by writing
$A:X\to Y,$ if for every sequence $x=(x_{k})\in X$ the sequence
$Ax=\\{(Ax)_{n}\\}$, the $A$-transform of $x$, is in $Y$; where

(1.1) $\displaystyle(Ax)_{n}=\sum_{k}a_{nk}x_{k}.$

For simplicity in notation, here and after, the summation without limits runs
from 1 to $\infty$. By $(X:Y)$, we denote the class of all matrices $A$ such
that $A:X\to Y$. Thus, $A\in(X:Y)$ if and only if the series on the right side
of (1.1) converges for each $n\in\mathbb{N}$. Furthermore, the sequence $x$ is
said to be $A$-summable to $a\in\mathbb{C}$ if $Ax$ converges to $a$ which is
called the $A$-limit of $x$. The reader can refer to [6, 7] and [26] for
recent results and related topics in summability.

We say that the series $\sum_{k}x_{k}$ in a normed space $X$ is
unconditionally convergent ($uc$) or unconditionally Cauchy ($uC$) if the
series $\sum_{k}x_{\pi(k)}$ converges or is a Cauchy series for every
permutation $\pi$ of $\mathbb{N}$. It is called weakly unconditionally Cauchy
($wuC$) if for every permutation $\pi$ of $\mathbb{N}$, the sequence
$\big{(}\sum_{k=1}^{n}x_{\pi(k)}\big{)}$ is a weakly Cauchy sequence or as a
useful result, $\sum_{k}x_{k}$ is $wuC$ if and only if
$\sum_{k}|x^{*}(x_{k})|<\infty$ for all $x^{*}\in X^{*}$, the space of all
linear and bounded (continuous) functionals defined on $X$. It is well known
that every $wuC$ series in a Banach space $X$ is $uc$ if and only if $X$
contains no copy of $c_{0}$; (cf. [4, pp. 42, 44], [8, p. 44] and [19, p.
18]). The reader can refer to the Diestel’ s famous monograph [8] given on the
theory of sequences and series in Banach spaces, Albiac and Kalton [4] for
specific investigations of Banach spaces and Marti’s [19] for basic sequences
and fundamentals of bases.

Let $X$ and $Y$ be two normed spaces and $\omega(X)$ be the space of all
$X$-valued sequences. By $\ell_{\infty}(X)$, $c(X)$ and $c_{0}(X)$, we denote
the spaces of all $X$-valued bounded, convergent and null sequences,
respectively. $\phi(X)$ also denotes the space of $X$-valued finitely non-zero
sequences. Let $\mathcal{V}$ be a vector space of $X$-valued sequences
equipped with a locally convex Hausdorff topology. If the maps
$x=(x_{k})\mapsto x_{k}$ from $\mathcal{V}$ into $X$ are continuous for all
$k\in\mathbb{N}$, then $\mathcal{V}$ is called as a $K$ space. If $x\in X$,
then by $e^{k}\otimes x$, we denote the sequence whose only non-zero term is
$x$ in the $k^{th}$ place for all $k\in\mathbb{N}$. If
$\phi(X)\subset\mathcal{V}$ and $T_{k}\in B(X:Y)$, the space of all bounded
and linear operators defined from $X$ into $Y$ for all $k\in\mathbb{N}$, then
we say that the series $\sum_{k}T_{k}$ is $\mathcal{V}$-multiplier convergent
or $\mathcal{V}$-multiplier Cauchy if the series $\sum_{k}T_{k}x_{k}$
converges in $Y$ or is a Cauchy series in $Y$, i.e., the partial sums of the
series $\sum_{k}T_{k}x_{k}$ form a norm Cauchy sequence in $Y$ for all
$x=(x_{k})\in\mathcal{V}$. The reader may refer to [22] for the vector valued
multiplier spaces associated with operator valued series (OVS) and more
detailed information on multiplier convergent series.

The multiplier form of a series $\sum_{k}x_{k}$ in a Banach space $X$
associated with an arbitrary real or complex sequence $a=(a_{k})$ is given as
$\sum_{k}a_{k}x_{k}$ and is also important to understand the behavior of the
series $\sum_{k}x_{k}$ in $X$. The series $\sum_{k}x_{k}$ in $X$ is $wuC$ (or
$uc$) series if and only if $\sum_{k}a_{k}x_{k}$ is convergent for every null
(or bounded) sequence $a=(a_{k})$, that is, $\sum_{k}x_{k}$ is a $c_{0}$-(or
an $\ell_{\infty}$-) multiplier convergent series. As another example, a
series $\sum_{k}x_{k}$ is subseries convergent if and only if it is
$m_{0}$-multiplier convergent series, where $m_{0}$ is the space of all finite
range sequences. Let also recall that a series $\sum_{k}x_{k}$ is subseries
convergent if and only if $\sum_{k}x_{n_{k}}$ is convergent for all
subsequences $(n_{k})$ of $\mathbb{N}$. There is also an important result on
subseries convergence which states that in a normed space $X$, a subseries
convergent series is $\ell_{\infty}$-multiplier Cauchy, if also $X$ is
sequentially complete then the series is $\ell_{\infty}$-multiplier
convergent.

The studies of characterizations of Banach spaces, and obtaining new
multiplier spaces by means of summability methods have been an interesting
field of research in the theory of modern analysis. In [20], McArthur
investigated some closed linear subspaces of the space of the weakly
unconditional summable sequences in a Banach space $X$ and gave some inclusion
relations between these spaces. One can refer to the Aizpuru et al. [1, 3],
Pérez-Fernández et al. [21], Kama and Altay [10], Kama et al. [11], Kama [9,
12], Karakuş [13], Karakuş and Başar [14, 15, 16] and Swartz [22, 23] for
recent studies on multiplier convergent series.

Through the rest of the paper, we deal with the Riesz transformation which is
a generalization of Cesàro mean $C_{1}$ of order one given as

(1.4)
$\displaystyle\begin{array}[]{clccl}C_{1}&:&\omega&\longrightarrow&\omega\\\
&&x=(x_{k})&\longmapsto&C_{1}x=\left(\frac{1}{n}\sum_{k=1}^{n}x_{k}\right)_{n\in\mathbb{N}}.\end{array}$

A series $\sum_{k}x_{k}$ in a normed space $X$ is said to be Cesàro or weak
Cesàro convergent to $x_{0}\in X$ and is denoted by
$C_{1}-\sum_{k}x_{k}=x_{0}$ or $wC_{1}-\sum_{k}x_{k}=x_{0}$, if

$\displaystyle\lim_{n\to\infty}\left[\frac{1}{n}\sum_{k=1}^{n}(n-k+1)x_{k}\right]=x_{0}$

or

$\displaystyle\lim_{n\to\infty}\left[\frac{1}{n}\sum_{k=1}^{n}(n-k+1)x^{*}(x_{k})\right]=x^{*}(x_{0})$

for all $x^{*}\in X^{*}$. Riesz transformation is also given by

$\displaystyle
Rx=\left(\frac{1}{R_{n}}\sum_{k=1}^{n}r_{k}x_{k}\right)_{n\in\mathbb{N}}$

for any sequence $x=(x_{k})_{k\in\mathbb{N}}\in\omega$, where $r=(r_{k})$ is a
sequence of nonnegative reals with $r_{1}>0$ such that

(1.5) $\displaystyle R_{n}=\sum_{k=1}^{n}r_{k}~{}\textrm{ with
}~{}\lim_{n\to\infty}R_{n}=\infty.$

The Riesz method is regular under the condition (1.5) and is reduced for $r=e$
to the Cesàro mean of order one. The corresponding matrix $R=(r_{nk})$ to the
Riesz method with respect to the sequence $r=(r_{k})$ can be given, as
follows;

$\displaystyle r_{nk}:=\left\\{\begin{array}[]{ccl}\frac{r_{k}}{R_{n}}&,&1\leq
k\leq n,\\\ 0&,&k>n\end{array}\right.$

for all $k,n\in\mathbb{N}$.

In this study, our main purpose is to give some results related to the some
new multiplier by using Riesz summability method as a generalization of Cesàro
summability, and construct some new classes of sequence spaces associated to
an OVS in a Banach space having similar properties with the classes defined in
[5, 13, 15, 16] and [23]. Among other results, we give some new
characterizations of $c_{0}(X)$\- and $\ell_{\infty}(X)$-multiplier convergent
series through this method.

The following lemma states a classical result of the $wuC$ series in a normed
space $X$, and gives an example of multiplier convergent series which
characterizes unconditionally convergent series and weakly unconditionally
Cauchy series with $\ell_{\infty}$-multiplier convergent series and
$c_{0}$-multiplier convergent series, respectively; [8, p. 44].

######  Lemma 1.1.

The following statements hold:

  1. (a)

In a normed space $X$, a formal series $\sum_{n}x_{n}$ is a $wuC$ series if
and only if there exists a positive real $H$ such that

$\displaystyle
H=\sup_{n\in\mathbb{N}}\left\\{\left\|\sum_{k=1}^{n}a_{k}x_{k}\right\|:|a_{k}|\leq
1,~{}k=1,2,\ldots,n\right\\}.$

  2. (b)

A formal series $\sum_{n}x_{n}$ in a Banach space $X$ is $uc$ (respectively
$wuC$) series if and only if for any $(t_{n})\in\ell_{\infty}$ (respectively
for any $(t_{n})\in c_{0}$), $\sum_{n}t_{n}x_{n}$ converges, that is,
$\sum_{n}x_{n}$ is an $\ell_{\infty}$-(respectively a $c_{0}$-) multiplier
convergent series.

##  2\. Vector Valued Multiplier Spaces Through Riesz Summability and compact
summing operator

In this section, we give some definitions on Riesz convergent or Riesz
summable sequences in a real normed space.

######  Definition 2.1.

A sequence $x=(x_{k})$ in a real normed space $X$ is said to be Riesz
convergent ($R$-convergent) and weakly Riesz convergent ($wR$-convergent) to
$x_{0}\in X$ which is called the $R$-limit and $wR$-limit of $x$, and is
denoted by $R-\lim_{k\to\infty}x_{k}=x_{0}$ and
$wR-\lim_{k\to\infty}x_{k}=x_{0}$, if

$\displaystyle\lim_{n\to\infty}\left\|\frac{1}{R_{n}}\sum_{k=1}^{n}r_{k}x_{k}-x_{0}\right\|=0$

and

$\displaystyle\lim_{n\to\infty}\left|\frac{1}{R_{n}}\sum_{k=1}^{n}r_{k}x^{*}(x_{k})-x^{*}(x_{0})\right|=0~{}\textrm{
for all }~{}x^{*}\in X^{*}$

hold, respectively.

Riesz summability of a sequence $x=(x_{k})$ (or $R$\- and $wR$-convergence of
a series) in a real normed space $X$ is also given, as follows:

######  Definition 2.2.

A series $\sum_{k}x_{k}$ in a real normed space $X$ is said to be Riesz
convergent ($R$-convergent) and weakly Riesz convergent ($wR$-convergent) to
$x_{0}\in X$ which is called the $R$-sum and $wR$-sum of $x$, and is denoted
by $R-\sum_{k}x_{k}=x_{0}$ and $wR-\sum_{k}x_{k}=x_{0}$, if

$\displaystyle\lim_{n\to\infty}\left\|\frac{1}{R_{n}}\sum_{k=1}^{n}r_{k}s_{k}-x_{0}\right\|=0$

and

$\displaystyle\lim_{n\to\infty}\left|\frac{1}{R_{n}}\sum_{k=1}^{n}r_{k}x^{*}(s_{k})-x^{*}(x_{0})\right|=0~{}\textrm{
for all }~{}x^{*}\in X^{*}$

hold, respectively. Here, $s_{k}=\sum_{j=1}^{k}x_{j}$ is the sequence of
partial sums of the series $\sum_{k}x_{k}$.

