Validity of the simplified operator E-norm formula (13) for arbitrary positive operators G

Determine whether the simplified variational characterization of the operator E-norm given in equation (13) remains valid for every positive (semidefinite) operator G on a separable Hilbert space H. Specifically, for any VG-bounded linear operator A: H → H' and any energy bound E > 0, ascertain whether the norm defined by the supremum over vectors y ∈ D(VG) with ||y|| ≤ 1 and ||VG y||^2 ≤ E equals the operator E-norm defined via states p with Tr(Gp) ≤ E (equations (9), (11), and (12).

Background

The operator E-norm is introduced for VG-bounded operators A: H → H' in terms of supremums over states with bounded energy (equation (9)) and equivalent forms (equations (11) and (12)). When G is a discrete operator, these expressions simplify to a vector-based supremum (equation (13)).

The paper notes that extending this simplified formula (13) beyond discrete G to arbitrary positive operators G is not established. A footnote further indicates that the conjectured equality of the norm from (13) with the operator E-norm is equivalent to concavity of the function E ↦ ||A||_{GE} for any VG-bounded operator A, highlighting the significance of the question for norm equivalences and continuity analyses.