Majorization and semidoubly stochastic operators on $L^{1}(X)$

Abstract

This paper is devoted to a study of majorization based on semidoubly stochastic operators (denoted by $S\mathcal{D}(L^{1})$) on $L^{1}(X)$ when X is a σ-finite measure space. We answer Mirsky’s question and characterize the majorization by means of semidoubly stochastic maps on $L^{1}(X)$. We prove some results on semidoubly stochastic operators such as a strong relation of semidoubly stochastic operators and integral stochastic operators and relatively weak compactness of $S_{f}=\{Sf : S\in S\mathcal{D}(L^{1})\}$ for a fixed element $f\in L^{1}(X)$ by proving the equiintegrability of $S_{f}$. We present a full characterization of majorization on a σ-finite measure space X.

1 Introduction

Until recent decades, the main attention in majorization theory was paid to finite-dimensional spaces. However, recently, because of its significant applications in a broad spectrum of fields, especially in quantum physics, there arose a considerable mathematical and physical interest in infinite-dimensional spaces [4, 10, 12, 13, 19].

The aim of this paper is to study the notion of majorization on $L^{1}(X,\mu )$, that is, the space of all absolutely integrable functions $f:X\to \mathbb{R}$ when $(X,\mu )$ is a σ-finite measure space. Our motivation to work on this space is its application in quantum information theory; for details, see [13, 18]. We start with a short history.

Hardy, Littlewood, and Pólya extended an equivalent condition of Muirhead’s inequality from nonnegative integer vectors to real vectors and called it vector majorization as follows. Let $X,Y\in \mathbb{R}^{n}$ with the similar total of the whole components. A vector X is said to be majorized by Y (denoted by $X\prec Y$) if for each $k\in \{1,2,\ldots , n\}$, the sum of the k largest components of X is less than or equal to the sum of the k largest components of Y.

To avoid difficulty of decreasing rearrangement of components, Hardy, Littlewood, and Pólya proved equivalent conditions for $X\prec Y$ independent of decreasing rearrangements.

Theorem 1.1

([14])

Let $X,Y\in \mathbb{R}^{n}$. Then the following statements are equivalent.

• (1) $X\prec Y$.

• (2) There exists a doubly stochastic matrix $D=[d_{ij}]$ (an n-square matrix with $d_{ij}\geq 0$, $\sum_{i=1}^{n} d_{ij}=1$, and $\sum_{j=1}^{n} d_{ij}=1$ for all $i,j\in \{1,2,\ldots , n\}$) such that $X=DY$.

• (3) The following inequality holds for all convex functions g:

  $$ \sum_{i=1}^{n} g(x_{i})\leq \sum _{i=1}^{n} g(y_{i}). $$

  Here $x_{i}$ and $y_{i}$ for all $i=1,\dots , n$ are the components of X and Y, respectively.

As an infinite counterpart of Hardy, Littlewood, and Pólya’s results, Mirsky proposed the following question [15, Sect. 4, p. 328]:

“The introduction of infinite doubly stochastic matrices raises the question whether there exists an infinite analogue of Hardy, Littlewood, and Pólya’s results?”

In this work, by using semidoubly stochastic operators, we completely answer Mirsky’s question (extension of Hardy, Littlewood, and Pólya’s results, Theorem 1.1) on $L^{1}(X,\mu )$ when $(X,\mu )$ is a σ-finite measure space. We denote the dual space of $L^{1} (X,\mu )$, which is the space of essentially bounded functions on X, by $L^{\infty }(X,\mu )$, and to shorten notation, we write $L^{1}(X)$ and $L^{\infty}(X)$ when there is no confusion.

Section 2 contains a brief historical summary of results and mathematical preliminaries on majorization on $L^{1}(X)$. In Sect. 3, we look more closely at semidoubly stochastic operators and provide a method of constructing them in Theorem 3.2, and for fixed $f\in L^{1}(X)$, we will prove the equiintegrability of $S_{f}=\{Sf: S\in S\mathcal{D}(L^{1}(X))\}$ in Theorem 3.8 for arbitrary measure space X. This theorem leads to Corollary 3.10, which states the relatively weak compactness of $S_{f}$ when X is a finite measure space. In Sect. 4, we review some recent results like a relation between majorization and integral operators in a more general setting by using semidoubly stochastic operators, and we answer Mirsky’s question by giving a full characterization of majorization in Theorem 4.13.

It is worth noticing that for the nonlinear case, quadratic stochastic operators that were originally introduced by Bernstein [5] in the early 20th century through his work on population genetics have been significantly developed for decades [1, 3, 8, 9, 17, 21]. In [8] the concept of doubly (bistochastic) quadratic stochastic operators is defined in terms of the majorization notion, and in [9] necessary and sufficient conditions were obtained for those operators. Recently, in [1] a consensus problem in multiagent systems based on doubly stochastic quadratic operators motivated by the prospective applications in unmanned aerial vehicles, sensor networks fields, distributed computing in computer science, medicine, environmental monitoring, and military reconnaissance [1, 3, 21] has been studied.

2 Mathematical preliminaries

There are two main ways for generalizing the majorization concept on measurable functions, an extension based on decreasing rearrangement and an extension based on stochastic operators.

2.1 Decreasing rearrangement

In 1963, Ryff [20] defined the decreasing rearrangement for measurable functions on $([0,1],m)$, where m is the Lebesgue measure, and then, in 1974, Chong [6] extended it to $L^{1}(X,\mu )$ for an arbitrary measure space $(X,\mu )$ as follows.

Theorem 2.1

([6])

Let f be a measurable (respectively, nonnegative integrable) function defined on a finite (respectively, infinite) measure space $( X, \mu )$. Then there exists a unique right-continuous decreasing function $f^{\downarrow}$ on the interval $[0, \mu (X)]$, called the decreasing rearrangement of f, defined by

$$\begin{aligned} f^{\downarrow }(s) =& \inf \bigl\{ t : d_{f}(t)\leq s\bigr\} \\ =& \sup \bigl\{ t : d_{f}(t)> s\bigr\} , \quad 0 \leq s \leq \mu (X), \end{aligned}$$

where $d_{f}$, called the distribution function of f, is defined for all real t by

$$ d_{f}(t)= \mu \bigl\{ x : f(x)>t\bigr\} . $$

Then Chong generalized the notion of majorization to $L^{1}(X,\mu )$ for an arbitrary measure space X as follows.

Definition 2.2

([6])

Let $(X, \mu )$ be an arbitrary measure space, and let $f,g\in L^{1}(X)$ (notice that for an infinite measure space, we have to suppose that f, g are nonnegative). Then we say that f is weak majorized by g and write $f\prec _{w} g$ if

$$ \int _{0}^{s} f^{\downarrow }\,dm \leq \int _{0}^{s} g^{\downarrow }\,dm,\quad 0\leq s\leq \mu (X). $$

Furthermore, we say that f is majorized by g and write $f\prec g$ if

$$ \int _{0}^{\mu (X)} f^{\downarrow }\,dm = \int _{0}^{\mu (X)} f^{ \downarrow }\,dm, $$

where dm is the Lebesgue measure on the interval $[0,\infty )$.

The following theorems are proved by Chong; we will apply them in the following sections.

Theorem 2.3

([6, Theorem 1.6])

If $f,g\in L^{1}(X,\mu )$ are nonnegative integrable functions and $(X,\mu )$ is a measure space (finite or infinite), then

$$ \int _{0}^{t}f^{\downarrow}\,dm\leq \int _{0}^{t}g^{\downarrow}\,dm,\quad t\in [0,\infty ), $$

if and only if

$$ \int _{u}^{\infty}d_{f}\,dm\leq \int _{u}^{\infty}d_{g}\,dm \quad \textit{for all } u\in \mathbb{R}. $$

Theorem 2.4

([6, Corollary 1.2])

If f is a measurable function on $(X,\mu )$, where $\mu (X)$ is finite or infinite, then

$$ \int _{X}\max \bigl\{ (f-u),0\bigr\} \,d\mu = \int _{u}^{\infty}d_{f}(t)\,dt. $$

2.2 Semidoubly stochastic operators

To avoid the difficulty of decreasing rearrangement in the definition of majorization, Ryff [20] introduced and characterized an important class of linear operators $T:L^{1}([0,1],m)\to L^{1}([0,1],m)$, which are known as doubly stochastic operators such that Tf is majorized by f for all $f\in L^{1}([0,1],m)$. Ryff’s characterization cannot be extended to $L^{1}(X, \mu )$ when $(X, \mu )$ is a σ-finite measure space (see [13, Example II.7] as a counterexample), but it can be extended by using semidoubly stochastic operators, which is a new class of operators introduced by Manjegani et al. [13] on the space $l^{1}$ and extended by Bahrami and et al. [4] to $L^{1}(X, \mu )$, where $(X, \mu )$ is a σ-finite measure space.