In the following, we introduce the spaces of $R$-convergence and weakly
$R$-convergence of multipliers with association of an OVS, and obtain some new
characterizations of $c_{0}(X)$\- and $\ell_{\infty}(X)$-multiplier convergent
(Cauchy) series. We also present the necessary and sufficient conditions for
the continuity and (weak) compactness of the summing operator $\mathcal{S}$
defined from these multiplier spaces to another normed space $Y$, and derive
some results by the previous works [5, 15] and [23].

######  Definition 2.3.

Suppose that $X$ and $Y$ are two normed spaces and $T_{k}\in B(X:Y)$ for all
$k\in\mathbb{N}$. We define the space
$M^{\infty}_{R}\big{(}\sum_{k}T_{k}\big{)}$ of $R$-convergence associated to
the $\sum_{k}T_{k}$ as;

(2.1) $\displaystyle
M^{\infty}_{R}\big{(}\sum_{k}T_{k}\big{)}:=\left\\{x=(x_{k})\in\ell_{\infty}(X):\sum_{k}T_{k}x_{k}~{}\textrm{is
}R\textrm{-convergent}\right\\},$

and endowed with the sup norm.

It can be easily checked that the inclusions

(2.2) $\displaystyle\phi(X)\subseteq
M^{\infty}_{R}\big{(}\sum_{k}T_{k}\big{)}\subseteq\ell_{\infty}(X)$

hold.

Firstly, we give the following theorem related to the completeness of the
normed space $M^{\infty}_{R}\big{(}\sum_{k}T_{k}\big{)}$ through
$c_{0}(X)$-multiplier convergence.

######  Theorem 2.4.

Regarding two Banach spaces $X$ and $Y$, and a series $\sum_{k}T_{k}$, where
$T_{k}\in B(X:Y)$ for all $k\in\mathbb{N}$, the following assertions are
equivalent:

  1. (i)

The series $\sum_{k}T_{k}$ is $c_{0}(X)$-multiplier convergent.

  2. (ii)

The space $M^{\infty}_{R}\big{(}\sum_{k}T_{k}\big{)}$ is a Banach space.

###### Proof.

(i)$\Rightarrow$(ii): Let us suppose that the series $\sum_{k}T_{k}$ is
$c_{0}(X)$-multiplier convergent. Then, from Part (a) of Lemma 1.1 there
exists $H>0$ such that

$\displaystyle
H=\sup_{n\in\mathbb{N}}\left\\{\left\|\sum_{k=1}^{n}T_{k}x_{k}\right\|:\|x_{k}\|\leq
1~{}\textrm{ for all }~{}k\in\\{1,2,\ldots,n\\}\right\\}.$

Now suppose that $(x^{m})=(x_{k}^{m})_{k\in\mathbb{N}}$ is a Cauchy sequence
in $M^{\infty}_{R}\big{(}\sum_{k}T_{k}\big{)}$. Then, since $\ell_{\infty}(X)$
is a Banach space and the inclusion relation (2.2) holds, one can find
$x^{0}=(x_{k}^{0})\in\ell_{\infty}(X)$ such that $x^{m}\to x^{0}$, as
$m\to\infty$. Now, we shall prove that $x^{0}\in
M^{\infty}_{R}\big{(}\sum_{k}T_{k}\big{)}$. For an arbitrary $\epsilon>0$, we
have $m_{0}\in\mathbb{N}$ satisfying

$\displaystyle\|x^{m}-x^{0}\|<\frac{\epsilon}{3H}$

for every $m>m_{0}$. Now, it is immediate by easy calculation that

$\displaystyle\frac{3H}{\epsilon}\left\|\frac{1}{R_{n}}\sum_{k=1}^{n}r_{k}\sum_{j=1}^{k}T_{j}(x_{j}^{m}-x_{j}^{0})\right\|\leq
H$

for every $m>m_{0}$ and $k\in\mathbb{N}$. Therefore, for every $\epsilon>0$
there exists $m_{0}\in\mathbb{N}$ such that

$\displaystyle\left\|\frac{1}{R_{n}}\sum_{k=1}^{n}r_{k}\sum_{j=1}^{k}T_{j}(x_{j}^{m}-x_{j}^{0})\right\|<\frac{\epsilon}{3}$

for all $m>m_{0}$ and $k\in\mathbb{N}$. Since $(x^{m})$ is a Cauchy sequence
in $M^{\infty}_{R}\big{(}\sum_{k}T_{k}\big{)}$, we have a sequence $(y_{m})$
in the Banach space $Y$ for which the following inequality

$\displaystyle\left\|\frac{1}{R_{n}}\sum_{k=1}^{n}r_{k}\sum_{j=1}^{k}T_{j}(x_{j}^{m}-y_{m})\right\|<\frac{\epsilon}{3}$

holds for every $n>n_{0}$. Then, since $Y$ is a Banach space and $(y_{m})$ is
a Cauchy sequence we can find $y_{0}\in Y$ such that $y_{m}\to y_{0}$, as
$m\to\infty$, and so for every $\epsilon>0$,

$\displaystyle\|y_{m}-y_{0}\|<\frac{\epsilon}{3}$

holds. Finally, for every $\epsilon>0$ there exists $n_{0}\in\mathbb{N}$ such
that

$\displaystyle\left\|\frac{1}{R_{n}}\sum_{k=1}^{n}r_{k}\sum_{j=1}^{k}T_{j}x_{j}^{0}-y_{0}\right\|$
$\displaystyle\leq$
$\displaystyle\left\|\frac{1}{R_{n}}\sum_{k=1}^{n}r_{k}\sum_{j=1}^{k}T_{j}(x_{j}^{0}-x_{j}^{m})\right\|+$
$\displaystyle+$
$\displaystyle\left\|\frac{1}{R_{n}}\sum_{k=1}^{n}r_{k}\sum_{j=1}^{k}T_{j}x_{j}^{m}-y_{m}\right\|+\|y_{m}-y_{0}\|$
$\displaystyle<$ $\displaystyle\epsilon$

holds for every $n>n_{0}$. Hence, $x^{0}\in
M^{\infty}_{R}\big{(}\sum_{k}x_{k}\big{)}$.

(ii)$\Rightarrow$(i): Let us suppose that the multiplier space
$M^{\infty}_{R}\big{(}\sum_{k}T_{k}\big{)}$ is complete and let us take
$x=(x_{k})\in c_{0}(X)$. Then, we have $c_{0}(X)\subseteq
M^{\infty}_{R}(\sum_{k}T_{k})$ since the space $M^{\infty}_{R}(\sum_{k}T_{k})$
is closed and the inclusion $\phi(X)\subset M^{\infty}_{R}(\sum_{k}T_{k})$
holds. Therefore, the series $\sum_{k}T_{k}x_{k}$ is $R$-convergent for all
$x=(x_{k})\in c_{0}(X)$. From the monotonicity of $c_{0}(X)$, the series
$\sum_{k}T_{k}x_{k}$ is subseries $R$-convergent, and so is weakly subseries
$R$-convergent. From a version of Orlicz-Pettis theorem for regular matrices
[3, Theorem 4.1], $\sum_{k}T_{k}x_{k}$ is subseries norm convergent, that is,
the series $\sum_{k}T_{k}$ is $\ell_{\infty}(X)$-multiplier convergent, and so
is $c_{0}(X)$-multiplier convergent. ∎

######  Corollary 2.5.

Let $X$ and $Y$ be any given two Banach spaces and a series $\sum_{k}T_{k}$
with $T_{k}\in B(X:Y)$ for all $k\in\mathbb{N}$. Then, the following
statements are equivalent:

  1. (i)

The series $\sum_{k}T_{k}$ is $c_{0}(X)$-multiplier convergent.

  2. (ii)

The inclusion $c_{0}(X)\subseteq M_{R}^{\infty}\big{(}\sum_{k}T_{k}\big{)}$
holds.

######  Remark 2.6.

Let $X$ and $Y$ be any given two Banach spaces, and $T_{k}\in B(X:Y)$ for all
$k\in\mathbb{N}$. The multiplier spaces
$M^{\infty}\big{(}\sum_{k}T_{k}\big{)}$,
$M^{\infty}_{C}\big{(}\sum_{k}T_{k}\big{)}$ and
$M^{\infty}_{f}\big{(}\sum_{k}T_{k}\big{)}$ are introduced in [23], [5] and
[15], respectively, as follows;

(2.3) $\displaystyle M^{\infty}\big{(}\sum_{k}T_{k}\big{)}$ $\displaystyle:=$
$\displaystyle\left\\{x=(x_{k})\in\ell_{\infty}(X):\sum_{k}T_{k}x_{k}\textrm{
is convergent}\right\\},$ (2.4) $\displaystyle
M^{\infty}_{C}\big{(}\sum_{k}T_{k}\big{)}$ $\displaystyle:=$
$\displaystyle\left\\{x=(x_{k})\in\ell_{\infty}(X):\sum_{k}T_{k}x_{k}\textrm{
is Ces\\`{a}ro convergent}\right\\},$ (2.5) $\displaystyle
M^{\infty}_{f}\big{(}\sum_{k}T_{k}\big{)}$ $\displaystyle:=$
$\displaystyle\left\\{x=(x_{k})\in\ell_{\infty}(X):\sum_{k}T_{k}x_{k}\textrm{
is almost convergent}\right\\}.$

Now, by considering the definitions of the multiplier spaces
$M^{\infty}_{R}\big{(}\sum_{k}T_{k}\big{)}$,
$M^{\infty}\big{(}\sum_{k}T_{k}\big{)}$,
$M^{\infty}_{C}\big{(}\sum_{k}T_{k}\big{)}$ and
$M^{\infty}_{f}\big{(}\sum_{k}T_{k}\big{)}$ respectively given by (2.1),
(2.3), (2.4)and (2.5), we have the following inclusions:

$\displaystyle M^{\infty}\big{(}\sum_{k}T_{k}\big{)}\subseteq
M_{f}^{\infty}\big{(}\sum_{k}T_{k}\big{)}\subseteq
M_{C}^{\infty}(\sum_{k}T_{k})\subseteq M_{R}^{\infty}(\sum_{k}T_{k}).$

By bearing in mind Theorem 2.4 together with previous results given in [5] and
[15], we have the following:

######  Corollary 2.7.

Let $X$ and $Y$ be Banach spaces, and $T_{k}\in B(X:Y)$ for all
$k\in\mathbb{N}$. Then, the following assertions are equivalent:

  1. (i)

The series $\sum_{k}T_{k}$ is $c_{0}(X)$-multiplier convergent.

  2. (ii)

$M^{\infty}_{f}\big{(}\sum_{k}T_{k}\big{)}$ is a Banach space.

  3. (iii)

$M^{\infty}_{C}\big{(}\sum_{k}T_{k}\big{)}$ is a Banach space.

