One of the fun things about the lag operator (applied to a vector space of infinite sequences) is that every element of the field over which the vector space is defined is an eigenvalue. All sequences of the form x_(n+1) = a*x_n are eigenvectors, and their corresponding eigenvalues are a^-1, where a^-1 is a's multiplicative inverse (unless a = 0, in which case the corresponding eigenvalue is 0). In other words, any time series with constant growth (or shrinkage) is an eigenvector, and the corresponding eigenvalue is the inverse of the growth factor.
You can extend this logic to double-lags, in which case the eigenvectors remain the same, and the corresponding eigenvalues are a^-2, or n-lags, in which case the corresponding eigenvalues are a^-n. You can even extend this to the n-forward-shift operator, in which case the eigenvalues are a^n.