The class of semidoubly stochastic operators is larger than the class of doubly stochastic operators and smaller than the class of integral-preserving operators which is known as Markov operators or stochastic operators in the literature. It is worth noting that the theory of Markov operators is extremely rich. One of its applications is an examination of the eventual behavior of densities in dynamical systems; for more details, see [11].

We recall that each bounded linear map $T : X\to Y$ between two normed linear spaces X and Y induces a bounded linear operator $T^{*}:Y^{*}\to X^{*}$ between the dual spaces, defined for all $g\in Y^{*}$ and $x\in X$ by $\langle x, T^{*}g\rangle =\langle Tx, g\rangle$, where $\langle \cdot , \cdot \rangle$ denotes the dual pairing between the dual spaces.

Definition 2.5

Let $(X,\mu )$ be a measure space.

• (a) A positive operator $T:L^{1}(X)\to L^{1}(X)$ is a Markov operator if for all $f\in L^{1}(X)$,

  $$ \int _{X} Tf \,d\mu = \int _{X} f \,d\mu . $$

  In ordinary terms $T^{*}(1)=1$. The set of all Markov operators on $L^{1}(X)$ is denoted by $\mathcal{M}(L^{1}(X))$.

• (b) $T\in \mathcal{M}(L^{1}(X))$ is a semidoubly stochastic operator if for each E in the σ-algebra $\mathcal{A}$ on X with $\mu (E)<\infty$,

  $$ \int _{X} T^{*}\chi _{E}\,d\mu \leq \mu (E), $$

  where $\chi _{E}$ stands for the characteristic function of the set E.

  The set of all semidoubly stochastic operators on $L^{1}(X)$ denoted by $S\mathcal{D}(L^{1}(X))$.

• (c) $T\in S\mathcal{D}(L^{1}(X))$ is a doubly stochastic operator if for each $g\in L^{\infty}(X)\cap L^{1}(X)$, $T^{*}g\in L^{1}(X,\mu )$, and

  $$ \int _{X} T^{*}g \,d\mu = \int _{X} g \,d\mu . $$

  The set of all doubly stochastic operators on $L^{1}(X)$ is denoted by $\mathcal{D}(L^{1}(X))$.

In the following lemma, we easily prove that for arbitrary finite or infinite measure space X, the Markov operators are bounded, and therefore all semidoubly stochastic and doubly stochastic operators are bounded.

Lemma 2.6

Let $T\in \mathcal{M}(L^{1}(X))$. Then $\|T\|=1$, where

$$ \Vert T \Vert =\sup \bigl\{ \Vert Tf \Vert _{1}: \Vert f \Vert _{1}=1\bigr\} . $$

Proof

By the definition of a Markov operator

$$ \int _{X} Tf \,d\mu = \int _{X} f \,d\mu $$

for all $f\in L^{1}(X)$. Therefore

$$\begin{aligned} \Vert T \Vert &= \sup \bigl\{ \Vert Tf \Vert _{1}: \Vert f \Vert _{1}=1\bigr\} \\ &= \sup \biggl\{ \int _{X} Tf \,d\mu : \Vert f \Vert _{1}=1 \biggr\} \\ &=\sup \biggl\{ \int _{X} f \,d\mu : \Vert f \Vert _{1}=1 \biggr\} = \sup \bigl\{ \Vert f \Vert _{1}: \Vert f \Vert _{1}=1 \bigr\} =1. \end{aligned}$$

 □

By adopting the counting measure μ on the space of natural numbers $\mathbb{N}$ we rewrite the previous definition for the Banach space $l^{1}$ consisting of all summable sequences. In fact, majorization theory on $l^{1}$ plays a key role in quantum information theory. For instance, the generalization of Nielsen’s result for infinite-dimensional quantum system states that a quantum states ϕ is convertible to a quantum state ψ if and only if the sequence of Schmidt coefficients of ϕ is majorized by the sequence of Schmidt coefficients of ψ [19, Theorem 1], and by the infinite-dimensional version of the Schmidt decomposition theorem the sequences of Schmidt coefficients belong to the space $l^{1}$. Therefore rewriting of the above definition would be useful to avoid confusion in quantum information theory. Before that, we define the sequence $e_{n}\in l^{1}$ by the well-known Kronecker delta $\delta _{mn}=(e_{n})_{m}$ for $m,n\in \mathbb{N}$; we also denote the dual pairing between $l^{1}$ and its dual space $l^{\infty}$ by $\langle \cdot ,\cdot \rangle : l^{1}\times l^{\infty}\rightarrow \mathbb{R}\mathbbm{,}$ which is defined by $\langle f,g\rangle =\sum_{i\in \mathbb{N}}f_{i}g_{i}$ for $f\in l^{1}$ and $g\in l^{\infty}$,. If we consider $e_{i}$ as an element of $l^{\infty}$, then $\sum_{i\in \mathbb{N}} \langle f, e_{i}\rangle =\sum_{i\in \mathbb{N}} f_{i}$ for all $f\in l^{1}$. We are now ready to rewrite the above definition on the space $l^{1}$.

Theorem 2.7

A positive operator $T : l^{1} \rightarrow l^{1}$ is

• (a) a Markov operator if and only if $\sum_{j=1}^{\infty }\langle Te_{j}, e_{i}\rangle =1$;

• (b) a semidoubly stochastic operator if and only if $\sum_{i=1}^{\infty }\langle Te_{j}, e_{i}\rangle =1$ and $\sum_{j=1}^{\infty }\langle Te_{j}, e_{i}\rangle \leq 1$;

• (c) a doubly stochastic operator if and only if $\sum_{i=1}^{\infty }\langle Te_{j}, e_{i}\rangle =1$ and $\sum_{j=1}^{\infty }\langle Te_{j}, e_{i}\rangle =1$.

Proof

By Definition 2.5, for σ-finite measure space $(\mathbb{N},\mu )$, when μ is the counting measure on the space of natural numbers, $T : l^{1} \rightarrow l^{1}$ is

• a Markov operator if for all $f\in l^{1}$,

  $$ \sum_{i\in \mathbb{N}}(Tf) (i)=\sum _{i\in \mathbb{N}}f(i); $$

• a semidoubly stochastic operator if for all $f\in l^{1}$ and $g\in l^{\infty}(X)\cap l^{1}(X)$, $T^{*}g\in l^{1}$, and

  $$ \sum_{i\in \mathbb{N}}(Tf) (i)=\sum _{i\in \mathbb{N}}f(i),\qquad \sum_{i\in \mathbb{N}} \bigl(T^{*}g\bigr) (i)\leq \sum_{i\in \mathbb{N}}g(i); $$

• a doubly stochastic operator if for all $f\in l^{1}$ and $g\in l^{\infty}(X)\cap l^{1}(X)$, $T^{*}g\in l^{1}$, and

  $$ \sum_{i\in \mathbb{N}}(Tf) (i)=\sum _{i\in \mathbb{N}}f(i),\qquad \sum_{i\in \mathbb{N}} \bigl(T^{*}g\bigr) (i)=\sum_{i\in \mathbb{N}}g(i). $$

Therefore it is sufficient to prove that for all $f\in l^{1}$,

$$ \sum_{i\in \mathbb{N}}(Tf) (i)=\sum _{i\in \mathbb{N}}f(i) \quad \text{if and only if}\quad \sum _{i=1}^{\infty }\langle Te_{j}, e_{i} \rangle =1; $$