  4. (iv)

$M^{\infty}_{R}\big{(}\sum_{k}T_{k}\big{)}$ is a Banach space.

######  Remark 2.8.

Let $X$ and $Y$ be normed spaces and $T_{k}\in B(X:Y)$ for all
$k\in\mathbb{N}$. Consider the vector valued multiplier Cauchy space
$CM^{\infty}\big{(}\sum_{k}T_{k}\big{)}$ and the vector valued multiplier
Riesz-Cauchy space $CM_{R}^{\infty}\big{(}\sum_{k}T_{k}\big{)}$ which are
associated to an OVS, and defined by

(2.6) $\displaystyle
CM^{\infty}\big{(}\sum_{k}T_{k}\big{)}:=\left\\{x=(x_{k})\in\ell_{\infty}(X):\sum_{k}T_{k}x_{k}\textrm{
is a Cauchy series}\right\\}$

and

(2.7) $\displaystyle
CM_{R}^{\infty}\big{(}\sum_{k}T_{k}\big{)}:=\left\\{x=(x_{k})\in\ell_{\infty}(X):\sum_{k}T_{k}x_{k}\textrm{
is a Riesz-Cauchy series}\right\\}.$

By using the spaces given in (2.6) and (2.7), we have the following corollary
as an analogue of corresponding result given in [1, Remark 2.4].

######  Corollary 2.9.

Let $X$ and $Y$ be normed spaces and $T_{k}\in B(X:Y)$ for all
$k\in\mathbb{N}$. The following statements are equivalent:

  1. (i)

$\sum_{k}T_{k}$ is $c_{0}(X)$-multiplier convergent.

  2. (ii)

$CM^{\infty}\big{(}\sum_{k}T_{k}\big{)}$ is a Banach space.

  3. (iii)

$CM_{R}^{\infty}\big{(}\sum_{k}T_{k}\big{)}$ is a Banach space.

The following theorem is an analogue of Theorem 2.5 given in [5], and
characterizes the completeness of a normed space by the vector valued
multiplier space $M^{\infty}_{R}\big{(}\sum_{k}T_{k}\big{)}$. Since the proof
is similar to the case $M^{\infty}_{C}\big{(}\sum_{k}T_{k}\big{)}$ given in
[5], we omit details; (see also [23, Corollary 1.8]).

######  Theorem 2.10.

Let $X$ and $Y$ be given any normed spaces such that $X$ is complete and
$T_{k}\in B(X:Y)$ for all $k\in\mathbb{N}$. Then, the space $Y$ is also
complete if and only if $M^{\infty}_{R}\big{(}\sum_{k}T_{k}\big{)}$ is
complete for every $c_{0}(X)$-multiplier Cauchy series.

In the following theorem, we give a new characterization of continuity of the
certain summing operator by using Riesz summability method and
$c_{0}(X)$-multiplier Cauchy series.

######  Theorem 2.11.

Let $X$ and $Y$ be any normed spaces and $T_{k}\in B(X:Y)$ for all
$k\in\mathbb{N}$. Then, the summing operator $\mathcal{S}$ defined by

(2.10)
$\displaystyle\begin{array}[]{ccccl}\mathcal{S}&:&M^{\infty}_{R}\big{(}\sum_{k}T_{k}\big{)}&\longrightarrow&Y\\\
&&x=(x_{k})&\longmapsto&\mathcal{S}x=R-\sum_{k}T_{k}x_{k}\end{array}$

is continuous if and only if the series $\sum_{k}T_{k}$ is
$c_{0}(X)$-multiplier Cauchy.

###### Proof.

Let us suppose that $\mathcal{S}$ is continuous and define the set $G$ by

(2.11) $\displaystyle
G:=\left\\{\left\|\sum_{k=1}^{n}T_{k}x_{k}\right\|:\|x_{k}\|\leq 1~{}\textrm{
for all }~{}k\in\\{1,2,\ldots,n\\}\right\\}.$

Since the inclusion $\phi(X)\subset M^{\infty}_{R}(\sum_{k}T_{k})$ holds, the
series $\sum_{k}T_{k}$ is $c_{0}(X)$-multiplier Cauchy from the inequality
$H=\sup_{n\in\mathbb{N}}G\leq\|\mathcal{S}\|$.

Conversely, let us suppose that $\sum_{k}T_{k}$ is $c_{0}(X)$-multiplier
Cauchy series. Therefore, the set $G$ defined by (2.11) is bounded (see [23,
Theorem 1.3]) and so, $H=\sup_{n\in\mathbb{N}}G$. If $x=(x_{k})\in
M^{\infty}_{R}\big{(}\sum_{k}T_{k}\big{)}$, then the proof follows from the
inequality

$\displaystyle\|\mathcal{S}x\|=\left\|R-\sum_{k}T_{k}x_{k}\right\|\leq
H\|x\|.$

∎

Combining Theorem 2.11 with the results due to Altay and Kama [5] and Karakuş
and Başar [15]; we have the following:

######  Corollary 2.12.

Let $X$, $Y$ be any normed spaces and $T_{k}\in B(X:Y)$ for all
$k\in\mathbb{N}$. Then, the following statements are equivalent:

  1. (i)

The series $\sum_{k}T_{k}$ is $c_{0}(X)$-multiplier Cauchy.

  2. (ii)

$\mathcal{S}:M_{f}^{\infty}\big{(}\sum_{k}T_{k}\big{)}\to Y$ is continuous.

  3. (iii)

$\mathcal{S}:M^{\infty}_{C}\big{(}\sum_{k}T_{k}\big{)}\to Y$ is continuous.

  4. (iv)

$\mathcal{S}:M^{\infty}_{R}\big{(}\sum_{k}T_{k}\big{)}\to Y$ is continuous.

Now, we give the characterization of the (weakly) compactness for the summing
operator $\mathcal{S}$ by using Riesz summability method and
$\ell_{\infty}(X)$-multiplier convergent series. Let us note that $X$ need not
to be complete.

######  Theorem 2.13.

Let $X$ be any normed space, $Y$ be a Banach space and $T_{k}\in B(X:Y)$ for
all $k\in\mathbb{N}$. Then the series $\sum_{k}T_{k}$ is
$\ell_{\infty}(X)$-multiplier convergent if and only if the summing operator
$\mathcal{S}$ defined by (2.10) is compact (weakly compact).

###### Proof.

Let us suppose that $\mathcal{S}$ is compact. If
$x=(x_{k})\in\ell_{\infty}(X)$, then the set

$\displaystyle H:=\left\\{\sum_{k\in\mathcal{F}}e^{k}\otimes
x_{k}|\mathcal{F}\textrm{ finite and }\|x_{k}\|\leq 1\right\\}\subset
M^{\infty}_{R}\big{(}\sum_{k}T_{k}\big{)}$

is bounded. By the hypothesis,

$\displaystyle\mathcal{S}(H):=\left\\{R-\sum_{k\in\mathcal{F}}T_{k}x_{k}|\mathcal{F}\textrm{
finite and }\|x_{k}\|\leq 1\right\\}$

is relatively compact. Therefore, the series $\sum_{k}T_{k}x_{k}$ is subseries
norm $R$-convergent, and so is weakly subseries $R$-convergent [22, Theorem
2.48]. Further, by a consequence of Orlicz-Pettis theorem for regular matrices
[3], the series $\sum_{k}T_{k}x_{k}$ is subseries norm convergent; that is the
series $\sum_{k}T_{k}$ is $\ell_{\infty}(X)$-multiplier convergent.

Conversely, suppose that the series $\sum_{k}T_{k}$ is
$\ell_{\infty}(X)$-multiplier convergent. We define the operators
$\mathcal{S}^{R}_{n}$ by

$\displaystyle\begin{array}[]{ccccl}\mathcal{S}^{R}_{n}&:&M^{\infty}_{R}\big{(}\sum_{k}T_{k}\big{)}&\longrightarrow&Y\\\
&&x=(x_{k})&\longmapsto&\mathcal{S}^{R}_{n}(x)=R-\sum_{k=1}^{n}T_{k}x_{k}\end{array}$

for all $n\in\mathbb{N}$. It is sufficient to prove that
$\|\mathcal{S}^{R}_{n}-\mathcal{S}\|\to 0$, as $n\to\infty$. Since
$\sum_{k}T_{k}$ is $\ell_{\infty}(X)$-multiplier convergent, then the series
$\sum_{k}T_{k}x_{k}$ is uniformly $R$-convergent for $\|x_{k}\|\leq 1$ [22,
Corollary 11.11]. Therefore,

$\displaystyle\lim_{n\to\infty}\left\|\mathcal{S}^{R}_{n}-\mathcal{S}\right\|$
$\displaystyle=$
$\displaystyle\lim_{n\to\infty}\left\|\left(R-\sum_{k=1}^{n}T_{k}x_{k}\right)-\left(R-\sum_{k}T_{k}x_{k}\right)\right\|$
$\displaystyle=$
$\displaystyle\lim_{n\to\infty}\left\|R-\sum_{k=n+1}^{\infty}T_{k}x_{k}\right\|=0$

with $\|x_{k}\|\leq 1$. This step completes the proof. ∎

Combining the related results in [5] and [15] with Theorem 2.13, we have the
following:

######  Corollary 2.14.

Let $X$ be any normed space, $Y$ be a Banach space and $T_{k}\in B(X:Y)$ for
all $k\in\mathbb{N}$. Then, the following statements are equivalent:

  1. (i)

The series $\sum_{k}T_{k}$ is $\ell_{\infty}(X)$-multiplier convergent.

  2. (ii)

$\mathcal{S}:M_{f}^{\infty}(\sum_{k}T_{k})\to Y$ is compact (weakly compact).

  3. (iii)

$\mathcal{S}:M_{C}^{\infty}(\sum_{k}T_{k})\to Y$ is compact (weakly compact).

  4. (iv)

$\mathcal{S}:M^{\infty}_{R}\big{(}\sum_{k}T_{k}\big{)}\to Y$ is compact
(weakly compact).

Now, we may introduce the multiplier space of weak $R$-convergence associated
to the series $\sum_{k}T_{k}$ and obtain the corresponding results similar to
the previous theorems and corollaries.

######  Definition 2.15.

Let $X$ and $Y$ be normed spaces, and $T_{k}\in B(X:Y)$ for all
$k\in\mathbb{N}$. The vector valued multiplier space
$M^{\infty}_{wR}\big{(}\sum_{k}T_{k}\big{)}$ of weakly $R$-convergence
associated to the series $\sum_{k}T_{k}$ is defined by

$\displaystyle
M^{\infty}_{wR}\big{(}\sum_{k}T_{k}\big{)}:=\left\\{x=(x_{k})\in\ell_{\infty}(X):\sum_{k}T_{k}x_{k}\textrm{
is $wR$-convergent}\right\\}$

and endowed with the sup norm.

Since the inclusion $M^{\infty}_{R}\big{(}\sum_{k}T_{k}\big{)}\subseteq
M^{\infty}_{wR}(\sum_{k}T_{k})$ clearly holds, we have the following
inclusions which are similar to the relation in (2.2):

(2.13) $\displaystyle\phi(X)\subseteq
M^{\infty}_{R}\big{(}\sum_{k}T_{k}\big{)}\subseteq
M^{\infty}_{wR}\big{(}\sum_{k}T_{k}\big{)}\subseteq\ell_{\infty}(X).$

Since the completeness of the multiplier space
$M_{wR}^{\infty}\big{(}\sum_{k}x_{k}\big{)}$ is given by means of
$c_{0}(X)$-multiplier convergent series by using the similar technique for
proving the completeness of $M_{R}^{\infty}\big{(}\sum_{k}x_{k}\big{)}$, we
omit the proof of following theorem.

