Let
$c\,=\,\left( {{c}_{1}},\ldots ,{{c}_{n}} \right)$
be such that
${{c}_{1}}\,\ge \,\cdots \,\ge \,{{c}_{n}}$
. The
$c$
-numerical range of an
$n\,\times \,n$
matrix
$A$
is defined by
$${{W}_{c}}\left( A \right)\,=\,\left\{ \sum\limits_{j=1}^{n}{{{c}_{j}}\left( A{{x}_{j}},\,{{x}_{j}} \right)\,:\,\left\{ {{x}_{1}},\ldots ,{{x}_{n}} \right\}\,\text{an}\,\text{orthonormal basis for }{{\mathbf{C}}^{n}}} \right\}\,,$$
and the
$c$
-numerical radius of
$A$
is defined by
${{r}_{c}}\left( A \right)\,=\,\max \left\{ \left| z \right|\,:\,z\,\in \,{{W}_{c}}\left( A \right) \right\}$
. We determine the structure of those linear operators
$\phi$
on algebras of block triangular matrices, satisfying
$${{W}_{c}}\left( \phi \left( A \right) \right)={{W}_{c}}\left( A \right)\text{for}\,\,\text{all}\,\,A\,\text{or}\,\,{{r}_{c}}\left( \phi \left( A \right) \right)={{r}_{c}}\left( A \right)\text{for}\,\,\text{all}\,A.$$