Jessen’s type inequality and exponential convexity for positive \(C_{0}\)-semigroups

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.

Thus \(\psi:V\rightarrow V\) is positive if and only if \(\vert \psi f\vert \leq\psi \vert f\vert \) holds for any \(f\in V\).

Here \(B(V)\) denotes the space of all bounded linear operators defined on a Banach space V.

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.

Example 1

Let \(X:= C_{0}(\mathbb{R})\), \(\{Z(t)\}_{t\geq0}\) be the left shift semigroup defined on X and ϕ taking the mirroring along y-axis. The identity function does not contain a compact support and therefore is not in X. We now take a bell-shaped curve like \(f(x):= e^{-x^{2}}\), \(x\in\mathbb{R}\). Then f is positive, \(\phi f=f\), and \(Z(t)(\phi f)\) has maximum at \(x=-t\), and it is between 0 and 1 elsewhere. On the other hand, \(\phi(Z(t)f)\) has a maximum at \(s=t\) and it is immediate that we cannot compare the two functions in the usual ordering. See Figure 1(a).

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.

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.

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.

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.