Abstract
Motivated by recent investigations of ergodic optimization for matrix cocycles, we study the measures of the maximum top Lyapunov exponent for pairs of bounded weighted shift operators on a separable Hilbert space. We prove that, for generic pairs of weighted shift operators, the Lyapunov-maximizing measure is unique, and show that there exist pairs of operators whose unique Lyapunov-maximizing measure takes any prescribed value less than $\log 2$ for its metric entropy. We also show that, in contrast to the matrix case, the Lyapunov-maximizing measures of pairs of bounded operators are, in general, not characterized by their supports: we construct explicitly a pair of operators and a pair of ergodic measures on the 2-shift with identical supports, such that one of the two measures is Lyapunov-maximizing for the pair of operators and the other measure is not. Our proofs make use of the Ornstein $\overline{d}$-metric to estimate differences in the top Lyapunov exponent of a pair of weighted shift operators as the underlying measure is varied.
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