Rotation methods in operator ergodic theory

Let E(·) : R → B(X) be the spectral decomposition of a trigonometrically well-bounded operator U acting on the arbitrary Banach space X, and suppose that the bounded function φ : T → C has the property that for each z ∈ T, the spectral integral R [0,2π] φ(e)dEz(t) exists, where Ez(·) denotes the spectral decomposition of the (necessarily) trigonometrically well-bounded operator (zU). We show this implies that for each z ∈ T, the spectral integral with respect to E(·) of the rotated function φz(·) ≡ φ((·)z) exists. In particular, these considerations furnish the preservation under rotation of spectral integration for the Marcinkiewicz r-classes of multipliers Mr(T), which are not themselves rotation-invariant. In the setting of an arbitrary super-reflexive space, we pursue a different aspect of the impact of rotations on the operator ergodic theory framework by applying the rotation group to the spectral integration of functions of higher variation so as to obtain strongly convergent Fourier series expansions for the operator theory counterparts of such functions. This vector-valued Fourier series convergence can be viewed as an extension of classical Calderón–Coifman–Weiss transference without being tied to the need of the latter for power-boundedness assumptions.