(1)

for all $g\in l^{\infty}(X)\cap l^{1}(X)$, $T^{*}g\in l^{1}$, and

$$ \sum_{i\in \mathbb{N}}\bigl(T^{*}g \bigr) (i)\leq \sum_{i\in \mathbb{N}}g(i)\quad \text{if and only if}\quad \sum_{j=1}^{\infty }\langle Te_{j}, e_{i} \rangle \leq 1; $$

(2)

and

$$ \sum_{i\in \mathbb{N}}\bigl(T^{*}g \bigr) (i)=\sum_{i\in \mathbb{N}}g(i) \quad \text{if and only if}\quad \sum _{j=1}^{\infty }\langle Te_{j}, e_{i} \rangle = 1. $$

(3)

Let $T:l^{1}\to l^{1}$ be a linear operator with $\sum_{i=1}^{\infty }\langle Te_{j}, e_{i}\rangle =1$. Each $f\in l^{1}$ can be written as $\sum_{j\in \mathbb{N}}f(j)e_{j}$. Then

$$ \langle Tf, e_{i}\rangle = \biggl\langle T \biggl( \sum_{j\in \mathbb{N}}f(j)e_{j} \biggr), e_{i} \biggr\rangle = \biggl\langle \biggl( \sum _{j\in \mathbb{N}}f(j)Te_{j} \biggr), e_{i} \biggr\rangle =\sum_{j\in \mathbb{N}}f(j)\langle Te_{j}, e_{i}\rangle . $$

(4)

Since $f\in l^{1}$ and $\sum_{i=1}^{\infty }\langle Te_{j}, e_{i}\rangle =1$, it follows that

$$ \sum_{j\in \mathbb{N}}\sum_{i\in \mathbb{N}} \bigl\vert f(j) \bigr\vert \langle Te_{j}, e_{i} \rangle =\sum_{j\in \mathbb{N}} \bigl\vert f(j) \bigr\vert < \infty . $$

Summing over i in equation (4) and using Fubini’s theorem, we obtain

$$ \sum_{i\in \mathbb{N}}\langle Tf, e_{i}\rangle =\sum _{i\in \mathbb{N}}\sum_{j\in \mathbb{N}}f(j) \langle Te_{j}, e_{i}\rangle = \sum _{j\in \mathbb{N}}\sum_{i\in \mathbb{N}}f(j)\langle Te_{j}, e_{i} \rangle . $$

Finally, by using $\sum_{i=1}^{\infty }\langle Te_{j}, e_{i}\rangle =1$ we have

$$ \sum_{i\in \mathbb{N}}(Tf) (i)=\sum _{i\in \mathbb{N}}\langle Tf, e_{i} \rangle =\sum _{j\in \mathbb{N}}f(j). $$

The converse is clear by setting $f=e_{j}$.

Let $T^{*}:l^{\infty}\to l^{\infty}$ be the adjoint map of T with $\sum_{j=1}^{\infty }\langle e_{j}, T^{*}e_{i}\rangle \leq 1$. Since $g\in l^{\infty}(X)\cap l^{1}(X)$, $T^{*}g\in l^{1}$, g can be written as $\sum_{i\in \mathbb{N}}g(i)e_{i}$. Then

$$ \langle Te_{j}, g\rangle = \biggl\langle Te_{j}, \sum_{i\in \mathbb{N}}g(i)e_{i} \biggr\rangle =\sum_{i\in \mathbb{N}}g(i) \langle Te_{j},e_{i} \rangle . $$

(5)

Since $g\in l^{1}$ and $\sum_{j=1}^{\infty }\langle Te_{j}, e_{i}\rangle =\sum_{j=1}^{ \infty }\langle e_{j}, T^{*}e_{i}\rangle \leq 1$, we have

$$ \sum_{i\in \mathbb{N}}\sum_{j\in \mathbb{N}} \bigl\vert g(i) \bigr\vert \langle Te_{j}, e_{i} \rangle \leq \sum_{j\in \mathbb{N}} \bigl\vert g(i) \bigr\vert < \infty . $$

Summing over j in equation (5) and using Fubini’s theorem, we obtain

$$ \sum_{j\in \mathbb{N}}\langle Te_{j}, g\rangle =\sum _{j\in \mathbb{N}}\sum_{i\in \mathbb{N}}g(i) \langle Te_{j}, e_{i}\rangle = \sum _{i\in \mathbb{N}}\sum_{j\in \mathbb{N}}g(i)\langle Te_{j}, e_{i} \rangle . $$

Finally, by using $\sum_{j=1}^{\infty }\langle Te_{j}, e_{i}\rangle \leq 1$ we have

$$ \sum_{j\in \mathbb{N}}\bigl(T^{*}g\bigr) (j)=\sum _{j\in \mathbb{N}}\langle Te_{j}, g\rangle \leq \sum _{i\in \mathbb{N}}g(i). $$

The converse is clear.

If instead of $\sum_{j=1}^{\infty }\langle Te_{j}, e_{i}\rangle \leq 1$, we use $\sum_{j=1}^{\infty }\langle Te_{j}, e_{i}\rangle = 1$, then we can prove the equivalence relation (3) in the same way as (2). □

In general, it is obvious that

$$ \mathcal{D}\bigl(L^{1}(X)\bigr)\subseteq S\mathcal{D} \bigl(L^{1}(X)\bigr)\subseteq \mathcal{M}\bigl(L^{1}(X) \bigr).$$

(6)

It is worth referring to an important result by Bahrami et al. [4, Proposition 2.6] that if $\mu (X)<\infty$, then on $L^{1}(X, \mu )$ the semidoubly stochastic operators coincide with the doubly stochastic operators. We provide a counterexample that shows that in general the converse of inclusions (6) is not true.

Example 2.8

Let μ be the counting measure on $X=\mathbb{N}$. Then define the positive operators $T_{1},T_{2},T_{3}:l^{1}\to l^{1}$ for each sequence $(a_{n})_{n\in \mathbb{N}}\in l^{1}$ by

$$\begin{aligned}& T_{1}(a_{n})=\Biggl(\sum_{i=1}^{\infty }a_{n},0,0, \dots \Biggr),\\& T_{2}(a_{n})=(0,a_{1},a_{2},\dots ),\\& T_{3}(a_{n})=(a_{1},a_{2},a_{3}, \dots ). \end{aligned}$$

We can easily see that $T_{1}\in \mathcal{M}(l^{1})$, $T_{2}\in S\mathcal{D}(l^{1})$, $T_{3}\in \mathcal{D}(l^{1})$, $T^{*}_{1}:l^{\infty}\to l^{\infty}$ is defined by $T^{*}_{1}((b_{n}))=(b_{1},b_{1},\dots )$, and therefore $T^{*}_{1}(\chi _{\{1\}})=T^{*}_{1}(e_{1})=(1,1,\dots )$. Hence $T_{1}\notin S\mathcal{D}(l^{1})$, and $T^{*}_{2}:l^{\infty}\to l^{\infty}$ is the left shift operator defined as $T^{*}_{2}((b_{n}))=(b_{2},b_{3},\dots )$. Hence $T^{*}_{2}(e_{1})=(0,0,\dots )$ and $\int T^{*}_{2}(e_{1})\,d\mu =0<\int _{\mathbb{N}}e_{1}\,d\mu =1$, and thus $T_{2}\notin \mathcal{D}(l^{1})$.

Now the following theorem brings together two areas of Sects. 2.1 and 2.2; it is an equivalent condition for majorization notion based on decreasing rearrangement (Definition 2.2) by using semidoubly stochastic operators (Definition 2.5).

Theorem 2.9

([4, Corollary 2.10])

Let X be a σ-finite measure space, and let f, g be nonnegative functions belonging to $L^{1}(X)$. Then $g \prec f$ if and only if there is a sequence $(S_{n})_{n\in \mathbb{N}}$ in $S\mathcal{D}( L^{1}(X))$ such that $S_{n}f \to g$ in $L^{1}(X)$.

3 Some results on semidoubly stochastic operators

3.1 Method of constructing semidoubly stochastic operators

In the following theorem, we present a method for constructing doubly (and semidoubly) stochastic operators. To this end, we assume that $(X,\mathcal{A},\mu )$ is a σ-finite measure space. Also, we assume that $\{A_{n}; n\in \mathbb{N}\}$ is a sequence of measurable sets with finite measure such that $X=\bigcup_{n=1}^{\infty }A_{n}$ and $A_{n}\subseteq A_{n+1}$ for each $n\in \mathbb{N}$. We denote by $\mathcal{S}$ the set of all measurable simple functions in $L^{1}(X)$.

