The aim of the paper is to give a principle which enables one to transfer inequalities with semi-inner products to inequalities containing positive semidefinite symmetric bilinear operators with values in a vector lattice.
For \(N\) in \(\mathbb N\), let \({\mathcal H}(\mathbb {R}^{N})\) denote the vector lattice of all continuous functions \(\phi : \mathbb {R}^{N} \rightarrow \mathbb N\) which satisfy \(\phi(\lambda t) = \lambda\phi(t)\) \((t\in \mathbb {R}^{N}\), \(0< \lambda \in \mathbb R)\).
For a finite collection \(x_1,\dots ,x_N \) in \(E\) and \(\phi \in {\mathcal H}(\mathbb {R}^{N})\), \(\widehat{\phi}(x_1,\dots ,x_N)\) is defined on \(E\) if there exists \(y\in E\) such that \(w(y) = \phi( w(x_1),\dots ,w(x_n))\) for every real valued lattice homomorphism \(w\) on the vector sublattice of \(E\) generated by \(\{x_1,\dots ,x_N\}\). Denote by \({\mathcal H} \vee (\mathbb {R}^{N})\) and \({\mathcal H} \wedge (\mathbb {R}^{N})\), respectively, the set of all continuous sublinear and continuous superlinear functions on \(\mathbb R^{N}\).
If \(E\) is a uniformly complete vector lattice and \(x_1,\dots ,x_N \in E\). Then \(\widehat{\phi}\{x_1,\dots ,x_N\}\) exists for every \(\phi\in {\mathcal H} (\mathbb R^{N})\). The map \( \phi \rightarrow \widehat{\phi}(x_1,\dots ,x_N)\), \(\phi \in {\mathcal H}(\mathbb R^{N})\) is a unique lattice homomorphism from \({\mathcal H}(\mathbb R^{N})\) into \(E\) with \(\widehat{dx_j}(x_1,\dots x_N) = x_j\) for \(1,\dots ,N\). Denote \(I = \{ 1,\dots ,n\}\) and \(J = \{1,\dots ,m\}\) and \(N = nm\). For a fixed bijection \(\sigma\) from \(I\times J\) onto \(\{1,\dots ,N\}\), consider positive homogeneous continuous mappings \(\Phi,\Phi':\mathbb {R}^{N} \rightarrow \mathbb {R}^{k}\), \(\Psi,\Psi' :\mathbb {R}^{N} \rightarrow \mathbb {R}^{j}\) and denote (for \(t\in \mathbb {R}^{N}\), \(u\in E^N)\):
\[ \begin{aligned} \Phi(t) &= (\phi_1(t),\dots ,\phi_k(t)), \quad \Phi'(t) = (\phi'_1(t),\dots ,\phi'_k(t)),\\ \Phi(t) \Phi'(t) &= (\phi_1(t)\phi'_1(t),\dots ,\phi_k(t)\phi'_k(t)),\\ \widehat{\Phi}(u)\circ\widehat{\Phi}(u) &= (\widehat{\phi_1}(u) \circ\widehat{\phi'_1}(u),\dots ,\widehat{\phi_k}(u)\circ \widehat{\phi'_k}(u)), \end{aligned} \]
and analogously for \(\Psi(t)\Psi'(t)\) and \(\widehat\Psi(u)\circ\Psi'(u)\).
The main theorem of the paper is as follows.
Theorem (Transfer Principle). Let \(E,F\) be uniformly complete vector lattices, let \(X\) be a real vector space and \(x_i,y_j\in X\) \((i\in I, j\in J)\). Let \(\phi \in {\mathcal H}(\mathbb {R}^{N})\wedge \mathbb {R}^{N}\) and \(\psi\in {\mathcal H}(\mathbb {R}^{N})\vee \mathbb {R}^{N}\). If for any semi-inner product \((\cdot,\cdot)\) on \(X\), the inequality
\[ \psi(\Psi(t)\Psi'(t)) \leq \phi(\Phi(t) \Phi'(t)) \]
holds with \(t = (t_1,\dots ,t_N)\in \mathbb R^N\), \(t_{\sigma(i,j)} = (x_i,x_j)\), then for any positive semidefinite symmetric bilinear operator \(\left\langle\cdot,\cdot\right\rangle \) from \(X \times X\) to \(E\) and any positive orthosymmetric bilinear operator \(\circ : E\times E \rightarrow F \), the inequality
\[ \widehat{\phi} (\widehat{\Psi}(u)\circ\widehat{\Psi'}(u)) \leq \widehat{\phi}(\widehat{\Phi}(u) \circ \widehat{\Phi'}(u)) \]
holds with \(u = (u_1,\dots ,u_N) \in E^N\), \(u_{\sigma(i,j)} = \left\langle x_i,y_j\right\rangle\), \((i,j)\in I\times J\).
There are plenty of examples.