PIETOOLS 2020a: User Manual

The manual’s Theorem 9 asserts that the 4-PI operator built from the data (P,Q1,Q2,R0,R1,R2) defined via the block matrix T is positive semidefinite when T<0 (negative definite) and g(s)≥0, see the statement and formulas (3.1) and (3.6) in Chapter 3.4 . However, expanding the quadratic form shows that ⟨[x;y],P[·][x;y]⟩ equals ∫ g(s) ξ(s)^T T ξ(s) ds for a suitable stacked ξ(s). Hence the operator inherits the sign of T: T⪰0 ⇒ operator ⪰0, T⪯0 ⇒ operator ⪯0. Under T<0, the operator is nonpositive—not nonnegative. Positivity is obtained by replacing T with −T or assuming T⪰0. Therefore, the paper’s sign is flipped; the model’s correction matches the correct Gram representation and sign logic.