We describe some invariant or reducing subspaces for a concave operator T on a complex Hilbert space which satisfies the regularity condition Δ_TT=Δ_T^1/2TΔ_T^1/2, where Δ_T = T*T-I. We consider those subspaces on which T acts as a 2-isometry and show that T has some Brownian type properties on them. Among other, the Brownian unitary part and the Brownian isometric (reducing or invariant) parts are investigated. In the case when T is a Brownian operator or even a general 2-isometry we determine the Brownian unitary reducing parts on which T has the maximal covariance.

The authors are grateful to the anonymous referee for his thorough reading of the manuscript and his very helpful comments which led to the improvement of the presentation. The second named author was supported by a Project financed from Lucian Blaga University of Sibiu research grants LBUS-IRG-2018-03.