Corollary 30 ([Logarithmic modular cuts]). For a prepared pair with (M,N)≤γ , put L_P=Δ_P , P=M,N . There is an absolute constant C such that, for every a∈ , 1_ (-∞,a-1/2] (L_N) 1_ [a+1/2,∞) (L_M) ≤Cγ. The same estimate holds with M,N …

The upper-left block of the projection onto a closed operator graph is (1+A^*A)^ -1 . Applied to, this gives (1+c^2Δ_P)^ -1 . With g(s)=(1+e^s)^ -1 , Proposition 29 ([Uniform graph comparison]) consequently yields g(L_M-a)-g(L_N-a)≤Cγ (a∈). Let p=1_ (-∞,a-1/2] (L_N) and q=1_ [a+1/2,∞) (L_M) , and regard pq as an operator qH→pH . Let A and B be the restrictions of g(L_N-a) and g(L_M-a) to those respective subspaces. Then A≥g(-1/2)1, B≤g(1/2)1, (pq)-(pq)B≤Cγ. For D=A(pq)-(pq)B , integration of the derivative of e^ -tA (pq)e^ tB gives pq=∫_0^∞e^ -tA D e^ tB \,dt, pqg(-1/2)-g(1/2). The denominator is a fixed positive number. Reversing the two algebras proves the other estimate.