  A locally compact group G is said to be Hermitian if every selfadjoint
element of L^1(G) has real spectrum. Using Halmos' notion of capacity in Banach
algebras and a result of Jenkins, Fountain, Ramsay and Williamson we will put a
bound on the growth of Hermitian groups. In other words, we will show that if G
has a subset that grows faster than a certain constant, then G cannot be
Hermitian. Our result allows us to give new examples of non-Hermitian groups
which could not tackled by the existing theory. The examples include certain
infinite free Burnside groups, automorphism groups of trees, and p-adic general
and special linear groups.
