5.2 The upper verifier passes on average
For a square-integrable \(b\) with values in a separable space, the Gram operator \(J = \int _\Omega b(x) b(x)^* \, d\mu (x)\) is positive and of finite trace (Lemma 3.3(a)), so its upper potential is defined.
Let \((\Omega , \mu )\) be a measure space, let \(H\) be a nonzero separable complex Hilbert space, let \(b : \Omega \to H\) be square-integrable with injective Gram operator \(J = \int _\Omega b(x) b(x)^* \, d\mu (x)\), let \(B\) be a strictly positive operator on \(H\), and let \(\zeta {\gt} 0\). Then \(x \mapsto U_B^{\zeta }(b(x))\) is integrable and
Both summands of the verifier are quadratic forms in \(b(x)\), bounded by a multiple of \(\| b(x)\| ^2\), so the verifier is integrable. Write \(W = B^{-1}\) and \(X = (B + \zeta \cdot J)^{-1}\), and let \(E = \Psi _J(B) - \Psi _J(B + \zeta \cdot J)\) be the gain of Lemma 5.2(a). By Lemma 3.3(b) for \(Q = X J X\) and \(Q = X\) the average equals \(\operatorname {Tr}(J X J X) / E + \Psi _J(B + \zeta \cdot J)\). Since \(X \preceq W\) by the antitonicity of the inverse, as in Lemma 2.1(c), we have \(J X J \preceq J W J\), and Lemma 2.6(c) bounds the numerator by \(\operatorname {Tr}(J W J X) = E / \zeta \). So the first summand is at most \(1/\zeta \), and the second is strictly smaller than \(\Psi _J(B)\) by Lemma 5.2(a).