######  Theorem 2.16.

Let $X$ and $Y$ be any given Banach spaces, and $T_{k}\in B(X:Y)$ for all
$k\in\mathbb{N}$. Then, the series $\sum_{k}T_{k}$ is $c_{0}(X)$-multiplier
convergent if and only if $M^{\infty}_{wR}\big{(}\sum_{k}T_{k}\big{)}$ is a
Banach space.

######  Corollary 2.17.

Let $X$ and $Y$ be Banach spaces, and $T_{k}\in B(X:Y)$ for all
$k\in\mathbb{N}$. Then, the series $\sum_{k}T_{k}$ is $c_{0}(X)$-multiplier
convergent if and only if the inclusion $c_{0}(X)\subseteq
M_{wR}^{\infty}\big{(}\sum_{k}T_{k}\big{)}$ holds.

######  Remark 2.18.

Let us suppose that $X$ and $Y$ are Banach spaces and $T_{k}\in B(X:Y)$ for
all $k\in\mathbb{N}$. Then, the multiplier space
$M_{w}^{\infty}\big{(}\sum_{k}T_{k}\big{)}$ is introduced in [23] as

$\displaystyle
M_{w}^{\infty}\big{(}\sum_{k}T_{k}\big{)}:=\left\\{x=(x_{k})\in\ell_{\infty}(X):\sum_{k}T_{k}x_{k}\textrm{
is weakly convergent}\right\\}.$

Now, if the series $\sum_{k}T_{k}$ is a $c_{0}(X)$-multiplier convergent, then
the series $\sum_{k}y^{*}(T_{k}x_{k})$ is convergent for all $x=(x_{k})\in
c_{0}(X)$ and for all $y^{*}\in Y^{*}$, that is, the series is weakly
convergent. It is known by Corollary 2.5 that $x=(x_{k})\in
M_{R}^{\infty}\big{(}\sum_{k}T_{k}\big{)}$, and so $x=(x_{k})\in
M_{wR}^{\infty}\big{(}\sum_{k}T_{k}\big{)}$. This means that there exists
$y_{0}\in Y$ with $wR-\sum_{k}T_{k}x_{k}=y_{0}$ such that

$\displaystyle\sum_{k}y^{*}(T_{k}x_{k})=R-\sum_{k}y^{*}(T_{k}x_{k})=y^{*}(y_{0}).$

Therefore, the inclusion $M_{R}^{\infty}\big{(}\sum_{k}T_{k}\big{)}\subseteq
M_{w}^{\infty}(\sum_{k}T_{k})$ holds. However, we have no an idea on the
sufficient conditions for the reverse inclusion.

By combining the previous results and Theorem 2.16, we derive the following
for the analogue of Corollary 2.7 in the weak topology:

######  Corollary 2.19.

Let $X$ and $Y$ be Banach spaces, and $T_{k}\in B(X:Y)$ for all
$k\in\mathbb{N}$. Then, the following assertions are equivalent:

  1. (i)

The series $\sum_{k}T_{k}$ is $c_{0}(X)$-multiplier convergent.

  2. (ii)

$M^{\infty}_{wf}\big{(}\sum_{k}T_{k}\big{)}$ is a Banach space.

  3. (iii)

$M^{\infty}_{wC}\big{(}\sum_{k}T_{k}\big{)}$ is a Banach space.

  4. (vi)

$M^{\infty}_{wR}\big{(}\sum_{k}T_{k}\big{)}$ is a Banach space.

Following theorem is the analogue of Theorem 3.3 of [5]. Since the proof is
similar to the case $M^{\infty}_{wC}\big{(}\sum_{k}T_{k}\big{)}$, we omit
details.

######  Theorem 2.20.

Let $X$ be a Banach space, $Y$ be any normed space and $T_{k}\in B(X:Y)$ for
all $k\in\mathbb{N}$. Then, $Y$ is complete if and only if the multiplier
space $M^{\infty}_{wR}\big{(}\sum_{k}T_{k}\big{)}$ is complete for every
$c_{0}(X)$-multiplier Cauchy series.

It is well-known that if $Y$ is a Banach space, then $B(X:Y)$ is a Banach
space. So, Theorem 2.20 and also Theorem 2.10 can be used for proving
completeness of $B(X:Y)$.

Now, we give the following theorem which characterizes the continuity of weak
summing operator with $c_{0}(X)$-multiplier Cauchy series.

######  Theorem 2.21.

Let $X$ and $Y$ be normed spaces, and $T_{k}\in B(X:Y)$ for all
$k\in\mathbb{N}$. Then, the summing operator $\mathcal{S}$ defined by

(2.16)
$\displaystyle\begin{array}[]{ccccl}\mathcal{S}&:&M^{\infty}_{wR}\big{(}\sum_{k}T_{k}\big{)}&\longrightarrow&Y\\\
&&x=(x_{k})&\longmapsto&\mathcal{S}x=wR-\sum_{k}T_{k}x_{k}.\end{array}$

is continuous if and only if the series $\sum_{k}T_{k}$ is
$c_{0}(X)$-multiplier Cauchy.

###### Proof.

Let us suppose that the summing operator $\mathcal{S}$ defined by (2.16) is
continuous and consider the set $G$ given by (2.11). Then, the desired result
follows from the inequality

$\displaystyle\sup_{n\in\mathbb{N}}G=\left|wR-\sum_{k}T_{k}x_{k}\right|\leq\|\mathcal{S}\|,$

since the inclusion $\phi\subset M_{wR}^{\infty}\big{(}\sum_{k}T_{k}\big{)}$
holds.

Conversely, if $\sum_{k}T_{k}$ is $c_{0}(X)$-multiplier Cauchy series, then
the set $G$ is bounded (see [23, Theorem 1.3]) and so
$H=\sup_{n\in\mathbb{N}}G$. If $x=(x_{k})\in
M^{\infty}_{wR}\big{(}\sum_{k}T_{k}\big{)}$, then the proof follows from the
inequality

$\displaystyle\|\mathcal{S}x\|=\left|R-\sum_{k}y^{*}\left(T_{k}x_{k}\right)\right|\leq
H\|x\|$

for all $y^{*}\in B_{Y}$. ∎

From Theorem 2.21 and the conclusions due to Altay and Kama [5] and Karakuş
and Başar [15], we have the following:

######  Corollary 2.22.

Let $X$ and $Y$ be two normed spaces, and $T_{k}\in B(X:Y)$ for all
$k\in\mathbb{N}$. Then, the following statements are equivalent:

  1. (i)

The series $\sum_{k}T_{k}$ is $c_{0}(X)$-multiplier Cauchy.

  2. (ii)

$\mathcal{S}:M_{wf}^{\infty}\big{(}\sum_{k}T_{k}\big{)}\to Y$ is continuous.

  3. (iii)

$\mathcal{S}:M_{wC}^{\infty}\big{(}\sum_{k}T_{k}\big{)}\to Y$ is continuous.

  4. (iv)

$\mathcal{S}:M^{\infty}_{wR}\big{(}\sum_{k}T_{k}\big{)}\to Y$ is continuous.

######  Theorem 2.23.

Let $X$ be any normed space, $Y$ be a Banach space and $T_{k}\in B(X:Y)$ for
all $k\in\mathbb{N}$. Then, the series $\sum_{k}T_{k}$ is
$\ell_{\infty}(X)$-multiplier convergent if and only if the summing operator
$\mathcal{S}$ defined by (2.16) is compact (weakly compact).

###### Proof.

Since the proof can be given by the similar way used in proving Theorem 2.13,
we omit details. ∎

By Theorem 2.23 and the results in [5] and [15], we have the following:

######  Corollary 2.24.

Let us suppose that $X$ and $Y$ are any normed spaces such that $Y$ is
complete, and $T_{k}\in B(X:Y)$ for all $k\in\mathbb{N}$. Then the following
statements are equivalent:

  1. (i)

The series $\sum_{k}T_{k}$ is $\ell_{\infty}(X)$-multiplier convergent.

  2. (ii)

$\mathcal{S}:M_{wf}^{\infty}(\sum_{k}T_{k})\to Y$ is compact (weakly compact).

  3. (iii)

$\mathcal{S}:M_{wC}^{\infty}(\sum_{k}T_{k})\to Y$ is compact (weakly compact).

  4. (iv)

$\mathcal{S}:M^{\infty}_{wR}\big{(}\sum_{k}T_{k}\big{)}\to Y$ is compact
(weakly compact).

Prior to passing to the next section, we present Proposition 2.25 which
states, besides the inclusion given in (2.13), the iclusion
$M^{\infty}_{wR}\big{(}\sum_{k}T_{k}\big{)}\subseteq
M^{\infty}_{R}\big{(}\sum_{k}T_{k}\big{)}$ also holds.

######  Proposition 2.25.

Let $X$ and $Y$ be normed spaces. If $\sum_{k}T_{k}$ is
$\ell_{\infty}(X)$-multiplier Cauchy, then
$M^{\infty}_{wR}\big{(}\sum_{k}T_{k}\big{)}=M^{\infty}_{R}\big{(}\sum_{k}T_{k}\big{)}$.

###### Proof.

Let $X$ and $Y$ be normed spaces, and $x=(x_{k})\in
M^{\infty}_{wR}\big{(}\sum_{k}T_{k}\big{)}$. Then, there exists $y\in Y$ such
that $R-\sum_{k}y^{*}(T_{k}x_{k})=y^{*}(y)$ for every $y^{*}\in Y^{*}$. From
hypothesis, since the partial sums of the series $\sum_{k}T_{k}x_{k}$ form a
Cauchy sequence in $Y$, there exists $y^{**}\in Y^{**}$ such that
$R-\sum_{k}T_{k}x_{k}=y^{**}.$ If we consider the uniqueness of the limit,
then we have $y^{**}=y$. Therefore, $x=(x_{k})\in
M^{\infty}_{R}\big{(}\sum_{k}T_{k}\big{)}$. ∎

##  3\. A version of Orlicz-Pettis Theorem for Riesz summability

The Orlicz-Pettis Theorem is one of the important results in the Theory of
Functional Analysis. Many generalizations and applications of this theorem can
be found in [5, 17, 18, 22, 24, 25, 27]. Before stating and proving the
Orlicz-Pettis Theorem by means of Riesz convergence, we will give the
following definition (see [22]):

######  Definition 3.1.

The space $\lambda$ has the infinite gliding hump property ($\infty$-GHP) if
whenever $x\in\lambda$ and $\\{\sigma_{m}\\}$ is an increasing sequence of
intervals, there exist a subsequence $\\{p_{m}\\}$ and $t_{p_{m}}>0$,
$t_{p_{m}}\to\infty$ such that every subsequence of $\\{p_{m}\\}$ has a
further subsequence $\\{q_{m}\\}$ such that the coordinatewise sum of the
series $\sum_{m}t_{q_{m}}\chi_{\sigma_{q_{m}}}x\in\lambda$.

