  In this paper, exponential stability of discrete-time Markov jump linear
systems (MJLSs) with the Markov chain on a Borel space $(\Theta,
\mathcal{B}(\Theta))$ is studied, and bounded real lemmas (BRLs) are given. The
work generalizes the results from the previous literature that considered only
the Markov chain taking values in a countable set to the scenario of an
uncountable set and provides unified approaches for describing exponential
stability and $H_{\infty}$ performance of MJLSs. This paper covers two kinds of
exponential stabilities: one is exponential mean-square stability with
conditioning (EMSSy-C), and the other is exponential mean-square stability
(EMSSy). First, based on the infinite-dimensional operator theory, the
equivalent conditions for determining these two kinds of stabilities are shown
respectively by the exponentially stable evolutions generated by the
corresponding bounded linear operators on different Banach spaces, which turn
out to present the spectral criteria of EMSSy-C and EMSSy. Furthermore, the
relationship between these two kinds of stabilities is discussed. Moreover,
some easier-to-check criteria are established for EMSSy-C of MJLSs in terms of
the existence of uniformly positive definite solutions of Lyapunov-type
equations or inequalities. In addition, BRLs are given separately in terms of
the existence of solutions of the $\Theta$-coupled difference Riccati equation
for the finite horizon case and algebraic Riccati equation for the infinite
horizon case, which facilitates the $H_{\infty}$ analysis of MJLSs with the
Markov chain on a Borel space.
