higher order exponentially isometric operators

For a positive integer m, a bounded linear operator t on ahilbert space is called an exponentially m-isometric operator ifmpk=0(????1)m????k????mk
ektekt = 0. we show that for every non-empty com-pact subset k of pure imaginary axis, there exits an exponentially m-isometric operator t whose spectrum is k. moreover, if (tn)n1 isa sequence of operators in this class that converges to t in the strongoperator topology, then t is also an exponentially m-isometric operator.