Markov Chains: Analytic and Monte Carlo Computations

1.3 Natural duality: algebraic approach

An algebraic study based on the natural duality between the space of signed measures and the space of bounded functions will provide some structural results. These results are quite complete for finite state spaces . The complete study for arbitrary discrete will be done later using probabilistic techniques. A reader for which this is the main interest may go directly to Section 1.4.

1.3.1 Complex eigenvalues and spectrum

1.3.1.1 Some reminders

The eigenvalues of the operator

are given by all such that is not injective as an operator on .

The eigenspace of is the kernel

and its nonzero elements are called eigenvectors. Hence, in is an eigenvector ...