1 Introduction and preliminaries
A significant theory regarding inequalities and exponential convexity for real-valued functions has been developed [
1,
2]. The intention to generalize such concepts for the
\(C_{0}\)-semigroup of operators is motivated from [
3].
In the present article, we shall derive a Jessen type inequality and the corresponding adjoint inequality, for some \(C_{0}\)-semigroup and the adjoint semigroup, respectively.
The notion of Banach lattice was introduced to get a common abstract setting, within which one could talk about the ordering of elements. Therefore, the phenomena related to positivity can be generalized. It had mostly been studied in various types of spaces of real-valued functions, e.g. the space \(C(K)\) of continuous functions over a compact topological space K, the Lebesque space \(L^{1}(\mu)\) or even more generally the space \(L^{p}(\mu)\) constructed over measure space \((X,\Sigma,\mu)\) for \(1\leq p\leq\infty\). We shall use without further explanation the terms: order relation (ordering), ordered set, supremum, infimum.
First, we shall go through the definition of a vector lattice.
The axiom
\(O_{1}\), expresses the translation invariance and therefore implies that the ordering of an ordered vector space
V is completely determined by the positive part
\(V_{+}=\{f\in V: f\geq0\}\) of
V. In other words,
\(f\leq g\) if and only if
\(g-f\in V_{+}\). Moreover, the other property,
\(O_{2}\), reveals that the positive part of
V is a convex set and a cone with vertex 0 (mostly called the
positive cone of
V).
-
An ordered vector space V is called a vector lattice, if any two elements \(f,g\in V\) have a supremum, which is denoted by \(\sup(f,g)\) and an infimum denoted by \(\inf(f,g)\).
It is trivially understood that the existence of supremum of any two elements in an ordered vector space implies the existence of supremum of finite number of elements in V. Furthermore, \(f\geq g\) implies \(-f\leq-g\), so the existence of finite infima is therefore implied.
-
Some important quantities are defined as follows:
$$\begin{aligned}& \sup(f,-f) = \vert f\vert \quad (\mathit{absolute\ value\ of\ }f), \\& \sup(f,0) = f^{+}\quad (\mathit{positive\ part\ of\ }f), \\& \sup(-f,0) = f^{-} \quad (\mathit{negative\ part\ of\ }f). \end{aligned}$$
-
Some compatibility axiom is required between norm and order. This is given in the following short way:
$$ \vert f\vert \leq \vert g\vert \quad \mbox{implies}\quad \Vert f\Vert \leq \Vert g\Vert . $$
(1)
The norm defined on a vector lattice is called a lattice norm.
Now, we are in a position to define a Banach lattice in a formal way.
A linear mapping
ψ from an ordered Banach space
V into itself is
positive (denoted
\(\psi\geq0\)) if
\(\psi f\in V_{+}\), for all
\(f\in V_{+}\). The set of all positive linear mappings forms a convex cone in the space
\(L(V)\) of all linear mappings from
V into itself, defining the natural ordering of
\(L(V)\). The absolute value of
ψ, if it exists, is given by
$$\vert \psi \vert (f)=\sup\bigl\{ \psi h: \vert h\vert \leq f\bigr\} \quad (f\in V_{+}). $$
Thus \(\psi:V\rightarrow V\) is positive if and only if \(\vert \psi f\vert \leq\psi \vert f\vert \) holds for any \(f\in V\).
An operator
A on
V satisfies the positive minimum principle if for all
\(f\in D(A)_{+}=D(A)\cap V_{+}\),
\(\phi\in V_{+}'\)
$$ \langle f,\phi\rangle= 0 \quad \mbox{implies}\quad \langle Af,\phi\rangle \geq0. $$
(2)
Here \(B(V)\) denotes the space of all bounded linear operators defined on a Banach space V.
Let
\(\{Z(t)\}_{t\geq0}\) be the strongly continuous positive semigroup, defined on a Banach lattice
V. The positivity of the semigroup is equivalent to
$$\bigl\vert Z(t)f\bigr\vert \leq Z(t)\vert f\vert ,\quad t\geq0, f\in V. $$
Here, for positive contraction semigroups
\(\{Z(t)\}_{t\geq0}\), defined on a Banach lattice
V, we have
$$\bigl\Vert \bigl(Z(t)f\bigr)^{+}\bigr\Vert \leq\bigl\Vert f^{+}\bigr\Vert ,\quad \mbox{for all }f\in V. $$
Reference [
4] guarantees the existence of the strongly continuous positive semigroups and positive contraction semigroups on a Banach lattice
V, with some conditions imposed on the generator. A very important among them is that it must always satisfy (
2).
A Banach algebra
X, with the multiplicative identity element
e, is called the
unital Banach algebra. We shall call the strongly continuous semigroup
\(\{Z(t)\}_{t\geq0}\) defined on
X a
normalized semigroup whenever it satisfies
$$ Z(t) (e)=e, \quad \mbox{for all }t>0. $$
(3)
The notion of normalized semigroup is inspired by normalized functionals [
2]. The theory presented in the next section is defined on such semigroups of positive linear operators defined on a Banach lattice
V.
2 Jessen’s type inequality
In 1931, Jessen [
5] gave the generalization of the Jensen’s inequality for a convex function and positive linear functionals. See [
6], p.47. We shall prove this inequality for a normalized positive
\(C_{0}\)-semigroup and a convex operator defined on a Banach lattice.
Throughout the present section, V will always denote a unital Banach lattice algebra, endowed with an ordering ≤.
Let \(\mathfrak{D}_{c}(V)\) denotes the set of all differentiable convex functions \(\phi:V\rightarrow V\).
