On the exponential dichotomy and spectral properties of difference operators related to the Howland semigroup

We consider the linear difference operator (Dx)(t) = x(t) − B(t)x(t − h), t ∈ ℝ, where h ∈ ℝ is a nonzero number and B is a continuous bounded operator-valued function on ℝ ranging in the algebra of linear bounded operators on a Banach space. We obtain necessary and sufficient conditions for the continuous invertibility of D in the space of continuous bounded vector functions and present a formula for the inverse operator.