  For a large class of integral operators or second order differential
operators, their isospectral (or cospectral) operators are constructed
explicitly in terms of $h$-transform (duality). This provides us a simple way
to extend the known knowledge on the spectrum (or the estimation of the
principal eigenvalue) from a smaller class of operators to a much larger one.
In particular, an open problem about the positivity of the principal eigenvalue
for birth--death processes is solved in the paper.
