Abstract
In this paper, we present the necessary and sufficient conditions for which the given pair of closed subspaces has a pair of non-trivial principal invariant subspaces. Using this we show that every pair of closed subspaces with or has a pair of non-trivial principal invariant subspaces. Also we prove that the pair of closed subspaces with has a pair of non-trivial principal invariant subspaces if and only if Similar results are proved for non-degenerate principal invariant subspaces.
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