The existence of an identity element and condition (
3), imposed in the hypothesis of the above theorem, is necessary. We shall elaborate all this by the following examples.
3 Adjoint Jessen’s type inequality
In the previous section, a Jessen type inequality has been derived for a normalized positive \(C_{0}\)-semigroup \(\{Z(t)\}_{t\geq0}\). This gives us the motivation toward finding the behavior of its corresponding adjoint semigroup \(\{Z^{\ast}(t)\}_{t\geq0}\) on \(V^{\ast}\). As the theory for dual spaces gets more complicated, we do not expect to have the analogous results. One may ask for a detailed introduction toward a part of the dual space \(V^{\ast}\), for which an adjoint of Jessen’s type inequality makes sense.
For a strongly continuous positive semigroup
\(\{Z(t)\}_{t\geq0}\) on a Banach space
X, by defining
\(Z^{\ast}(t)=(Z(t))^{\ast}\) for every
t, we get a corresponding adjoint semigroup
\(\{Z^{\ast}(t)\}_{t\geq0}\) on the dual space
\(X^{\ast}\). In [
8], the result is obtained that the adjoint semigroup
\(\{ Z^{\ast}(t)\}_{t\geq0}\) fails in general to be strongly continuous. The investigation [
9], shows that
\(\{Z^{\ast}(t)\}_{t\geq0}\) acts in a strongly continuous way on
$$ X^{\bigodot} := \Bigl\{ x^{\ast}\in X^{\ast}: \lim_{t\rightarrow0}\bigl\Vert Z^{\ast}(t)x^{\ast}-x^{\ast}\bigr\Vert =0\Bigr\} . $$
This is the maximal such subspace on
\(X^{\ast}\). The space
\(X^{\bigodot}\) was introduced by Philips in 1955 and later has been studied extensively by various authors. At the present moment, we do not necessarily require the strong continuity of the adjoint semigroup
\(\{Z^{\ast}(t)\} _{t\geq0}\) on
\(X^{\ast}\).
If X is an ordered vector space, we say that a functional \(x^{\ast}\) on X is positive if \(x^{\ast}(x)\geq0\), for each \(x\in X\). By the linearity of \(x^{\ast}\), this is equivalent to \(x^{\ast}\) being order preserving; i.e.
\(x\leq y\) implies \(x^{\ast}(x)\leq x^{\ast}(y)\). The set P of all positive linear functionals on X is a cone in \(X^{\ast}\).
We are mainly interested in the study of the space \(V^{\ast}\), where in our case V is a Banach lattice algebra. Let us consider the regular ordering among the elements of \(V^{\ast}\), i.e.
\(v_{1}^{\ast}\geq v_{2}^{\ast}\), whenever \(v_{1}^{\ast}(v)\geq v_{2}^{\ast}(v)\), for each \(v\in V\).
Consider the convex operator (
4). In the case of equality,
F is simply a linear operator and the adjoint
F can be defined as above. But how can it be defined in the other case? This question has already been answered.
In [
10], some kind of adjoint has been associated to a nonlinear operator
F. In fact, this is possible for Lipschitz continuous operators only. Consider the Banach space
\(\mathfrak{Lip}_{0}(X,Y)\) of all Lipschitz continuous operators
\(F : X \rightarrow Y\) satisfying
\(F(\theta) = \theta\), equipped with the norm
$$ [F]_{\mathrm{Lip}}= \sup_{x_{1}\neq x_{2}}\frac{\Vert F(x_{1})-F(x_{2})\Vert }{\Vert x_{1}-x_{2}\Vert },\quad x_{1},x_{2}\in X. $$
Here
\(\theta\in X\) is the identity. It is easy to see that the space
\(L(X, Y )\) of all bounded linear operators from
X to
Y is a closed subspace of
\(\mathfrak{Lip}_{0}(X,Y)\). In particular, we set
$$ \mathfrak{Lip}_{0}(X,\mathbb{K}):= X^{\sharp} $$
and call
\(X^{\sharp}\) the pseudo-dual space of
X; this space contains the usual dual space
\(X^{\ast}\) as a closed subspace.
This is, of course, a straightforward generalization of (
6); in fact, for linear operators
L we have
\(L^{\sharp}|_{Y^{\ast}} = L^{\ast}\);
i.e. the restriction of the pseudo-adjoint to the dual space is the classical adjoint.
For the sake of convenience, we shall denote the adjoint of the operator F by \(F^{\ast}\) throughout the present section. Either it is a classical adjoint or the pseudo-adjoint (depending upon the operator F).
Similarly, the considered dual space of the vector lattice algebra V will be denoted by \(V^{\ast}\), which can be the intersection of the pseudo-dual and classical dual spaces in the case of a nonlinear convex operator.
4 Exponential convexity
In this section we shall define the exponential convexity of an operator. Moreover, some complex structures, involving the operators from a semigroup, will be proved to be exponentially convex.
For
\(U\subseteq V\), let us assume that
\(F:U\rightarrow V\) is continuously differentiable on
U,
i.e. the mapping
\(F': U\rightarrow\mathfrak{L}(V)\), is continuous. Moreover,
\(F''(f)\), will be a continuous linear transformation from
V to
\(\mathfrak{L}(V)\). A bilinear transformation
B defined on
\(V\times V\) is symmetric if
\(B(f,g)=B(g,f)\) for all
\(f,g\in V\). Such a transformation is
positive definite (nonnegative definite), if for every nonzero
\(f\in V\),
\(B(f,f)>0\) (
\(B(f,f)\geq0\)). Then
\(F''(f)\) is symmetric wherever it exists. See [
7], p.69.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.