In the following, we give a new version of Orlicz-Pettis theorem for vector
valued multiplier Riesz convergent space of OVS by using Antosik-Mikusinski
matrix theorem.

######  Theorem 3.2.

Let $\lambda$ have $\infty$-GHP and $T_{k}\in B(X:Y)$ for all
$k\in\mathbb{N}$. If the series $\sum_{k}T_{k}$ is $\lambda-$multiplier Riesz
convergent with respect to weak topology of $Y$, then the series
$\sum_{k}T_{k}$ is $\lambda-$multiplier Riesz convergent with respect to
strong topology of $Y$.

###### Proof.

Let $\epsilon>0$. If the conclusion is false, there exists $x\in\lambda$,
$(y_{n}^{*})\subset Y^{*}$ bounded sequence and an increasing sequence of
intervals $\\{\sigma_{n}\\}$ such that

(3.1)
$\displaystyle\left|R-\sum_{k\in\sigma_{n}}y_{n}^{*}(T_{k}x_{k})\right|>\epsilon$

for all $n\in\mathbb{N}$. Since $\lambda$ has $\infty$-GHP, there exist a
subsequence $\\{p_{n}\\}$ and $t_{p_{n}}>0$, $t_{p_{n}}\to\infty$ such that
every subsequence of $\\{p_{n}\\}$ has a further subsequence $\\{q_{n}\\}$
such that $\sum_{n}t_{q_{n}}\chi_{\sigma_{q_{n}}}x\in\lambda$. Now, we
consider the matrix $H=[h_{ij}]$ defined by

$\displaystyle
h_{ij}=\sum_{m\in\sigma_{p_{j}}}\frac{y^{*}_{p_{i}}}{t_{p_{i}}}(T_{m}(t_{p_{j}}x_{m}))$

for all $i,j\in\mathbb{N}$. Since $(y_{i}^{*})$ is bounded and
$t_{p_{i}}\to\infty$, the columns of $H$ converge to 0. On the other hand,
since $\lambda$ has $\infty$-GHP, we have
$\sum_{n}t_{q_{j}}\chi_{\sigma_{q_{j}}}x\in\lambda$ for every subsequence
$\\{q_{j}\\}$ and hence the subseries

$\displaystyle\sum_{j}\sum_{m\in\sigma_{q_{j}}}T_{m}(t_{q_{j}}x_{m})$

is weakly Riesz convergent. Then, we have

$\displaystyle\lim_{i\to\infty}\sum_{j}a_{iq_{j}}=\lim_{i\to\infty}\sum_{j}\sum_{m\in\sigma_{q_{j}}}\frac{y^{*}_{p_{i}}}{t_{p_{i}}}(T_{m}(t_{q_{j}}x_{m})).$

From Antosik-Mikusinski theorem ([22] Appendix D), the matrix $H$ is Riesz
convergent to zero as diagonal, which is contradiction with (3.1). ∎

##  4\. Conclusion

Pérez-Fernández et al. [21] gave some properties of a normed space $X$ like
completeness and barrelledness through the $wuC$ series in $X$ and weakly*
unconditionally Cauchy series in $X^{*}$. Aizpuru et al. [1] gave a new
characterization of $wuC$ and $uc$ series through Cesàro summability method
and studied the new spaces associated to the series in a Banach space for
proving completeness and barrelledness of normed spaces, and obtained a new
version of Orlicz-Pettis theorem in scalar case. They also characterized the
$wuC$ series by means of continuity of a linear mapping from these new spaces
to a normed space. Aizpuru et al. [2] gave a new characterization of $wuC$ and
$uc$ series via the completeness of some new subspaces of $\ell_{\infty}$
obtained from almost convergence and also presented a new version of Orlicz-
Pettis theorem. In [3], they also generalized the results given in [1] to any
regular matrix. In recent times, Karakuş and Başar [14] introduced a slight
generalization of almost convergence and gave some new multiplier spaces
associated to the series $\sum_{k}x_{k}$ in a normed space by means of this
new summability method. Swartz constructed some new versions of the Orlicz-
Pettis theorem for multiplier convergent series under continuity assumptions
on some linear operators, and gave applications to spaces of continuous linear
operators, [24, 25]. In [27], Yuanghong and Ronglu proved that multiplier
convergence of OVS is closely related to the $AK$ property of the sequence
spaces, and obtained corresponding versions of Orlicz-Pettis theorems. Kama
and Altay [10] obtained some new multiplier space by using Fibonacci sequence
spaces, and Kama et al. [11] gave similar results for multiplier spaces which
were obtained from backward difference matrix. Most recently, León-Saavedra et
al. [17, 18] also obtained some new versions of Orlicz-Pettis theorem by using
$w_{p}$-summability and a general summability methods.

Swartz [23] extended these studies (which are concerned with the scalar valued
multiplier spaces) to the case of operator valued series and vector valued
multipliers. Later, Altay and Kama [5] introduced the vector valued multiplier
spaces by means of Cesàro convergence and a sequence of continuous linear
operators. Karakuş gave the vector valued multiplier spaces
$S_{\Lambda}(\mathcal{T})$ and $S_{w\Lambda}(\mathcal{T})$ by means of
$\Lambda$-convergence and a series of bounded linear operators together with
the characterization the completeness of both of the normed spaces
$S_{\Lambda}(\mathcal{T})$ and $S_{w\Lambda}(\mathcal{T})$ via
$c_{0}(X)$-multiplier convergence of an operator series, [13]. Quite recently,
Karakuş and Başar [15, 16] and Kama [9] introduced the vector valued
multiplier spaces of almost summability, a generalized almost summability and
statistical Cesàro summability, respectively.

It is worth noting to the reader that one can obtain corresponding results of
the present paper by using any regular matrix
$A=(\alpha_{nk})_{n,k\in\mathbb{N}}$ instead of the Riesz matrix.

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# Frequency Estimation in the Shuffle Model
with (Almost) a Single Message

Qiyao Luo, Yilei Wang, Ke Yi
HKUST
(qluoak, ywanggq<EMAIL_ADDRESS>

###### Abstract

We present a protocol in the shuffle model of differential privacy for the
frequency estimation problem, in which each user only sends $1+o(1)$ messages
in expectation, while achieving a polylogarithmic error.

##  1 Introduction

###  1.1 Frequency Estimation

#### Problem definition.

In frequency estimation problem, each of the $n$ users holds an element from a
domain $[B]\coloneqq\\{1,\dots,B\\}$ and the goal is to estimate the number of
users holding element $j$, for any query element $j\in[B]$.

####  $(\alpha,\beta)$-accuracy.

Suppose we have a randomized protocol $P:[B]^{n}\rightarrow\mathbb{N}^{B}$ for
frequency estimation. The protocol is $(\alpha,\beta)$-accurate if for any
dataset $\bm{D}=(x_{1},\cdots,x_{n})\in[B]^{n}$, where user $i$ holds $x_{i}$,
we have

$\Pr\left[\max_{j\in[B]}\left|P(\bm{D})_{j}-\sum_{i=1}^{n}\mathbbm{1}[x_{i}=j]\right|>\alpha\right]<\beta,$

where $P(\bm{D})_{j}$ denotes the estimated frequency of element $j$. In other
words, the maximum estimation error exceeds $\alpha$ with probability at most
$\beta$.

###  1.2 Shuffle Differential Privacy

#### Differential privacy (DP).

A randomized mechanism $\mathcal{M}:\mathcal{X}^{n}\rightarrow\mathcal{Y}$ is
$(\varepsilon,\delta)$-differentially private if for any two neighbouring
datasets $\bm{D}\sim\bm{D}^{\prime}$ (i.e., $\bm{D}$ differs from
$\bm{D}^{\prime}$ by one element), and any set of outputs
$Y\subseteq\mathcal{Y}$,

$\Pr[\mathcal{M}(\bm{D})\in Y]\leq
e^{\varepsilon}\cdot\Pr[\mathcal{M}(\bm{D}^{\prime})\in Y]+\delta$

for some privacy parameters $\varepsilon>0$ and $0\leq\delta<n^{-\Omega(1)}$.
Note that an equivalent definition using the privacy loss random variable is
the following:

$\Pr_{y\sim{\mathcal{M}}(\bm{D})}\left[\frac{\Pr[{\mathcal{M}}(\bm{D})=y]}{\Pr[{\mathcal{M}}(\bm{D^{\prime}})=y]}\geq
e^{\varepsilon}\right]\leq\delta.$

#### The shuffle model.

In the standard definition of DP (a.k.a. the central model), the mechanism
relies on a trusted curator who has direct access to the private dataset. The
local model of differential privacy considers the case where each user $i$
locally privatizes their own data $x_{i}$ using a local randomizer
$\mathcal{R}$, and sends $\mathcal{R}(x_{i})$ to an untrusted analyzer
$\mathcal{A}$ for analysis.

In the shuffle model, the data analyzer receives anonymous messages from
users, which can be realized by a trusted shuffler $\mathcal{S}$ that randomly
permutes the messages before giving them to the analyzer. Equivalently, the DP
definitions of the local model and shuffle model are the same as above, except
that (1) in the central model, $\mathcal{M}(\bm{D})$ is defined as the final
output; (2) in the local model,
$\mathcal{M}(\bm{D})=(\mathcal{R}(x_{1}),\dots,\mathcal{R}(x_{n}))$ is the
vector of the messages; (3) in the shuffle model,
$\mathcal{M}(\bm{D})=\\{\mathcal{R}(x_{1}),\cdots,\mathcal{R}(x_{n})\\}$ is
the multiset of the messages. Several recent papers have shown that, for a
number of fundamental problems, there is a polynomial lower bound on the error
achievable if each user is only allowed to send a single message [1, 3, 4].
Thus, the multi-message shuffle model has gained interests, where each
$\mathcal{R}(x_{i})$ is a multiset, and
$\mathcal{M}(\bm{D}):=\uplus_{i}\mathcal{R}(x_{i})$.

The first problem studied in the multi-message shuffle model is private
aggregation, i.e., to estimate $\sum_{i}x_{i}$. The optimal error in the
central model of DP for this problem is $O(B)$ (for constant $\varepsilon)$.
In the single-message shuffle model, there is a lower bound of $\Omega(B\cdot
n^{1/6})$ [1]. [6] first shows that $O(B)$ error can be achieved with $O(\log
n)$ messages (assuming $\log n\approx\log B\approx\log(1/\delta)$), which is
later improved to $O(1)$ (but at least $3$) messages [2]. Interestingly, a
recent result [5] further reduces the number of messages to merely $1+o(1)$.

For the frequency estimation problem, there is a single-message lower bound of
$\Omega(n^{1/4})$ [4], in which the authors also show that, with $O(\log n)$
messages, the error can be brought down to polylogarithmic. In this paper, we
show that $1+o(1)$ messages are also enough for the frequency estimation
problem while achieving a polylogarithmic error.

When stating the results, we sometimes use $\tilde{O}(\cdot)$ and
$\tilde{\Omega}(\cdot)$ to suppress $\mathrm{poly}(\varepsilon,\log B,\log
n,\log(1/\delta),\log(1/\beta))$ factors.