Theorem 3.1

Let $D:\mathcal{S}\to L^{1}(X)$ be a linear function. Then D has a unique extension to a doubly stochastic operator if and only if D is nonnegative and the following equalities hold for each $E\in \mathcal{A}$ with $\mu (E)<\infty$:

$$\begin{aligned}& \int _{X} D\chi _{E}\,d\mu =\mu (E), \end{aligned}$$

(7)

$$\begin{aligned}& \lim_{n\to \infty} \int _{X} \chi _{E} D\chi _{A_{n}}\,d\mu =\mu (E). \end{aligned}$$

(8)

Proof

Let $\varphi =\sum_{i=1}^{n} a_{i} \chi _{E_{i}}$ be a simple function. Then according to (7),

$$ \int _{X} D\varphi \,d\mu = \int _{X} D\Biggl(\sum_{i=1}^{n} a_{i}\chi _{E_{i}}\Biggr)\,d\mu =\sum _{i=1}^{n} a_{i} \int _{X} D\chi _{E_{i}}\,d\mu =\sum _{i=1}^{n} a_{i}\mu (E_{i})= \int _{X} \varphi \,d\mu . $$

(9)

Therefore, for each $\varphi \in S$,

$$ \Vert D \varphi \Vert _{1}= \int _{X} \vert D\varphi \vert \,d\mu \leq \sum _{i=1}^{n} \vert a_{i} \vert \int _{X} D\chi _{E_{i}}\,d\mu =\sum _{i=1}^{n} \vert a_{i} \vert \mu (E_{i})= \int _{X} \vert \varphi \vert \,d\mu = \Vert \varphi \Vert _{1}. $$

Hence the operator $D:\mathcal{S}\to L^{1}(X)$ is bounded, and $\|D\|\leqslant 1$. Since $\mathcal{S}$ is dense in $L^{1}(X)$, D has a unique extension to $L^{1}(X)$, which we again denote by D.

Due to (9), for each $f\in L^{1}(X)$,

$$ \int _{X} Df \,d\mu = \int _{X} f \,d\mu , $$

and by (8), for each $\varphi \in \mathcal{S}\subset L^{1}(X)\cap L^{\infty}(X)$, we have

$$\begin{aligned} \int _{X} D^{*}\varphi \,d\mu &= \lim _{n\to \infty} \int _{X} D^{*} \varphi \chi _{A_{n}}\,d\mu = \lim_{n\to \infty} \int _{X}\varphi D \chi _{A_{n}}\,d\mu \end{aligned}$$

(10)

$$\begin{aligned} &= \lim_{n\to \infty}\sum_{i=1}^{m} a_{i} \int _{X} \chi _{E_{i} } D \chi _{A_{n}}\,d\mu = \sum_{i=1}^{m} a_{i}\mu (E_{i}) = \int _{X} \varphi \,d\mu . \end{aligned}$$

(11)

Hence equalities in (10) hold for each $f\in S\subset L^{1}(X)\cap L^{\infty}(X)$, which means that

$$ \int _{X} D^{*}f \,d\mu = \int _{X} f \,d\mu . $$

Therefore $D:\mathcal{S}\to L^{1}(X)$ has a unique extension to a doubly stochastic operator on $L^{1}(X)$. The reverse is easily verifiable. □

It is obvious that by the same proof with a very slight modification we have the the following semidoubly stochastic version of Theorem 3.1.

Theorem 3.2

Let $S:\mathcal{S}\to L^{1}(X)$ be a linear function. Then S has a unique extension to a semidoubly stochastic operator if and only if S is nonnegative and the following inequalities hold for each measurable set E with $\mu (E)<\infty$:

$$\begin{aligned}& \int _{X} S\chi _{E}\,d\mu =\mu (E), \end{aligned}$$

(12)

$$\begin{aligned}& \lim_{n\to \infty} \int _{X} \chi _{E} S\chi _{A_{n}}\,d\mu \leq \mu (E). \end{aligned}$$

(13)

In this part, we want to introduce another class of semidoubly stochastic operators on $L^{1}(X)$ given in Proposition 3.3, which we will use for characterization of majorization relation. From now on, unless otherwise stated, we will assume that $(X,\mathcal{A},\mu )$ is a σ-finite measure space and $P:=\{ A_{n} : n\in \mathbb{N}\}$ is a disjoint family of measurable sets such that $X=\bigcup_{n\in \mathbb{N}}A_{n}$ and $0<\mu (A_{n})<+\infty$ for all $n\in \mathbb{N}$. Then we easily see that the map $\Phi _{P}:L^{1}(X)\to l^{1}$ given by

$$\begin{aligned} \Phi _{P}(f)=\biggl( \int _{A_{1}} f \,d\mu , \int _{A_{2}} f \,d\mu , \dots , \int _{A_{n}} f \,d\mu , \dots \biggr), \quad f\in L^{1}(X), \end{aligned}$$

(14)

is a bounded linear map. Let ${\Phi}^{*}_{P} : l^{\infty}\to L^{\infty}(X)$ be its adjoint. Then for $(a_{n})\in l^{\infty}$ and $f\in L^{1}(X)$,

$$ \bigl\langle f, \Phi ^{*}_{P}(a_{n})\bigr\rangle =\bigl\langle \Phi _{P} (f), (a_{n}) \bigr\rangle =\sum _{n=1}^{\infty }a_{n} \int _{A_{n}} f \,d\mu = \int _{X} f \sum_{n=1}^{\infty }a_{n} \chi _{A_{n}}\,d\mu . $$

Therefore

$$ \Phi ^{*}_{P}(a_{n})=\sum _{n=1}^{\infty }a_{n} \chi _{A_{n}}\quad \forall (a_{n})\in l^{\infty}. $$

Similarly, if $\Psi _{P}:l^{1}\to L^{1}(X)$ is defined by

$$\begin{aligned} \Psi _{P}(a_{n})=\sum _{n=1}^{\infty }\frac{a_{n} }{\mu (A_{n})}\chi _{A_{n}},\quad (a_{n})\in l^{\infty}, \end{aligned}$$

(15)

then $\Psi _{P}$ is also a bounded linear map with the adjoint $\Psi _{P}^{*} : L^{\infty}(X)\to l^{\infty}$ defined for all $g\in L^{\infty}(X)$ by

$$ \Psi _{P}^{*}(g)=\biggl(\frac{1}{\mu (A_{1})} \int _{A_{1}} g \,d\mu , \frac{1}{\mu (A_{2})} \int _{A_{2}} g \,d\mu , \dots , \frac{1}{\mu (A_{n})} \int _{A_{n}} g \,d\mu , \dots \biggr). $$

The next proposition shows that $\Psi _{P}\Phi _{P}$ can be considered as a class of semidoubly stochastic operators, which plays an important role in the characterization of majorization.

Proposition 3.3

The bounded linear map $G_{P} : L^{1}(X)\to L^{1}(X)$ defined by $G_{P}=\Psi _{P}\Phi _{P}$ is a doubly stochastic operator and thus a semidoubly stochastic operator.

