A quadratic operator equation and a non-commutative uncertainty relation

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In this paper, we explore equivalent conditions for the positivity of a quadratic equation P ( x ) = A x 2 + Bx + C with operator coefficients. Specifically, we show that A x 2 + 2 Bx + C ≥ 0 for all x ∈ R if and only if A , C ≥ 0 , B = B ∗ , and there exists a self-adjoint operator H such that [ A + C + H B + i ( A − C ) B + i ( A − C ) A + C − H ] ≥ 0. Also, we show that if A and B are two unital C ∗ -algebras, Φ : A → B is a unital tracial positive linear map, and ϕ : A → A is a linear map satisfying ϕ ( X ) ∗ ϕ ( X ) ≤ ϕ ( X ∗ X ) for all X ∈ A , then V Ψ ( A ) ♯ V Ψ ( B ) ≥ ± 1 2 Ψ ( [ A , B ] ) , where A and B are self-adjoint elements in A , Ψ = Φ ∘ ϕ , and V Ψ ( X ) = Ψ ( X ∗ X ) − Ψ ( X ∗ ) Ψ ( X ) denotes the generalized variance of an observable X.