##  2 The Frequency Estimation Protocol

###  2.1 Balls-into-bins mechanism

The balls-into-bins mechanism $\mathcal{M}^{\mathrm{BIB}}$ is shown in
Algorithm 1. The parameters of the mechanism is explained as follows. We
consider a set of bins $[m]$, where $s$ of them will be marked as special. The
set of special bins $S$ is the input of $\mathcal{M}^{\mathrm{BIB}}$. We first
throw a ball into one of the bins in $S$ uniformly, and then throw $k$ balls
into one of all the bins uniformly. Finally we flip $n$ coins with probability
$p$ and throw a ball into one of all the bins uniformly if we get a heads.
Note that the output is a multiset which contains $1+k+np$ values in
expectation. We call the ball thrown into $S$ the real ball, while other
(expected) $k+np$ balls noisy balls, which are used to hide the information of
$S$.

Public Parameters: $m,s,k,n,p$

Input: A set $S\subseteq[m]$, where $|S|=s$

Output: A multiset $\mathcal{O}\subseteq[m]$

1 Choose $x\in S$ uniformly and independently at random

2 $\mathcal{O}\leftarrow\\{x\\}$

3 for  _$i\in 1,2,\cdots,k$_ do

4 Choose $x\in[m]$ uniformly and independently at random

5 $\mathcal{O}\leftarrow\mathcal{O}\uplus\\{x\\}$

6 end for

7for  _$i\in 1,2,\cdots,n$_ do

8 $y\leftarrow\mathrm{Ber}(p)$

9 if  _$y=1$_ then

10 Choose $x\in[m]$ uniformly and independently at random

11 $\mathcal{O}\leftarrow\mathcal{O}\uplus\\{x\\}$

12 end if

13

14 end for

return _$\mathcal{O}$_

Algorithm 1 Balls-into-bins Mechanism $\mathcal{M}^{\mathrm{BIB}}(m,s,k,n,p)$

######  Theorem 2.1.

For any $0<\varepsilon\leq 3$ and $0<\delta<1$, the mechanism
$\mathcal{M}^{\mathrm{BIB}}$ is $(\varepsilon,\delta)$-differentially private
if $k+np\geq\frac{32\ln(2/\delta)}{\varepsilon^{2}}\cdot\frac{m}{s}$. Here,
two inputs $S$ and $S^{\prime}$ are considered neighbors if
$|S|=|S^{\prime}|=s$ and $S^{\prime}\neq S$.

###### Proof.

Let $\mathcal{M}(S)$ be the output vector of
$\mathcal{M}^{\mathrm{BIB}}(m,s,k,n,p)$ with input $S$, where the $i$-th
element of it corresponds to the number of balls in the $i$-th bin. We prove
the result by the following inequality:

$\Pr_{\bm{W}\sim{\mathcal{M}}(S)}\left[\frac{\Pr[{\mathcal{M}}(S)=\bm{W}]}{\Pr[{\mathcal{M}}(S^{\prime})=\bm{W}]}\geq
e^{\varepsilon}\right]\leq\delta.$ (1)

We first consider mechanism $\mathcal{M}_{0}^{\mathrm{BIB}}$ that only throws
noisy balls (remove the first two lines in Algorithm 1). Let $\mathcal{M}_{0}$
be the output vector of $\mathcal{M}_{0}^{\mathrm{BIB}}$, where the $i$-th
element of it corresponds to the number of noisy balls in the $i$-th bin. For
any $\bm{W}=(w_{i})_{i=1}^{m}\in\mathbb{N}^{m}$, denote
$w=\sum_{i=1}^{m}w_{i}$. Then when $k\leq w\leq k+n$, we have

$\Pr[\mathcal{M}_{0}=\bm{W}]=\binom{n}{w-k}p^{w-k}(1-p)^{n-w+k}\frac{w!}{\prod_{i=1}^{m}w_{i}!}.$

Now consider $\mathcal{M}^{\mathrm{BIB}}$. Let $I$ be the bin chosen by the
real ball (line 1 in Algorithm 1) and $\bm{e_{i}}$ be a length-$m$ one-hot
vector where the $i$-th coordinator is 1 and other coordinators are all 0. For
any $\bm{W}=(w_{i})_{i=1}^{m}\in\mathbb{N}^{m}$, denote
$w=\sum_{i=1}^{m}w_{i}$. Then when $1+k\leq w\leq 1+k+n$, we have

$\displaystyle\Pr[\mathcal{M}(S)=\bm{W}]$ $\displaystyle=\sum_{i\in
S}\Pr[\mathcal{M}(S)=\bm{W}\mid I=i]\cdot\Pr[I=i]$ $\displaystyle=\sum_{i\in
S}\Pr[\mathcal{M}_{0}=\bm{W}-\bm{e_{i}}]\cdot s^{-1}$
$\displaystyle=\binom{n}{w-1-k}p^{w-1-k}(1-p)^{n-w+1+k}\cdot\frac{(w-1)!}{s\prod_{i=1}^{m}w_{i}!}\cdot\sum_{i\in
S}w_{i}.$

Therefore, Inequality (1) is equivalent to

$\Pr_{\bm{W}\sim{\mathcal{M}}(S)}\left[\frac{\sum_{i\in S}w_{i}}{\sum_{i\in
S^{\prime}}w_{i}}\geq e^{\varepsilon}\right]\leq\delta.$

Notice that when $\bm{W}\sim\mathcal{M}(S)$, $\sum_{i\in S}w_{i}\sim
1+\mathrm{Bin}(k,\frac{s}{m})+\mathrm{Bin}(n,\frac{ps}{m})$, while $\sum_{i\in
S^{\prime}}w_{i}\sim\mathrm{Ber}(|S\cap
S^{\prime}|/s)+\mathrm{Bin}(k,\frac{s}{m})+\mathrm{Bin}(n,\frac{ps}{m})$.
Therefore, it suffices to prove

$\Pr\left[\frac{1+\mathrm{Bin}(k,\frac{s}{m})+\mathrm{Bin}(n,\frac{ps}{m})}{\mathrm{Bin}(k,\frac{s}{m})+\mathrm{Bin}(n,\frac{ps}{m})}\geq
e^{\varepsilon}\right]\leq\delta.$ (2)

Let $X_{1},X_{2}\sim\mathrm{Bin}(k,\frac{s}{m})+\mathrm{Bin}(n,\frac{ps}{m})$.
Then
$\mu\coloneqq\mathbb{E}[X_{1}]=(k+pn)\frac{s}{m}\geq\frac{32\ln(2/\delta)}{\varepsilon^{2}}$.
By Chernoff bound,

$\Pr[X_{1}\leq(1+\varepsilon/3)\mu]\geq
1-\exp\left(-\frac{\mu}{3}\left(\frac{\varepsilon}{3}\right)^{2}\right)>1-\frac{\delta}{2},$
$\Pr[X_{2}\geq(1-\varepsilon/4)\mu]\geq
1-\exp\left(-\frac{\mu}{2}\left(\frac{\varepsilon}{4}\right)^{2}\right)>1-\frac{\delta}{2}.$

Since $\delta<1$, we have $\mu>(32\ln 2)/\varepsilon^{2}>18/\varepsilon^{2}$.
Also note that $e^{\varepsilon/3}\geq 1+\varepsilon/3+\varepsilon^{2}/18$ and
$1-\varepsilon/4>e^{-\varepsilon/2}$ for $0<\varepsilon\leq 3$, so we have

$\frac{1+(1+\varepsilon/3)\mu}{(1-\varepsilon/4)\mu}\leq\frac{\varepsilon^{2}/18+(1+\varepsilon/3)}{1-\varepsilon/4}\leq\frac{e^{\varepsilon/3}}{e^{-\varepsilon/2}}\leq
e^{\varepsilon}.$

Therefore,

$\Pr\left[\frac{1+X_{1}}{X_{2}}\leq
e^{\varepsilon}\right]\geq\Pr[X_{1}\leq(1+\varepsilon/3)\mu]\cdot\Pr[X_{2}\geq(1-\varepsilon/4)\mu]\geq\left(1-\frac{\delta}{2}\right)^{2}>1-\delta,$

which implies Inequality (2). ∎

Notice that the assumption $\varepsilon\leq 3$ is only required for simplicity
when applying Chernoff inequality. For general $\varepsilon$, a similar result
can be proved by replacing Chernoff bound with Bennett’s inequality (details
omitted).

###  2.2 The Protocol for a Small Domain

For the small-domain case $B<\tilde{O}(n)$, our protocol is a simple
modification of the single-message shuffle protocol of [1]. In their local
randomizer, each user sends its input with probability $\rho=\tilde{O}(B/n)$
(so their protocol only works when $B=O(n)$), and sends a blanket noise
otherwise. The blanket noise is a random element uniformly chosen from the
domain $[B]$. When $B=\sqrt{n}$, their analyzer incurs
$\tilde{O}(\sqrt[4]{n})$ error, matching the lower bound of single-message
protocols [4].

We extend their result to a $(1+\rho)$-message protocol
$\mathcal{P}^{\mathrm{FE}0}=(\mathcal{R}^{\mathrm{FE}0},\mathcal{S},\mathcal{A}^{\mathrm{FE}0})$
for better accuracy, where $\rho=\tilde{O}(B/n)$, which is $o(1)$ for
$B<\tilde{O}(n)$. The local randomizer works as follows: each user sends its
real input, and then with probability $\rho$ it also sends a blanket noise
(see Algorithm 2). We show that our local randomizer also satisfies shuffle-DP
constraint, while the error of our analyzer (see Algorithm 3) can be upper
bounded by $\tilde{O}(1)$.

Public Parameters: $B,n,\varepsilon,\delta$

Input: $x\in[B]$

Output: A multiset $\mathcal{T}\subseteq[B]$

1 $\mathcal{T}\leftarrow\\{x\\}$

2 $\rho\leftarrow\frac{32\ln(2/\delta)}{\varepsilon^{2}}\cdot\frac{B}{n}$

3 $y\leftarrow\mathrm{Ber}(\rho)$

4 if  _$y=1$_ then

5 Choose $x^{\prime}$ from $[B]$ uniformly

6 $\mathcal{T}\leftarrow\mathcal{T}\uplus\\{x^{\prime}\\}$

7

8 end if

return _$\mathcal{T}$_

Algorithm 2 Local Randomizer $\mathcal{R}^{\mathrm{FE}0}$

Public Parameters: $B,n,\varepsilon,\delta$

Input: A multiset $\mathcal{T}\subseteq[B]$; element $x$

Output: Estimated frequency $g_{x}$

1 $X\leftarrow$ the frequency of $x$ in $\mathcal{T}$

2 $\hat{g}(x)\leftarrow X-n\rho/B$

return _$\hat{g}(x)$_

Algorithm 3 Analyzer $\mathcal{A}^{\mathrm{FE}0}$

######  Lemma 2.2 (Privacy of $\mathcal{R}^{\mathrm{FE}0}$).

For any $n,B\in\mathbb{N}$, $0<\varepsilon\leq 3$, and $0<\delta<1$, let
$\rho=\frac{32\ln(2/\delta)}{\varepsilon^{2}}\cdot\frac{B}{n}$. If $\rho\leq
1$, then the mechanism $\mathcal{S}\circ(\mathcal{R}^{\mathrm{FE}0})^{n}$ is
$(\varepsilon,\delta)$-differentially private.

###### Proof.

For any input $\bm{D}=(x_{1},x_{2},\dots,x_{n})$, the view of the analyzer is
the multiset $\mathcal{T}$ which can be split into two parts:

  1. 1.

