Let H be a Hilbert space and P,Q orthogonal projections on B(H).It is proved that Mθ(P,Q)is non-empty if and only if the dimension of the intersection of PH and(I-Q)H is the same as the dimension of the intersection of QH and(I-P)H,where Mθ(P,Q)is the set of projections on Hwhich are within distance of sinθ from P and cosθ from Q.The opera-tor matrix decomposition form of Halmos'two projection theory is applied to prove the sufficiency.On the other hand,a linear bijection between the intersection of PH and(I-Q)H and the intersection of QH and(I-P)H is constructed to give the necessity.
projection operatormatrix decompositionunitary element
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