  We investigate the structure of norm-preserving and linear but not
necessarily surjective operators on variable-exponent, discrete Lebesgue
spaces. A certain class of isometries, novel to this work, are especially
considered; this class completely coincides with all isometries when the
Lebesgue space is classical, i.e. of a fixed-exponent. For said isometries it
is shown that their actions are completely determined by pairs consisting of
set-mappings and bounded functions on $\mathbb{N}$. This result recovers the
previously-known structure of isometries on fixed-exponent spaces as a special
case. In the second part, we show that another wide class of operators,
including shift operators, are only isometric under very restrictive conditions
on the exponent sequence. Together these results serve to highlight the
striking similarities and yet radical differences between isometric operators
on fixed- and variable-exponent spaces.
