On restrictions of operators on Hilbert space to a half-space

This paper is a sequel to Jung (Bull Aust Math Soc 97: 133–140, 2018) that was originally written concurrently with Jung (Bull Aust Math Soc 97: 133–140, 2018). In that paper we transferred the discussions in Androulakis (Int Eq Op Th 65: 473–484, 2009) and Popov (J Funct Anal 265: 257–265, 2013) concerning almost invariant half-spaces for operators on complex Banach spaces to the context of operators on Hilbert space, and we gave slightly simpler proofs of the main results in Androulakis (Int Eq Op Th 65: 473–484, 2009) and Popov (J Funct Anal 265: 257–265, 2013) in that context. In the present paper we discuss a consequence of the main construction in Jung (Bull Aust Math Soc 97: 133–140, 2018) for the restriction to a half-space of a certain large class of operators on Hilbert space.