Automorphisms of Hilbert space effect algebras

Let H be a Hilbert space and E (H) the effect algebra on H. A bijective map $\varphi :E\left(H\right)\to E\left(H\right)$ is called an ortho-order automorphism of E (H) if for every $A,B\in E\left(H\right)$ we have $A⩽B⟺\varphi \left(A\right)⩽\varphi \left(B\right)$ and $\varphi \left(A^{\perp }\right)=\varphi {\left(A\right)}^{\perp }$. The classical theorem of Ludwig states that every such ϕ is of the form $\varphi \left(A\right)=UA{U}^{\ast }$, $A\in E\left(H\right)$, for some unitary or antiunitary operator U. It is also known that each bijective map on E (H) preserving order and coexistency in both directions is of the same form. Can we improve these two theorems by relaxing the bijectivity assumption and/or replacing the above preserving properties by the weaker assumptions of preserving above relations in one direction only and still get the same conclusion? For both characterizations of automorphisms of effect algebras we will prove the optimal versions and give counterexamples showing the optimality of the obtained results. (paper)