The authors relate various properties of local and global Lyapunov exponents, which were used in the study of the Hausdorff dimension of the global attractor for the 2D Navier-Stokes equations [cf. \textit{P. Constantin} and \textit{C. Foias}, Commun. Pure Appl. Math. 38, 1-27 (1985; Zbl 0582.35092), \textit{P. Constantin}, \textit{C. Foias} and \textit{R. Temam}, Mem. Am. Math. Soc. 314 (1985; Zbl 0567.35070)]. This goal is achieved by posing their problem in the framework of flows of positive operators on a space of continuous functions over a compact set and utilizing the theory developed by \textit{G. Choquet} and \textit{C. Foias} [Ann. Inst. Fourier Grenoble 25 (1975), No.3-4, 109-129 (1976; Zbl 0303.47004)]. The key idea is to obtain a flow of positive operators from the nonlinear semigroup of solution operators \(S_ t\), acting on a compact invariant subset X of a Hilbert space H. The main content is as follows: Section 2. L-exponents associated with positive operators. Section 3. Semiflows on infinite-dimensional vector spaces and associated L-exponents. Section 4. Estimates on the dimension of attractors. Section 5. An estimate on the dimension of the Lorenz global attractor. Section 6. Evolution equations satisfying a dissipativity condition.