Abstract
Let B ( X ) be the space of all bounded linear operators on a Banach space X and let Lat A be the lattice of invariant subspaces of the operator A ∈ B ( X ) . We characterize some maps Φ : B ( X ) → B ( X ) with one of the following preserving properties: Lat ( Φ ( A ) + Φ ( B ) ) = Lat ( A + B ) , or Lat ( Φ ( A ) Φ ( B ) ) = Lat ( AB ) , or Lat ( Φ ( A ) Φ ( B ) + Φ ( B ) Φ ( A ) ) = Lat ( AB + BA ) , or Lat ( Φ ( A ) Φ ( B ) Φ ( A ) ) = Lat ( ABA ) , or Lat ( [ Φ ( A ) , Φ ( B ) ] ) = Lat ( [ A , B ] ) .
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