INEQUALITIES FOR EXPONENTIALS IN BANACH-ALGEBRAS

For commuting elements x, y of a unital Banach algebra B it is clear that parallel-to e(x + y) parallel-to less-than-or-equal-to parallel-to e(x) parallel-to parallel-to e(y) parallel-to. On the other hand, M. Taylor has shown that this inequality remains valid for a self-adjoint operator x and a skew-adjoint operator y, without the assumption that they commute. In this paper we obtain similar inequalities under conditions that lie between these extremes. The inequalities are used to deduce growth estimates of the form parallel-to e(i)<x,zeta> parallel-to less-than-or-equal-to c(1 + \zeta\)s for all zeta is-element-of R(m), where x = (x1,..., x(m)) is-element-of B(m) and c, s are constants.