Let K(s, t) be a complex-valued L2 kernel on the square ⋜ s, t ⋜ by which we mean
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0305004100049859/resource/name/S0305004100049859_eqnU1.gif?pub-status=live)
and let {λν}, perhaps empty, be the set of finite characteristic values (f.c.v.) of K(s, t), i.e. complex numbers with which there are associated non-trivial L2 functions øν(s) satisfying
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0305004100049859/resource/name/S0305004100049859_eqn1.gif?pub-status=live)
For such kernels, the iterated kernels,
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0305004100049859/resource/name/S0305004100049859_eqnU2.gif?pub-status=live)
are well-defined (1), as are the higher order traces
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0305004100049859/resource/name/S0305004100049859_eqn2.gif?pub-status=live)
Carleman(2) showed that the f.c.v. of K are the zeros of the modified Fredhoim determinant
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0305004100049859/resource/name/S0305004100049859_eqn3.gif?pub-status=live)
the latter expression holding only for |λ| sufficiently small (3). The δn in (3) may be calculated, at least in theory, by well-known formulae involving the higher order traces (1). For our later analysis of this case, we define
and
, respectively, as the minimum and maximum moduli of the zeros of
, the nth section of D*(K, λ).