Proof

Using the above considerations, we have

$$ \forall f\in L^{1}(X),\quad G_{P}(f)=\Psi _{P}\bigl( \Phi _{P}(f)\bigr)=\sum_{n=1}^{ \infty } \biggl(\frac{1 }{\mu (A_{n})} \int _{A_{n}} f \,d\mu \biggr)\chi _{A_{n}}, $$

and

$$ \forall g\in L^{\infty}(X),\quad G^{*}_{P}(g)=\Phi _{P}^{*}\bigl(\Psi _{P}^{*}(g)\bigr)= \sum_{n=1}^{\infty }\biggl(\frac{1 }{\mu (A_{n})} \int _{A_{n}} g \,d\mu \biggr) \chi _{A_{n}}. $$

Clearly, $G_{P}$ is a positive operator. For $f\in L^{1}(X)$, using the monotone convergence theorem, we have

$$\begin{aligned} \int _{X} G_{P}(f)\,d\mu & = \sum _{n=1}^{\infty }\biggl( \frac{1 }{\mu (A_{n})} \int _{A_{n}} f \,d\mu \biggr) \int _{X} \chi _{A_{n}}\,d\mu \\ & = \sum_{n=1}^{\infty} \int _{A_{n}} f \,d\mu = \int _{X} f \,d\mu . \end{aligned}$$

If $g\in L^{\infty}(X)\cap L^{1}(X)$, then

$$ \int _{X} \bigl\vert G^{*}_{P}(g) \bigr\vert \,d\mu \leq \sum_{n=1}^{\infty } \frac{1 }{\mu (A_{n})} \biggl\vert \int _{A_{n}} g \,d\mu \biggr\vert \int _{X} \chi _{A_{n}}\,d\mu \leq \int _{X} \vert g \vert \,d\mu < \infty , $$

that is, $G^{*}_{P}(g)\in L^{1}(X)$. Similarly,

$$ \int _{X} G^{*}_{P}(g)\,d\mu = \sum _{n=1}^{\infty } \int _{A_{n}} g \,d \mu = \int _{X} g \,d\mu . $$

By definition $G_{P}\in \mathcal{D}(L^{1}(X))$, and thus $G_{P}\in S\mathcal{D}(L^{1}(X))$. □

Let D be a semidoubly stochastic operator on $l^{1}$. Then as a general case of the previous proposition, we want to prove that $\Psi _{P} D \Phi _{P}$ is a semidoubly stochastic operator on $L^{1}(X)$. To this end, we need the extra assumption on the measure space X that

$$ \inf \bigl\{ \mu (A_{n}), n=1, 2 , \dots \bigr\} =a\ne 0. $$

To see why we need this extra assumption in the general case (Theorem 3.4), first, let $D:l^{1}\to l^{1}$ be a doubly stochastic operator, and let $\Phi _{P}:L^{1}(X)\to l^{1}$ and $\Psi _{P} : l^{1}\to L^{1}(X)$ be the maps defined in (14) and (15), corresponding to the family of measurable subsets $A=\{A_{n}: n\in \mathbb{N}\}$ of X such that $X=\bigcup_{n\in \mathbb{N}} A_{n}$ and $0<\mu (A_{n})<\infty$ for all $n\in \mathbb{N}$. If $G_{D}:=\Psi _{P}D\Phi _{P}$, then for each $f\in L^{1}(X)$,

$$ G_{D}(f)= \sum_{n=1}^{\infty } \frac{1 }{\mu (A_{n})}\Biggl(\sum_{j=1}^{ \infty }d_{jn} \int _{A_{j}} f \,d\mu \Biggr)\chi _{A_{n}}. $$

Hence

$$\begin{aligned} \int _{X} G_{D}(f)\,d\mu =& \sum _{n=1}^{\infty }\sum_{j=1}^{\infty }d_{jn} \int _{A_{j}} f \,d\mu =\sum_{j=1}^{\infty} \int _{A_{j}} f \,d\mu \Biggl(\sum_{n=1}^{ \infty }d_{jn} \Biggr)=\sum_{j=1}^{\infty} \int _{A_{j}} f \,d\mu \\ =& \int _{X} f \,d\mu . \end{aligned}$$

So to prove that $G_{D}$ is a doubly stochastic (resp., semidoubly stochastic) operator, we should obtain the same equality relation (resp., inequality relation) for the map $G^{*}_{D}=\Psi ^{*}_{A} D^{*}\Phi ^{*}_{A}$.

Theorem 3.4

Let (X, $\mathcal{A}$, μ) be σ-finite measure space and suppose

$$ \inf \bigl\{ \mu (A_{n}),n=1, 2 , \ldots \bigr\} =a\ne 0. $$

Then

• (i) if $D:l^{1}\to l^{1}$ is semidoubly stochastic operator, then the map $G_{D}:L^{1}(X)\to L^{1}(X)$ defined by $G_{D}=\Psi _{P} D \Phi _{P}$ is a semidoubly stochastic operator;

• (ii) if $D:L^{1}(X)\to L^{1}(X)$ is a semidoubly stochastic operator, then the map $G_{D}:=\Phi _{P} D \Psi _{P} : l^{1}\to l^{1}$ is also semidoubly stochastic on the sequence space $l^{1}$.

Proof

(i) For $g\in L^{1}(X)\cap L^{\infty}(X)$,

$$ G^{*}_{D}(g)= \sum_{n=1}^{\infty } \Biggl(\sum_{i=1}^{\infty }d_{in} \frac{1 }{\mu (A_{i})} \int _{A_{i}} g \,d\mu \Biggr)\chi _{A_{n}} $$

Therefore, using the assumption, we have

$$\begin{aligned} \int _{X} G^{*}_{D}(g)\,d\mu &= \sum _{n=1}^{\infty }\Biggl(\sum _{i=1}^{ \infty }d_{in}\frac{1 }{\mu (A_{i})} \int _{A_{i}} g \,d\mu \Biggr)\mu (A_{n})= \sum _{n=1}^{\infty }\sum_{i=1}^{\infty }d_{in} \int _{A_{i}} g \,d\mu \\ &= \sum_{i=1}^{\infty }\sum _{n=1}^{\infty }d_{in} \int _{A_{i}} g \,d \mu \leq \sum_{i=1}^{\infty } \int _{A_{i}} g \,d\mu = \int _{X} g \,d\mu . \end{aligned}$$

(ii) For each $n\in \mathbb{N}$,

$$\begin{aligned} De_{n} =& \Phi _{P} G \Psi _{P} (e_{n})\\ =&\Phi _{P} \biggl(\frac{1}{\mu (A_{n})} G (\chi _{A_{n}})\biggr)= \biggl( \frac{1}{\mu (A_{n})} \int _{A_{1}} G (\chi _{A_{n}})\,d\mu , \dots , \frac{1}{\mu (A_{n})} \int _{A_{n}} G (\chi _{A_{n}})\,d\mu , \dots \biggr) \end{aligned}$$

Hence

$$\begin{aligned} \sum_{m=1} ^{\infty }\langle De_{n} , e_{m}\rangle =& \frac{1}{\mu (A_{n})}\sum_{m=1} ^{\infty} \int _{A_{m}} G (\chi _{A_{n}})\,d\mu =\frac{1}{\mu (A_{n})} \int _{X} G (\chi _{A_{n}})\,d\mu \\ =&\frac{1}{\mu (A_{n})} \int _{X} \chi _{A_{n}}\,d\mu =1. \end{aligned}$$

Similarly,

$$\begin{aligned} \sum_{n=1} ^{\infty }\langle De_{n} , e_{m}\rangle &= \sum_{n=1} ^{ \infty } \frac{1}{\mu (A_{n})} \int _{A_{m}} G (\chi _{A_{n}})\,d\mu \\ & = \sum _{n=1} ^{\infty }\frac{1}{\mu (A_{n})} \int _{A_{n}} G^{*} (\chi _{A_{m}})\,d\mu \\ &\le \frac{1}{a}\sum_{n=1} ^{\infty} \int _{A_{n}} G^{*} (\chi _{A_{m}})\,d\mu = \frac{1}{a} \int _{X} G^{*} (\chi _{A_{m}})\,d\mu \\ &\leq \frac{1}{a} \int _{X} \chi _{A_{m}}\,d\mu = \frac{1}{a}\mu (A_{m}) \leq 1. \end{aligned}$$

 □

3.2 Equiintegrability

Bahrami et al. characterized semidoubly stochastic operators on $L^{1}(X)$ when X is a σ-finite measure space by using the notion of majorization as follows.

Theorem 3.5

([4, Theorem 2.4])

Let $(X,\mu )$ be a σ-finite measure space, and let $S : L^{1}(X)\to L^{1}(X)$ be a positive bounded linear operator. Then for every nonnegative integrable function f on X, $Sf\prec f$ if and only if $S\in S\mathcal{D}(L^{1}(X))$.

In this part, we prove that when $(X,\mu )$ is a σ-finite measure space, $S_{f}=\{Sf: S\in S\mathcal{D}\}$ is equiintegrable. This immediately will give us the relatively weak compactness of $S_{f}$ when $(X,\mu )$ is a probability measure.

