Suppose $A$ is a bounded operator on an infinite dimensional Hilbert space $H$ and $I$ is the identity operator.
Question (1): What conditions $A$ should have so that $(I-A)H$ is closed in $H$?
Clearly, if $A$ is onto this is true. Also, if $A$ is compact this is true. However, I am interested in the case where $A$ is a non-surjective isometry.
Question (2): What conditions a non-surjective isometry $A$ (with a fixed point) should have so that $(I-A)H$ is closed in $H$?
I don't expect this to be true for any isometry even if isometries have closed range.
Many thanks!