VARIATIONAL PRINCIPLES FOR SPECTRAL RADIUS OF WEIGHTED ENDOMORPHISMS OF C(X, D)

We give formulas for the spectral radius of weighted endomorphisms as : C(X,D) -> C(X,D), a is an element of C(X, D), where X is a compact Hausdorff space and D is a unital Banach algebra. Under the assumption that alpha generates a partial dynamical system (X, phi), we establish two kinds of variational principles for r(a alpha): using linear extensions of (X, phi) and using Lyapunov exponents associated with ergodic measures for (X, phi). This requires considering (twisted) cocycles over (X, phi) with values in an arbitrary Banach algebra D, and thus our analysis cannot be reduced to any of the multiplicative ergodic theorems known so far. The established variational principles apply not only to weighted endomorphisms but also to a vast class of operators acting on Banach spaces that we call abstract weighted shifts associated with alpha : C(X, D) -> C(X,D). In particular, they are far-reaching generalizations of formulas obtained by Kitover, Lebedev, Latushkin, Stepin, and others. They are most efficient when D = B(F), for a Banach space F, and endomorphisms of B(F) induced by a are inner isometric. As a by-product we obtain a dynamical variational principle for an arbitrary operator b is an element of B(F) and that its spectral radius is always a Lyapunov exponent in some direction v is an element of F when F is reflexive.