Let A be a (non-unital, in general) C*-algebra with center Z(M(A)) of its multiplier algebra and let \(\{ X, \langle .,. \rangle \}\) be a full Hilbert A-module. Then any bijective bounded module morphism T, for which every norm-closed A-submodule of X is invariant, is of the form \(T=d \cdot \textrm{id}_X\) where \(d \in Z(M(A))\) is invertible. As an example of a merely injective bounded module operator with that preserver property serves \(T =d \cdot \textrm{id}_X\) where \(|d| \in Z(M(A))\) has a positive spectrum, but not bounded away from zero. The same assertions are true if the restriction on the C*-submodules to be norm-closed is dropped. From a different point of view, for two given strongly Morita equivalent C*-algebras A and B and a Hilbert B-A bimodule \(\{ X, \langle .,. \rangle \}\) with faithful compact right action of B, for any two two-sided norm-closed ideals \(I \in A\) , \(J \in B\) , any full compatible norm-closed Hilbert J-I sub-bimodule of X is invariant for any left bounded B-module operator and any right bounded A-module operator. So these subsets of submodules of X cannot rule out any bounded module operator as a non-preserver of that subset collection, however any single element of this subset collection is preserved by any bounded module operator on X. For any B-A imprimitivity bimodule both the C*-valued inner product values are always preserved by bijective bounded module operators T on X iff \(T= u \cdot \textrm{id}_X\) for a unitary element \(u\in Z(M(A))\) .
