A Characterization Of The Product Of Two Orthogonal Projections And The Mapping Preserving Range Inclusion

The product of operators not only plays an important role in operator theory, but also in other many different areas as mathematical physics, signal processing, numerical analysis, and so on. Recently, Corach and Maestripieri have characterized the sets of the product of all two orthogonal projections in [1]. Thus, it is interesting to consider the problem that what the necessary and sufficient condition is for an operator as a product of two orthogonal projections. Besides, we know range inclusion gives several equivalent statements for an operator which can be represented as a product of two bounded operators. Motivated by this, we definite the R order invariant subspaces on operator algebras and characterize these subspaces, then get some results about mapping persevering range inclusion on operator algebras.Firstly, we give a necessary and sufficient condition for an operator which can be represented as a product of two orthogonal projections by the technique of block operator matrix. Let A∈B(H)be a contraction operator, the necessary and sufficient condition for A as a product of two orthogonal projections is PAAPA=PAAA*PA, where the projection PA is from H to R (A).Secondly, we mainly study the characterizations of R order invariant subspaces and the properties of the mapping persevering range inclusion. Prove that if M∈B(H) is an R order invariant and norm closed subspace, then there exists a projection P∈P(H) such that M∩K(H)=PK(H) and MWOT=PB(H).Lastly, we get some results about mapping persevering range inclusion on operator algebras. Let H be an infinite Hilbert space and φ be a WOT-continuous bijective mapping from B(H) to B(H) persevering range inclusion in both directions, then there are invertible operators A, B in B(H) such that φ(T)=ATB, for any T∈B(H). Besides, let A and B be nonzero operators. If△(X)=AXB for any X∈B(H), then△preserves range inclusion if and only if B is surjective.