  In this paper, we introduce operators that are represented by upper
triangular $2\times 2$ block matrices whose entries satisfy some algebraic
constraints. We call them Brownian-type operators of class $\mathcal Q,$
briefly operators of class $\mathcal Q.$ These operators emerged from the study
of Brownian isometries performed by Agler and Stankus via detailed analysis of
the time shift operator of the modified Brownian motion process. It turns out
that the class $\mathcal Q$ is closely related to the Cauchy dual subnormality
problem which asks whether the Cauchy dual of a completely hyperexpansive
operator is subnormal. Since the class $\mathcal Q$ is closed under the
operation of taking the Cauchy dual, the problem itself becomes a part of a
more general question of investigating subnormality in this class. This issue,
along with the analysis of nonstandard moment problems, covers a large part of
the paper. Using the Taylor spectrum technique culminates in a full
characterization of subnormal operators of class $\mathcal Q.$ As a
consequence, we solve the Cauchy dual subnormality problem for expansive
operators of class $\mathcal Q$ in the affirmative, showing that the original
problem can surprisingly be extended to a class of operators that are far from
being completely hyperexpansive. The Taylor spectrum approach turns out to be
fruitful enough to allow us to characterize other classes of operators
including $m$-isometries. We also study linear operator pencils associated with
operators of class $\mathcal Q$ proving that the corresponding regions of
subnormality are closed intervals with explicitly described endpoints.
