Approximation Properties and Analytic Semigroups

It is shown that if X is a separable Banach Space with the hermitian approximation property, then there exists a compact analytic semigroup, t--> a^t:H-->CL(X) such that (a^t X) = X and ||a^t||≤1 for all t G H, where H is the open right half of the Complex plane, a^t represents a compact operator on X and CL(X) denotes the set of all compact operators on X. Further, it is shown with a counter - example that the converse to the above result is false in general. In the process, we establish an enlightening result, namely, the scalar multiples of the identity operator are the only hermitians on the disc algebra.