Definition 3.6

A set $\mathcal{F}\subset L^{1}(X)$ is said to be equiintegrable if for every $\epsilon >0$, there exists $\delta >0$ such that for every $E\subset X$ with $\mu (E)<\delta$,

$$ \int _{E} \vert f \vert \,d\mu < \epsilon\quad \forall f\in \mathcal{F}. $$

To prove the equiintegrability of $S_{f}=\{Sf: S\in S\mathcal{D}(L^{1}(X))\}$ when $f\in L^{1}(X)$, we first prove the following lemma.

Lemma 3.7

Let $f,g\in L^{1}(X)$ be nonnegative functions with $f\prec g$, and let $g\in L^{\infty}(X)$. Then for each $E\in X$ with $\mu (E)<\infty$,

$$ \int _{E} f \,d\mu \leq \Vert g \Vert _{\infty}\mu (E). $$

Proof

By definition, for each $s>0$,

$$ d_{g}(s)=\mu \bigl(\bigl\{ x\in X; g(x)>s \bigr\} \bigr). $$

Therefore for each $s\geq \|g\|_{\infty}$, $d_{g}(s)=0$. On the other hand, by Theorem 2.3, since $f\prec g$, for each $s>0$, we have

$$ \int _{s}^{\infty} d_{f}(\tau )\,d\tau \leq \int _{s}^{\infty }d_{g}( \tau )\,d\tau . $$

Hence for each $s\geq \|g\|_{\infty}$,

$$ \int _{s}^{\infty }d_{f}(\tau )\,d\tau \leq \int _{s}^{\infty }d_{g}( \tau )\,d\tau =0, $$

and thus for each $s\geq \|g\|_{\infty}$, $d_{f}(\tau )=0$. On the other hand,

$$\begin{aligned} d_{f\chi _{E}}(s) &=\mu \bigl(\bigl\{ x\in X, f(x)\chi _{E}(x)>s \bigr\} \bigr) \\ &=\mu \bigl(\bigl\{ x\in E, f(x)>s\bigr\} \bigr) \\ &\leq \min \bigl\{ \mu (E), d_{f}(s)\bigr\} , \end{aligned}$$

Therefore for each $s\geq \|g\|_{\infty}$, $d_{f\chi _{E}}(s)=0$, and thus

$$ \int _{E}f \,d\mu = \int _{X}f\chi _{E}\,d\mu = \int _{0}^{\infty }d_{f \chi _{E}}(s)\,ds\leq \mu (E) \Vert g \Vert _{\infty}. $$

 □

Now we are ready for the following key theorem.

Theorem 3.8

Let $f\in L^{1}(X)$ be a nonnegative function. Then $S_{f}$ is equiintegrable.

Proof

For $\epsilon >0$, take a nonnegative function $g\in L^{1}(X)\cap L^{\infty}(X)$ such that $g\leq f$ and $\|f-g\|<\frac {\epsilon}{2}$. Now set $\delta = \frac {\epsilon}{2\|g\|_{\infty}}$. Tthen for each measurable subset $E\subset X$ with $\mu (e)<\delta$ and each $S\in S\mathcal{D}(L^{1}(X))$,

$$\begin{aligned} \int _{E} \vert Sf \vert \,d\mu &\leq \int _{E} \vert Sf-Sg \vert \,d\mu + \int _{E} \vert Sg \vert \,d\mu \\ &\leq \int _{X}S \vert f-g \vert \,d\mu + \int _{E} Sg \,d\mu \\ &= \Vert f-g \Vert + \int _{E} Sg \,d\mu < \frac {\epsilon}{2}+ \int _{E} Sg \,d\mu . \end{aligned}$$

Since $Sg\prec g$ by Lemma 3.7, $\int _{E}Sg\,d\mu \leq \|g\|_{\infty}\mu (E)$, and therefore

$$ \int _{E} \vert Sf \vert \,d\mu \leq \frac {\epsilon}{2} + \Vert g \Vert _{\infty}\mu (E) < \frac {\epsilon}{2} + \Vert g \Vert _{\infty}\delta < \epsilon . $$

 □

For probability space $(X,\mu )$, which has many applications in quantum sciences, the following theorem provides lots of significant equivalence conditions for the equiintegrability of $S_{f}=\{Sf: S\in S\mathcal{D}(L^{1}(X))\}$.

Theorem 3.9

([2, Theorem 5.2.9])

Let $(X, \mu )$ is a probability measure space. Let $\mathcal{F}$ be a bounded set in $L^{1}(X)$. Then the following conditions on $\mathcal{F}$ are equivalent:

• (i) $\mathcal{F}$ is relatively weakly compact;

• (ii) $\mathcal{F}$ is equiintegrable;

• (iii) $\mathcal{F}$ does not contain a basic sequence equivalent to the canonical basis of $l^{1}$;

• (iv) $\mathcal{F}$ does not contain a complemented basic sequence equivalent to the canonical basis of $l^{1}$;

• (v) for every sequence $(A_{n})_{n=1}^{\infty}$ of disjoint measurable sets,

  $$ \lim_{n\rightarrow \infty}\sup_{f\in F} \int _{A_{n}} \vert f \vert \,d\mu =0. $$

Without loss of generality, Theorem 3.9 holds under the more general assumption that $(X,\mu )$ is a finite measurable space.

Now we have the following corollary of Theorem 3.8.

Corollary 3.10

Let $f\in L^{1}(X)$ be a nonnegative function, where $(X,\mu )$ is a finite measure space. Then $S_{f}$ is relatively weakly compact.

Proof

For each $S\in S\mathcal{D}(L^{1}(X))$ and a fixed $f\in L^{1}(X)$, by the definition of $S\mathcal{D}(L^{1}(X))$ we have

$$ \int _{X} Sf \,d\mu = \int _{X} f \,d\mu . $$

Then for each $S\in S\mathcal{D}(L^{1}(X))$ and a fixed $f\in L^{1}(X)$, we have $\|Sf\|_{1}=\|f\|_{1}$, so $S_{f}$ is bounded. Therefore by Theorem 3.8 and the equivalence of items (i) and (ii) in Theorem 3.9, $S_{f}$ is relatively weakly compact. □

4 Characterization of majorization on $L^{1}(X)$

The goal of this section is to answer Mirsky’s question, which is a full characterization of majorization, by using semidoubly stochastic operators.

We first need to consider the relation between sublinear functions and convex functions with majorization on $L^{1}(X)$. We also need to discuss the relation of integral operators and semidoubly stochastic operators. Finally, as a conclusion, we present the main result of our work, which is a full characterization of majorization on $L^{1}(X)$ when X is a σ-finite measure space.

4.1 Sublinear and convex functions

In a matrix space, Dahl proved an equivalent condition for matrix majorization using sublinear functionals (i.e., convex and positively homogeneous maps). Moein et al. proved the following one-side extension of Dahl’s result.

Theorem 4.1

([16, Theorem 3.8])

If $(X, \mu )$ is σ-finite measure space and $f,g\in L^{1}(X)$ are such that f is matrix majorized by g (i.e., there exists a Markov operator M that $f=Mg$), then

$$ \int _{X}\varphi (f)\,d\mu \leq \int _{X} \varphi (g)\,d\mu $$

for all sublinear functionals $\varphi :\mathbb{R}\to \mathbb{R}^{+}$.

Since $S\mathcal{D}(L^{1}(X))\subset \mathcal{M}(L^{1}(X))$, we simply have the following corollary.

Corollary 4.2

Let $(X,\mu )$ be a σ-finite measure space, and let $f,g\in L^{1}(X,\mu )$. Suppose there exists $S\in S\mathcal{D}(L^{1}(X))$ such that $f=Sg$. Then for all sublinear functionals $\varphi :\mathbb{R}\to \mathbb{R}^{+}$,

$$ \int _{X}\varphi (f)\,d\mu \leq \int _{X} \varphi (g)\,d\mu . $$

(16)

As the following counterexample shows, the converse of the corollary does not hold.

