  For nonautonomous linear difference equations with bounded coefficients on
$\mathbb{N}$ which have a bounded inverse, we introduce two different notions
of spectra and discuss their relation to the well-known exponential dichotomy
spectrum. The first new spectral notion is called Bohl spectrum and is based on
an extended notion of the concept of Bohl exponents. The second new spectral
notion is called Bohl dichotomy spectrum and is based on a relaxed version of
exponential dichotomy called Bohl dichotomy. We prove spectral theorems and
show that the Bohl dichotomy spectrum is the closure of the Bohl spectrum and
also a subset of the exponential dichotomy spectrum. We discuss the spectra of
upper triangular systems and how they relate to the spectra of their diagonal
entries. An example illustrates the subtle differences between the different
notions of spectra.
