Curves of projections and operator inequalities

CORACH, GUSTAVO; ESTEBAN ANDRUCHOW

Operator theory: themes and variations

Theta Foundation

Lugar: Bucharest; Año: 2018; p. 25 - 44

Given two orthogonal projections P and Q in a complex Hilbertspace such thatR(P) ∩ N(Q) = N(P) ∩ R(Q) = {0},there exists a unique selfadjoint operator XP,Q, which is P-codiagonal, has normat most π/2 and satisfies that the curveδ(t) = eitXP,Q Pe?itXP,Qjoins δ(0) = P and δ(1) = Q, and has minimal length among all piecewise smoothcurves of projections joining P and Q. We use this fact to obtain operator inequalities in particular examples. Namely, given projections P, Q as above, anda path P(t), t ∈ [a, b], joining them, then one hasZ bakddtP(t)kdt ≥ kXP,Qk,where the right hand term is the length of δ.