Study On Certain Preserving Problems Of Partial Isometries On B(H)

The study on preserving problems is one of the most important research topics of the theory of operator algebras. As a special class of operators, partial isometries play a great role in the theory of both Polar Decomposition and von Neumann algebras. Let B(H) and PI(H) be the set of all bounded linear operators and of all partial isometries on a complex Hilbert space H, respectively. In this paper, it is mainly considered the characterization of maps which regard partial isometries as invariant preserver. The main results are as follows.1. Let dim H≥3. Suppose that φ is a bijective transformation on PI(H) which preserves the partial ordering and the orthogonality between partial isometries in both directions. Then φ can be written in one of the following forms:(1) There exist two unitary operators U and V on H such that φ(R)=URV, VR ∈PI(H);(2) There exist two unitary operators U and V on H such that φ(R)=UR*V, (?)R∈PI(H);(3) There exist two anti-unitary operators U and V on H such that φ(R)= URV, VR ∈PI(H);(4) There exist two anti-unitary operators U and V on H such that φ(R)= UR*V, VR ∈PI(H).2. Let dim H≥ 3. Then an additive surjection φ on B(H) preserves partial isometries in both directions if and only if one of the following assertions holds:(1) There exist two unitary operators U and V on H such that φ(X)=UXV , VX ∈B(H);(2) There exist two unitary operators U and V on H such that φ(X)=UX*V, (?)X∈B(H);(3) There exist two anti-unitary operators U and V on H such that φ(X)= UXV, (?)X∈B(H);(4) There exist two anti-unitary operators U and V on H such that φ(X)= UX*V, (?)X∈B(H).3. Assume that φ is a linear surjection on B(H) with φ(I)=I. Then φ preserves operator pairs whose products are nonzero partial isometries if and only if one of the following assertions holds:(1) There exists a unitary operator U on H such that φ(X)=UXU*, (?)X∈ B(H);(2) There exists an anti-unitary operator U on H such that φ(X)=UX*U*, (?)X∈B(H).