Abstract
1. Let I be a Hilbert space. A projection is a bounded idempotent linear operator. Orthogonal projections form a proper subclass of the class of all projections. A problem is to get a condition for there to exist commuting projections E1, * E,n with the same ranges as given orthogonal projections P1, * , P,n respectively. This will be settled in terms of properties of the sublattice, generated by P1, * * , P,, in the lattice of all orthogonal projections. In case n = 2, commuting projections with minimum norms are constructed. Another problem is to find commuting projections E1, * * , E,n in a Hilbert space S, containing I as a subspace, such that Pjx =PEjx for xCeSD, j= 1, 2, ... , n, where P is the orthogonal projection from 9 onto t. This will be proved to be always possible.
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