OBSERVATIONS ABOUT A SURPRISING SPECTRUM

A is a direct sum of finite dimensional blocks, each of them the sum o f a pure imaginary multiple of the identity and a Jordan chain. It is known that this operator generates a group with a rate of exponential growth greater than its spectral bound. A description of the spectrum of e(At) is attempted It turns out that for most values of t, this spe ctrum is the anulus e(-t) less-than-or-equal-to/lambda/less-than-or-eq ual-to e(t). For other values of t it is smaller, so that it is not an upper semi-continuous function of t.