Example 4.3

Let $X=[0,\infty )$, let μ be the Lebesgue measure, and let $f=3\chi _{[0,1)}+\frac {1}{2}\chi _{[1,2)}$ and $g=2\chi _{[0,2)}$. Then neither $f\prec _{w} g$, nor $g\prec _{w} f$. Now let $\varphi :\mathbb{R}\to \mathbb{R}^{+}$ be an arbitrary sublinear function. Then by its convexity and positively homogeneous property we have

$$ \varphi (f)=3\varphi (\chi _{[0,1)})+\frac {1}{2}\varphi (\chi _{[1,2)}), \quad \text{and}\quad \varphi (f)=2\varphi (\chi _{[0,2)}). $$

Then by the linearity of integral we have

$$ \int _{X}\varphi (f)\,d\mu =3 \int _{0}^{1}\varphi (1)\,d\mu +\frac {1}{2} \int _{1}^{2}\varphi (1)\,d\mu \leq 2 \int _{0}^{2}\varphi (1)= \int _{X} \varphi (g)\,d\mu . $$

Because each sublinear function is convex, a natural question can arise: is the converse of Corollary 4.2 true for convex functions? Or can we characterize the majorization relation in a σ-finite measure space with inequality in Corollary 4.2 based on convex functions? Chong answered this question positively for weak majorization with the restriction that convex functions have to be increasing.

Theorem 4.4

([6, Theorem 2.1])

Let $(X,\mu )$ be infinite measure space, and let $f,g\in L^{1}(X)$ be nonnegative. Then $f\prec _{w} g$, that is,

$$ \int _{0}^{s} f^{\downarrow }\,dm \leq \int _{0}^{s} g^{\downarrow }\,dm, \quad 0\leq s\leq \infty , $$

if and only if for all nonnegative increasing convex functions $\phi :\mathbb{R}^{+}\to \mathbb{R}^{+}$ with $\phi (0)=0$,

$$ \int _{X} \phi (f)\,d\mu \leq \int _{X} \phi (g)\,d\mu . $$

4.2 Semidoubly stochastic and integral operators

In this part, we first recall the definitions of stochastic integral operators and doubly stochastic integral operators.

Definition 4.5

([16])

A measurable function $K:X\times Y\to [0,\infty )$ is called a stochastic kernel if $\int _{X} K(x,y)\,d\mu (x)=1$ for almost all $y\in Y$, and it is called a doubly stochastic kernel if, in addition, $\int _{Y} K(x,y)\,d\nu (y)=1$ for almost all $x\in X$.

Definition 4.6

([16])

An integral operator $A:L^{1}(Y)\to L^{1}(X)$ defined by $Ag=\int _{Y}K(x,y) g(y)\,d\nu (y)$ is said to be a stochastic integral operator (resp., doubly stochastic integral operator) if $K(x,y)$ is a stochastic kernel (resp., doubly stochastic kernel).

Each stochastic integral operator is a Markov operator (stochastic operator), and each doubly stochastic integral operator is a doubly stochastic operator. However, by a simple example of the identity operator, which is a doubly stochastic operator, it is clear that the converses of both statements are false. In spite of that, in [16], it is proven that a Markov (resp., doubly stochastic) operator $D:L^{1}(Y)\to L^{1}(X)$ on a finite-dimensional subspace F of $L^{1}(Y)$ can be approximated by stochastic (resp., doubly stochastic) integral operators when $(X,\mu )$ and $(Y,\nu )$ are a σ-finite (resp., finite) measure spaces.

Theorem 4.7

([16, Theorem 3.7])

If $(X,\mu )$ and $(Y,\nu )$ are finite measure spaces, then D as a doubly stochastic operator from $L^{1}(Y)$ to $L^{1}(X)$ on a finite subspace F of $L^{1}(Y)$ can be approximated by doubly stochastic integral operators.

By using the fact from [4, Proposition 2.6] that for a finite measure space $(X,\mu )$, the sets of semidoubly stochastic operators and doubly stochastic operators coincide, a natural question, which is our first aim in this section, arises: Can we extend Theorem 4.7 to σ-finite measure spaces? For this purpose, we need a class of integral operators between stochastic integral operators and doubly stochastic integral operators.

Definition 4.8

A measurable functional $K:X\times Y\to [0,\infty )$ is called a semidoubly stochastic kernel if $\int _{X} K(x,y)\,d\mu (x)=1$ for almost all $y\in Y$ and $\int _{Y} K(x,y)\,d\nu (y)\leq 1$ for almost all $x\in X$.

Definition 4.9

An integral operator $A:L^{1}(Y)\to L^{1}(X)$ defined by $Ag=\int _{Y}K(x,y) g(y)\,d\nu (y)$ is said to be a semidoubly stochastic integral operator if $K(x,y)$ is a semidoubly stochastic kernel.

Lemma 4.10

([16, Lemma 3.5])

Let $(X,\mu )$ be a σ-finite measure space, and let F be a finite-dimensional subspace of $L^{1}(X)$. Then there exists a sequence of partitions $\{P_{n}\}_{n=1}^{\infty}$ of X into disjoint sets of finite measure such that $\{{G_{P}}_{n}f\}_{n=1}^{\infty}$ converges to f in the $L^{1}$ norm for all $f\in F$.

Theorem 4.11

Let $(X,\mu )$ be σ-finite measure space, and let $S:L^{1}(X)\to L^{1}(X)$ be a semidoubly stochastic operator. Then there exists a sequence of semidoubly stochastic integral operators on $L^{1}(X)$ that converge to S on a finite-dimensional subspace F of $L^{1}(X)$.

Proof

(The proof is a combination of the results of [4] and [16] after a modification in terms of semidoubly stochastic operators). From Proposition 3.3 we have that

$$ G_{P}(f)=\Psi _{P}\bigl(\Phi _{P}(f)\bigr)=\sum _{n=1}^{\infty }\biggl( \frac{1 }{\mu (A_{n})} \int _{A_{n}} f \,d\mu \biggr)\chi _{A_{n}},\quad f \in L^{1}(X), $$

is a semidoubly stochastic operator on $L^{1}(X)$. Since the composition of two semidoubly stochastic operators is a semidoubly stochastic operator, $G_{P}S$ is a semidoubly stochastic operator on $L^{1}(X)$, and we will show that it is a semidoubly stochastic integral operator. Since F is a finite-dimensional subspace of $L^{1}(X)$, the forward image $S(F)$ is a finite-dimensional subspace of $L^{1}(X)$ as well, and the result follows from Lemma 4.10.

Fix $x\in X$. There exists a unique $A_{k}\in P$ such that $x\in A_{k}$. The boundedness follows from

$$\begin{aligned} \bigl\vert \bigl(G_{P}S(f)\bigr) (x) \bigr\vert =& \biggl\vert \frac{1}{\mu (A_{k})} \int _{A_{k}}\bigl(Sf(t)\bigr)\,d\mu (t) \biggr\vert \\ \le & \frac{1}{\mu (A_{k})} \int _{A_{k}} \bigl\vert (Sf) (t) \bigr\vert \,d\mu (t) \\ \le & \frac{1}{\mu (A_{k})} \int _{X} \bigl\vert (Sf) (t) \bigr\vert \,d\mu (t) \\ \le & \frac{1}{\mu (A_{k})} \int _{X} \bigl\vert f(x) \bigr\vert \,d\mu (x) \\ = &\frac{1}{\mu (A_{k})} \Vert f \Vert _{1}. \end{aligned}$$

Hence by the Riesz representation theorem there exists a nonnegative function $h_{x}\in L^{\infty}(X)$ such that $(G_{P}S(f))(x)=\int _{X} f(y)h_{x}(y)\,d\mu$. Now let $K_{P}(x,y)=h_{x}(y)$ for $x,y\in X$. Then

$$ (G_{P}Sf) (x)= \int _{X} K_{P}(x,y)f(y)\,d\mu (y). $$

(17)

Since $G_{P}S$ is a semidoubly stochastic operator, then it preserves the integral

$$ \int _{X} \bigl((G_{P}S)f\bigr)\,d\mu = \int _{X} f\,d\mu \quad \forall f\in L^{1}(Y), $$

and by Fubini’s theorem

$$\begin{aligned} \int _{X} f(x)\,d\mu (x) =& \int _{X} \bigl((G_{P}S)f\bigr) (x)\,d\mu (x) \\ =& \int _{X} \int _{X} K_{P}(x,y)f(y)\,d\mu (y)\,d\mu (x) \\ =& \int _{X} f(y) \biggl( \int _{X} K_{P}(x,y)\,d\mu (x)\biggr)\,d\mu (y). \end{aligned}$$

Therefore $\int _{X} K_{P}(x,y)\,d\mu (x)=1$ for almost all $y\in X$.