$n$ real inputs from all the users $\\{x_{i}\\}_{i=1}^{n}$, and

  2. 2.

the multiset of expected $n\rho$ blanket noises
$\mathcal{T}_{\mathrm{Noise}}\subseteq[B]$.

Let $\bm{D}^{\prime}$ be any neighbouring databases of $\bm{D}$. Then we need
to prove:

$\Pr_{\bm{\tau}\sim\mathcal{T}(\bm{D})}\left[\frac{\Pr[\mathcal{T}(\bm{D})=\bm{\tau}]}{\Pr[\mathcal{T}(\bm{D^{\prime}})=\bm{\tau}]}\geq
e^{\varepsilon}\right]\leq\delta.$ (3)

Without loss of generality, we assume $\bm{D}$ and $\bm{D}^{\prime}$ differ in
the last element, i.e., $\bm{D}^{\prime}=(x_{1},x_{2},\dots,x_{n}^{\prime})$.
By factoring out the first $n-1$ users, it suffices to prove:

$\Pr_{\bm{\tau}\sim\mathcal{T}_{\mathrm{Noise}}}\left[\frac{\Pr[\mathcal{T}_{\mathrm{Noise}}=\bm{\tau}]}{\Pr[\mathcal{T}_{\mathrm{Noise}}\uplus\\{x^{\prime}_{n}\\}=\bm{\tau}\uplus\\{x_{n}\\}]}\geq
e^{\varepsilon}\right]\leq\delta.$ (4)

Note that (4) is actually stronger than (3), as the first $n-1$ common users
provide additional randomness. To prove (4), we note that without real inputs
from the first $n-1$ users, $\mathcal{S}\circ(\mathcal{R}^{\mathrm{FE}0})^{n}$
corresponds to the mechanism $\mathcal{M}^{\mathrm{BIB}}(B,1,0,n,\rho)$:

  * •

The set of bins in $\mathcal{M}^{\mathrm{BIB}}$ corresponds to $[B]$.

  * •

The input set in $\mathcal{M}^{\mathrm{BIB}}$ corresponds to $\\{x\\}$ where
$x$ is the input.

  * •

No fixed $k$ noisy balls, i.e., $k=0$.

  * •

The multiset of bins chosen by the expected $np$ noisy balls is
$\mathcal{T}_{\mathrm{Noise}}$.

Then the lemma immediately follows from Theorem 2.1. ∎

######  Lemma 2.3 (Accuracy of $\mathcal{P}^{\mathrm{FE}0}$).

The protocol $\mathcal{P}^{\mathrm{FE}0}$ is $(\alpha,\beta)$-accurate for any
$\beta\in(0,1]$ and

$\alpha=\max\left\\{3\ln(2B/\beta),\sqrt{3\ln(2B/\beta)\cdot\frac{32\ln(2/\delta)}{\varepsilon^{2}}}\right\\}.$

###### Proof.

For any element $x\in[B]$, let $g(x)$ be the true frequency of $x$ in
$\bm{D}$, and $X$ be the frequency of $x$ in the output multiset
$\mathcal{T}$, then $X=g(x)+\mathrm{Bin}(n,\rho/B)$, where the second term
comes from the blanket noises. Since $\hat{g}(x)=X-n\rho/B$, we conclude
$\mathbb{E}[\hat{g}(x)]=g(x)$, so $\mathcal{A}^{\mathrm{FE}0}$ provides an
unbiased estimation.

Let $Y=\mathrm{Bin}(n,\rho/B)$ which is a sum of $n$ independent Bernoulli
variables. Define $\mu\coloneqq\mathbb{E}[Y]$, then by Chernoff bound, for
$0<\eta\leq 1$,

$\Pr\left[|Y-\mu|>\eta\mu\right]<2\exp(-\frac{\eta^{2}\mu}{3}).$

And for $\eta>1$,

$\Pr\left[|Y-\mu|>\eta\mu\right]<2\exp(-\frac{\eta\mu}{3}).$

If $\mu\geq 3\ln(2B/\beta)$, by setting
$\eta=\sqrt{\frac{3\ln(2B/\beta)}{\mu}}\leq 1$ we get

$\Pr\left[|Y-\mu|>\sqrt{3\ln(2B/\beta)\mu}\right)<2\exp(-\frac{\eta^{2}\mu}{3})=\frac{\beta}{B}.$

Otherwise, by setting $\eta=\frac{3\ln(2B/\beta)}{\mu}>1$ we get

$\Pr\left[|Y-\mu|>3\ln(2B/\beta)\right)<2\exp(-\frac{\eta\mu}{3})=\frac{\beta}{B}.$

In conclusion,

$\Pr\left[|Y-\mu|>\max\left\\{3\ln(2B/\beta),\sqrt{3\ln(2B/\beta)\cdot\mu}\right\\}\right]<\frac{\beta}{B}.$

Therefore,

$\Pr\left[|\hat{g}(x)-g(x)|>\max\left\\{3\ln(2B/\beta),\sqrt{3\ln(2B/\beta)\cdot\frac{32\ln(2/\delta)}{\varepsilon^{2}}}\right\\}\right]<\frac{\beta}{B}.$
(5)

By union bound,

$\Pr\left[\bigvee_{x\in[B]}|\hat{g}(x)-g(x)|>\max\left\\{3\ln(2B/\beta),\sqrt{3\ln(2B/\beta)\cdot\frac{32\ln(2/\delta)}{\varepsilon^{2}}}\right\\}\right]<\beta,$

which implies the result. ∎

Finally, by Lemma 2.2 and 2.3, we conclude with the following theorem:

######  Theorem 2.4.

For $0<\varepsilon\leq 3$, $0<\delta<1$, and any $n,B\in\mathbb{N}$ such that
$B\leq\frac{\varepsilon^{2}n}{32\ln(2/\delta)}$, there exists a private-coin
$(\varepsilon,\delta)$-differentially private protocol
$\mathcal{P}^{\mathrm{FE}0}$ in the shuffle model for the frequency estimation
problem, in which each user sends
$1+O\left(\frac{\log(1/\delta)}{\varepsilon^{2}}\cdot\frac{B}{n}\right)$
messages in expectation, each consisting of $O(\log B)$ bits. The frequency
query of any element $x\in[B]$ is answered in expected time $O(n)$, and the
maximum error is $O\left(\log
B+\frac{1}{\varepsilon}\sqrt{\log(B)\log(1/\delta)}\right)$ with constant
probability.

######  Corollary 2.4.1.

For $B=o\left(\frac{\varepsilon^{2}n}{\log(1/\delta)}\right)$,
$\mathcal{P}^{\mathrm{FE}0}$ is a $(1+o(1))$-message protocol with error
$\tilde{O}(1)$.

###  2.3 The Protocol for a Large Domain

The condition $\rho\leq 1$ in Lemma 2.2 can actually be lifted, hence
supporting a large $B$: We could ask each user to send $\lfloor\rho\rfloor$
blanket noises, and then another blanket noise with probability
$\rho-\lfloor\rho\rfloor$. This still ensures privacy but the number of
messages per user will no longer be $1+o(1)$. In this section, we provide
another protocol $\mathcal{P}^{\mathrm{FE}1}$ for a large domain size
$B=\tilde{\Omega}(n)$, while maintaining the $1+o(1)$ messages and
$\tilde{O}(1)$ error of $\mathcal{P}^{\mathrm{FE}0}$.

The idea is to use hashing to reduce the domain size. Specifically, we use the
classical hash function $h_{u,v}:~{}[B]\to\mathbb{Z}_{b}$ to hash an element
in $[B]$ to a bin from $\mathbb{Z}_{b}\coloneqq\\{0,1,\dots,b-1\\}$, where
$h_{u,v}(x)=((ux+v)\bmod{q})\bmod{b}$, $q\in[B,2B)$ is a prime111This $q$ must
exist due to Bertrand’s postulate., and $b$ is a parameter that will control
the trade-off between communication cost and accuracy. Let
$\mathcal{H}=\\{h_{u,v}\mid(u,v)\in[q-1]\times[q]\\}$. It is well-known that
$\mathcal{H}$ is a universal family with collision probability
$p_{\mathrm{col}}(x,y)\coloneqq\Pr[h_{u,v}(x)=h_{u,v}(y)]<\frac{1}{b}$. In
fact, for this family, the collision probability is the same for all $x\neq
y$: $p_{\mathrm{col}}(\cdot,\cdot)\equiv\lfloor
q/b\rfloor[(q\bmod{b})+q-b]/[q(q-1)]$, which is simply denoted as
$p_{\mathrm{col}}$.

#### Local randomizer (Algorithm 4).

Assume a user holds a private input $x$. It uniformly randomly chooses
$(u,v)\in[q-1]\times[q]$, and sends $(u,v,h_{u,v}(x))$ to the shuffler, called
the real output. It also sends $\rho$ blanket noises to the shuffler, where
$\rho=\frac{32\ln(2/\delta)}{\varepsilon^{2}}\cdot\frac{b}{n}$. Each blanket
noise $(u,v,w)$ is uniformly chosen from $[q-1]\times[q]\times\mathbb{Z}_{b}$
independently. Note that when $\rho$ is not an integer, it sends
$\lfloor\rho\rfloor$ blanket noises and then with probability
$\rho-\lfloor\rho\rfloor$ sends another blanket noise, so that the expected
number of blanket noises is $\rho$. The total expected number of messages the
user sends is therefore $1+\rho$, including one real output and $\rho$ blanket
noises.

Public Parameters: $B,q,b,n,\varepsilon,\delta$

Input: $x\in[B]$

Output: A multiset $\mathcal{T}\subseteq[q-1]\times[q]\times\mathbb{Z}_{b}$

1 Choose $(u,v)$ from $[q-1]\times[q]$ uniformly

2 $\mathcal{T}\leftarrow\\{(u,v,f_{u,v}(x))\\}$

3 $\rho\leftarrow\frac{32\ln(2/\delta)}{\varepsilon^{2}}\cdot\frac{b}{n}$

4 for  _$i\leftarrow 1$ to $\lfloor\rho\rfloor$_ do

5 Choose $(u,v,w)$ from $[q-1]\times[q]\times\mathbb{Z}_{b}$ uniformly

6 $\mathcal{T}\leftarrow\mathcal{T}\uplus\\{(u,v,w)\\}$

7

8 end for

9$y\leftarrow\mathrm{Ber}(\rho-\lfloor\rho\rfloor)$

10 if  _$y=1$_ then

11 Choose $(u,v,w)$ from $[q-1]\times[q]\times\mathbb{Z}_{b}$ uniformly

12 $\mathcal{T}\leftarrow\mathcal{T}\uplus\\{(u,v,w)\\}$

13

14 end if

return _$\mathcal{T}$_

Algorithm 4 Local Randomizer $\mathcal{R}^{\mathrm{FE}1}$

#### Analyzer (Algorithm 5).

The analyzer’s view is a multiset of tuples $(u,v,w)$. To query the frequency
of an element $x$, it scans all tuples received and count the number of tuples
satisfying $h_{u,v}(x)=w$. Then we correct the bias caused by hash collisions
and the blanket noises.