Now let $\{B_{n};n\in \mathbb{N}\}$ be an increasing sequence of measurable sets such that $X= \bigcup_{n\in \mathbb{N}}B_{n}$ and $\mu (B_{n})<\infty$ for each $n\in \mathbb{N}$. Because of the equality of dual pairing

$$ \langle G_{P}S\chi _{B_{n}},\chi _{A} \rangle = \bigl\langle \chi _{B_{n}},\bigl((G_{P}S)^{*} \chi _{A}\bigr) \bigr\rangle , $$

we have

$$ \int _{X}(G_{P}S\chi _{B_{n}})\chi _{A}\,\mathrm{d}\mu (x)= \int _{X} \chi _{B_{n}}\bigl((G_{P}S)^{*} \chi _{A}\bigr)\,\mathrm{d}\mu (x)\rightarrow \int _{X}(G_{P}S)^{*}\chi _{A}\,\mathrm{d}\mu (x), $$

and by equation (17) for $f=\chi _{B_{n}}$ for each n, we have

$$\begin{aligned} \int _{X}(G_{P}S\chi _{B_{n}})\chi _{A}\,\mathrm{d}\mu (x) =& \int _{X} \int _{X} K_{P}(x,y)\chi _{B_{n}}\,d\mu (y) \chi _{A}\,d\mu (x)\\ =&\mu (A) \int _{B_{n}}K_{P}(x,y)\,\mathrm{d}\mu (y). \end{aligned}$$

Then taking the limit as $n\to \infty$, we have

$$ \int _{X}(G_{P}S)^{*}\chi _{A}\,\mathrm{d}\mu (y)=\mu (A) \int _{X} K_{P}(x,y) \,\mathrm{d}\mu (y). $$

Since $G_{P}S$ is semidoubly stochastic, according to its definition, for every measurable subset A of finite measure, we have

$$ \int _{X}(G_{P}S)^{*}\chi _{A}\,\mathrm{d}\mu \leq \mu (A). $$

Then

$$ \mu (A) \int _{X} K_{P}(x,y)\,\mathrm{d}\mu (y)\leq \mu (A). $$

Therefore $\int _{X} K_{P}(x,y)\,d\mu (y)\leq 1$ for almost all $x\in X$, and hence $G_{P}S$ is a semi-doubly stochastic integral operator. □

The following theorem is a combination of two well-known results for finite measurable spaces by Chong and Day.

Theorem 4.12

Let $f,g\in L^{1}(X,\mu )$ and $\mu (X)<\infty$. Then the following statements are equivalent:

• (a) $f\prec g$.

• (b) For all convex functions $\varphi :\mathbb{R}\to \mathbb{R}$,

  $$ \int _{X}\varphi (f)\,d\mu \leq \int _{Y} \varphi (g)\,d\mu . $$

• (c) There exists a doubly stochastic operator D on $L^{1}(X)$ such that $f=Dg$.

Proof

The equivalence of (a) and (b) is proved by Chong [6, Theorem 2.5], and the equivalence of (a) and (c) is proved by Day [7, Theorem 4.9]. □

Now we can summarize all equivalent conditions for majorization relation in the case of σ-finite measure spaces. Theorem 4.13, as an extension of Day–Chong’s result, Theorem 4.12, or Hardy, Littlewood, and Pólya’s result to σ-finite measure spaces states a strong relation between majorization based on decreasing rearrangement, semidoubly stochastic operators, semidoubly stochastic integral operators, and integral inequalities for convex functions. We should also mention an open problem in [13] (the converse of Corollary II.13), which can be solved by the following theorem using the counting measure on $\mathbb{N}$.

Theorem 4.13

Let $(X,\mu )$ be σ-finite measure space, and let $f,g\in L^{1}(X,\mu )$ be nonnegative. Then the followings statements are equivalent:

• (1) $g\prec f$;

• (2) There exists a sequence $(S_{k})_{n=1}^{\infty}$ of semidoubly stochastic operators on $L^{1}$ such that $S_{k}f\to g$ in $L^{1}(X)$;

• (3) There exists a sequence $(I_{n})_{n=1}^{\infty}$ of semidoubly stochastic integral operators on $L^{1}$ such that $I_{n}f\to g$ in $L^{1}(X)$.

• (4) $\int _{X} \phi (g)\,d\mu \leq \int _{X} \phi (f)\,d\mu$ for all increasing convex functions $\phi :\mathbb{R}^{+}\to \mathbb{R}^{+}$ such that $\phi (0)=0$ and $\int _{X} g \,d\mu =\int _{X} f \,d\mu$.

• (5) $\int _{X} (g-u)^{+}\,d\mu \leq \int _{X} (f-u)^{+}\,d\mu$ for each positive real number u, and $\int _{X} g \,d\mu =\int _{X} f \,d\mu$.

Proof

(1), (2), and (4) are equivalent by Theorems 2.9 and 4.4.

(2) implies (3): There exists a sequence of semidoubly stochastic operators $\{S_{k}\}_{k=1}^{\infty}$ such that $S_{k}g\to f$ in $L^{1}(X)$ by Theorem 4.11. Therefore, for each $k\in \mathbb{N}$, $S_{k}$ as a semidoubly stochastic operator on a finite-dimensional subspace $F=\text{span}\{g\}$ of $L^{1}(X)$ can be approximated by semidoubly stochastic integral operators. This means that for each $k\in \mathbb{N}\mathbbm{,}$ there exists a sequence of semidoubly stochastic integral operators $\{I_{n}\}_{n=1}^{\infty}$ on $L^{1}(X)$ that converge to $S_{k}$ on F. Therefore there exists a sequence of semidoubly stochastic integral operators $I_{n}$ such that $I_{n}g\to f$ in $L^{1}(x)$.

(3) implies (2): We show that each semidoubly stochastic integral operator is a semidoubly stochastic operator.

Let $A:L^{1}(X)\to L^{1}(X)$ be a semidoubly stochastic integral operator. Then by definition for each $f\in L^{1}(X)$ defined by a semidoubly kernel $K(x,y)$ as

$$ Ag= \int _{X}K(x,y)g(y)\,d\mu (y) $$

we have to show that $A\in S\mathcal{D}(L^{1}(X))$, that is, for all $f\in L^{1}(X)$ and $E\in \mathcal{A}$ with $\mu (E)<\infty$,

$$ \int _{X} Af \,d\mu = \int _{X} f \,d\mu , $$

(18)

and also

$$ \int _{X} A^{*}\chi _{E}\,d\mu \leq \mu (E). $$

(19)

We obtain (18) simply from the fact that $\int _{X} K(x,y)\,d\mu (x)=1$ for almost all $y\in X$. To get (19), first, note that for every measurable subset E of finite measure,

$$ \lim_{n\to \infty} \int _{X}(A\chi _{B_{n}})\chi _{E}\,\mathrm{d} \mu (x)= \int _{X}\chi _{B_{n}}\bigl(A^{*}\chi _{E}\bigr)\,\mathrm{d}\mu (x)= \int _{X}A^{*} \chi _{E}\,\mathrm{d}\mu (x). $$

Since

$$ A\chi _{B_{n}}= \int _{X} K(x,y)\chi _{B_{n}}\,d\mu (y) $$

and $\int _{Y} K(x,y)\,d\nu (y)\leq 1$ for almost all $x\in X$, taking the limit as $n\to \infty$, we obtain (19).

(4) implies (5) simply by choosing the convex function ϕ given by $\phi (g)=\max \{g-u,0\}=(g-u)^{+}$ for $g\in L^{1}(X)$.

Finally, by Theorems 2.4 and 2.3, (5) implies (1). □

Remark 4.14

As we can see in Definition 2.2, the nonnegativity of functions in $L^{1}(X)$ is a necessary condition for defining majorization based on decreasing rearrangement. However, Theorem 4.13 allows us to extend the definition of majorization to arbitrary, not necessarily nonnegative functions.