Public Parameters: $B,q,b,n,\varepsilon,\delta$

Input: A multiset $\mathcal{T}\subseteq[q-1]\times[q]\times\mathbb{Z}_{b}$;
element $x$

Output: Estimated frequency $g_{x}$

1 $X\leftarrow 0$

2 for  _$(u,v,w)\in\mathcal{T}$_ do

3 if  _$h_{u,v}(x)=w$_ then

4 $X\leftarrow X+1$

5 end if

6

7 end for

8$\hat{g}(x)\leftarrow(X-n\rho/b-np_{\mathrm{col}})/(1-p_{\mathrm{col}})$

return _$\hat{g}(x)$_

Algorithm 5 Analyzer $\mathcal{A}^{\mathrm{FE}1}$

######  Lemma 2.5 (Privacy of $\mathcal{R}^{\mathrm{FE}1}$).

Fix $n,B,b\in\mathbb{N}$, $0<\varepsilon\leq 3$ and $0<\delta<1$, and let
$\rho=\frac{32\ln(2/\delta)}{\varepsilon^{2}}\cdot\frac{b}{n}$. The mechanism
$\mathcal{S}\circ(\mathcal{R}^{\mathrm{FE}1})^{n}$ is
$(\varepsilon,\delta)$-differentially private.

###### Proof.

The proof is similar to the proof of Lemma 2.2: First we exclude the multiset
$\\{(u_{i},v_{i},f_{u_{i},v_{i}}(x_{i}))\\}_{i=1}^{n-1}$, i.e., the real
outputs from the first $n-1$ users, from its view. Then the privacy is
guaranteed by the mechanism
$\mathcal{M}^{\mathrm{BIB}}((q-1)qb,(q-1)q,n\lfloor\rho\rfloor,n,\rho-\lfloor\rho\rfloor)$.
∎

######  Lemma 2.6 (Accuracy of $\mathcal{P}^{\mathrm{FE}1}$).

The protocol $\mathcal{P}^{\mathrm{FE}1}$ is $(\alpha,\beta)$-accurate for any
$\beta\in(0,1]$ and

$\alpha=2\max\left\\{3\ln(2B/\beta),\sqrt{3\ln(2B/\beta)\cdot\left(\frac{n}{b}+\frac{32\ln(2/\delta)}{\varepsilon^{2}}\right)}\right\\}.$

###### Proof.

For any element $x\in[B]$, let $X$ be the number of tuples $(u,v,w)$ in
$\mathrm{View}(\bm{D})$ that satisfy $f_{u,v}(x)=w$. $X$ is contributed from
three parts:

  1. 1.

$g(x)$ real outputs from the users with input $x$,

  2. 2.

$n-g(x)$ real outputs from the users with input not equal to $x$, each
contributes 1 with probability $p_{\mathrm{col}}$ due to hash collision,

  3. 3.

$n$ blanket noises, each contributes 1 with probability $\rho/b$.

Therefore,
$X=g(x)+\mathrm{Bin}(n-g(x),p_{\mathrm{col}})+\mathrm{Bin}(n,\rho/b)$. Since
$\hat{g}(x)=(X-n\rho/b-np_{\mathrm{col}})/(1-p_{\mathrm{col}})$, we conclude
$\mathbb{E}[\hat{g}(x)]=g(x)$, so $\mathcal{P}^{\mathrm{FE}1}$ provides an
unbiased estimation.

Let $Y=\mathrm{Bin}(n-g(x),p_{\mathrm{col}})+\mathrm{Bin}(n,\rho/b)$ which is
a sum of $2n-g(x)$ independent Bernoulli variables. Define
$\mu\coloneqq\mathbb{E}[Y]$, then similar to the proof of Lemma 2.3, we have

$\Pr\left[|Y-\mu|>\max\left\\{3\ln(2B/\beta),\sqrt{3\ln(2B/\beta)\cdot\mu}\right\\}\right]<\frac{\beta}{B}.$

Therefore,

$\Pr\left[|\hat{g}(x)-g(x)|>\max\left\\{3\ln(2B/\beta),\sqrt{3\ln(2B/\beta)\cdot\mu}\right\\}/(1-p_{\mathrm{col}})\right]<\frac{\beta}{B}.$
(6)

Recall that $p_{\mathrm{col}}<1/b$, we have

$\mu=p_{\mathrm{col}}(n-g(x))+n\rho/b\leq\frac{n}{b}+\frac{32\ln(2/\delta)}{\varepsilon^{2}}.$

Also note that $p_{\mathrm{col}}\leq 1/2$, so inequality (6) can be relaxed to

$\Pr\left[|\hat{g}(x)-g(x)|>2\max\left\\{3\ln(2B/\beta),\sqrt{3\ln(2B/\beta)\left(\frac{n}{b}+\frac{32\ln(2/\delta)}{\varepsilon^{2}}\right)}\right\\}\right]<\frac{\beta}{B}.$

By union bound,

$\Pr\left[\bigvee_{x\in[B]}|\hat{g}(x)-g(x)|>2\max\left\\{3\ln(2B/\beta),\sqrt{3\ln(2B/\beta)\left(\frac{n}{b}+\frac{32\ln(2/\delta)}{\varepsilon^{2}}\right)}\right\\}\right]<\beta,$

which implies the result. ∎

######  Lemma 2.7 (Efficiency of $\mathcal{P}^{\mathrm{FE}1}$).

The following efficiency statements hold for $\mathcal{P}^{\mathrm{FE}1}$:

  1. 1.

Each user sends $1+\frac{b}{n}\cdot\frac{32\ln(2/\delta)}{\varepsilon^{2}}$
messages in expectation, each containing $2\log q+\log b$ bits.

  2. 2.

For any given $x\in[B]$, the analyzer can report the estimated frequency of
$x$ in expected time $n+b\cdot\frac{32\ln(2/\delta)}{\varepsilon^{2}}$.

  3. 3.

The analyzer can report estimated frequencies of all elements in $[B]$ in
expected time at most
$\frac{q}{b}\cdot(n+b\cdot\frac{32\ln(2/\delta)}{\varepsilon^{2}})$. Note that
this is faster than querying each element one by one using statement 2 by a
$b$ factor.

###### Proof.

Statements 1 and 2 follow directly from the description of the protocol. We
prove statement 3 below. For any $(u,v,w)\in\mathcal{T}$, it can be verified
that the set of solutions in $[B]$ to the equation $h_{u,v}(x)=w$ is

$\mathcal{X}(u,v,w)\coloneqq\left\\{u^{-1}(w+i\cdot
b-v)\bmod{q}~{}\middle|~{}i\in\left\\{0,1,\dots,\left\lfloor\frac{q-1-w}{b}\right\rfloor\right\\}\right\\}\cap[B].$

The estimated frequency of any element $x\in[B]$ is the number of occurrences
in $\\{\mathcal{X}(u,v,w)\mid(u,v,w)\in\mathcal{T}\\}$ (followed by a bias
reduction). Therefore, it suffices to compute the frequencies of the multiset
$\uplus_{(u,v,w)\in\mathcal{T}}\mathcal{X}(u,v,w)$. Since each set
$\mathcal{X}(u,v,w)$ has at most $q/b$ elements and the size of $\mathcal{T}$
is expected to be $n+n\rho=n+b\cdot\frac{32\ln(2/\delta))}{\varepsilon^{2}}$,
the total time is expected to be at most
$\frac{q}{b}\cdot(n+b\cdot\frac{32\ln(2/\delta)}{\varepsilon^{2}})$. ∎

Finally, by Lemma 2.5, 2.6 and 2.7, we conclude with the following theorem:

######  Theorem 2.8.

For any $0<\varepsilon\leq 3$, $0<\delta<1$, and $n,B,b\in\mathbb{N}$ such
that $2\leq b\leq B/2$, there exists a private-coin
$(\varepsilon,\delta)$-differentially private protocol
$\mathcal{P}^{\mathrm{FE}1}$ in the shuffle model, in which each user sends
$1+O\left(\frac{\log(1/\delta)}{\varepsilon^{2}}\cdot\frac{b}{n}\right)$
messages in expectation, each consisting of $O(\log B)$ bits. The frequency
query of any element $x\in[B]$ is answered in time
$O\left(n+\frac{\log(1/\delta)}{\varepsilon^{2}}\cdot b\right)$ in
expectation, the frequencies of all elements in $[B]$ can be found in time
$O\left(\left(\frac{n}{b}+\frac{\log(1/\delta)}{\varepsilon^{2}}\right)\cdot
B\right)$ in expectation, and the maximum error is $O\left(\log B+\sqrt{\log
B\left(\frac{n}{b}+\frac{\log(1/\delta)}{\varepsilon^{2}}\right)}\right)$ with
constant probability.

######  Corollary 2.8.1.

For $B=\Omega\left(\frac{\varepsilon^{2}n}{\log(1/\delta)}\right)$,
$\mathcal{P}^{\mathrm{FE}1}$ is a $(1+o(1))$-message protocol with error
$\tilde{O}(1)$ by setting $b=f(n)\cdot\frac{\varepsilon^{2}n}{\log(1/\delta)}$
for any $(\log n)^{-\Omega(1)}<f(n)<o(1)$.

We also compare the error and various costs of our protocols in Table 1 with
the start-of-the-art private-coin frequency estimation protocol [4]. In
particular, by taking $b=n/\log n$ in $\mathcal{P}^{\mathrm{FE}1}$, our
protocol achieves the same error as [4], but we reduce the number of messages
to $O(1)$, reduce the message size by a $\log n$ factor, and reduce the query
time of a single element by a $\log^{3}n$ factor.

Protocol | Messages per user | Message size | Error | Query time (single) | Query time (all)
---|---|---|---|---|---
[4] | $O(\log n)$ | $O(\log^{2}n)$ | $O(\log n)$ | $O(n\log^{3}n)$ | $O(n\log^{4}n+B\log^{2}n)$
This work (small $B$) | $1+O(B\log n/n)$ | $O(\log n)$ | $O(\log n)$ | $O(n)$ | $O(n)$
This work (large $B$) | $1+O(b\log n/n)$ | $O(\log n)$ | $O(\log n+\sqrt{\frac{n}{b}\log n})$ | $O(n+b\log n)$ | $O(Bn/b+B\log n)$
Above with | $c=1$ | $O(1)$ | $O(\log n)$ | $O(\log n)$ | $O(n)$ | $O(B\log n)$
$b=\frac{n}{\log^{c}n}$ | $c>1$ | $1+o(1)$ | $O(\log n)$ | $O(\log^{\frac{c+1}{2}}n)$ | $O(n)$ | $O(B\log^{c}n)$
Table 1: Comparison with prior work for constant $\varepsilon$, and
$\log(1/\delta)\approx\log n\approx\log B$.

#### Robust shuffle-DP.

In the robust $(\varepsilon,\delta,\gamma)$ shuffle-DP, a fraction of
$1-\gamma$ of the users may not follow the specified protocol. Our randomizer
splits (expected) $n\rho$ blanket noises into $n$ parts, so that each user
contributes (expected) $\rho$ blanket noises. Therefore, our protocol also
satisfies $(\varepsilon,\delta,\gamma)$ shuffle-DP, in which we set
$\rho=\frac{32\ln(2/\delta)}{\gamma\varepsilon^{2}}\cdot\frac{b}{n}$ instead.
The accuracy and efficiency should be updated accordingly, but the
complexities do not change when $\gamma$ is considered as a constant.

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