Corrigenda

Volume 92 (1982), 115–119 K. J. Falconer School of Mathematics, University of Bristol ‘Growth conditions on powers of Hermitian elements’ The above paper aimed to obtain Banach algebra equivalents of the elegant theorems of Roe [2] and Burkill [1] which gave characterizations of the sine function. Professor John Duncan has kindly pointed out that the paper contains two oversights, with the result that the theorems stated are not the strict Banach algebra equivalents as was claimed. (The assertion on p. 117 that T is Hermitian is false if μ > 0, and at the bottom of p. 117 it follows from Theorem 2 that F0(t) = c1eit + c2eit so that a2 – e = 0.) Whilst Theorems 1–3 are correct as stated, the conclusions of Theorems 1 and 3 must be strengthened to provide equivalent versions when μ > 0. In the basic case when μ = 0, the analysis remains as given.

The above paper aimed to obtain Banach algebra equivalents of the elegant theorems of Roe [2] and Burkill [l] which gave characterizations of the sine function. Professor John Duncan has kindly pointed out that the paper contains two oversights, with the result that the theorems stated are not the strict Banach algebra equivalents as was claimed. (The assertion on p. 117 that T is Hermitian is false if fi > 0, and at the bottom of p. 117 it follows from Theorem 2 that F 0 (t) = c 1 e it + c 2 e it so that a 2 -e = 0.) Whilst Theorems 1-3 are correct as stated, the conclusions of Theorems 1 and 3 must be strengthened to provide equivalent versions when fi > 0. In the basic case when /i = 0, the analysis remains as given.
I am most grateful to Professor Duncan for providing detailed comments on these points, and to Professor V. I. Istra^escu for drawing my attention to [3].
In the restatement of the results below, it is convenient to give two sets of equivalent theorems. In order to state the second set of theorems, we need to use quasi-Hermitian elements.
An element a e A is said to be quasi-Hermitian of order ji if For the sake of completeness, we state the following Theorem of Istra$escu [3] which is of a similar form. The proofs of these theorems and the equivalences follow the lines of those presented before, with the following amendments.
1. To prove Theorem 1', which is Theorem 1 with the conclusion strengthened to a z = e, we proceed as before to estimate the resolvent R{z) = (ze -a)" 1 near z = ± 1. If w = z -lorw> = z + l w e have || ^ clwl-*-1 (|Rew| > |InH), near w = 0, where c is independent of 2. Thus R{z) has a terminating Laurent expansion at 1 and -1. In particular, the growth of R(z) near these poles is determined by its growth in any segment. Thus (2) holds for any z near w = 0. It follows as before, but using this stronger estimate, that a 2 = e. 2. Theorem 3' follows from Theorem 1' exactly as before but again the conclusion has been strengthened.
3. Theorem 1' is equivalent to Roe's theorem, Theorem 2', with proofs as before. In this case, we need only consider differential equations of the form F" + F = 0.
For the second set of theorems we study properties of quasi-Hermitian elements. 5. To prove Theorem 1" we proceed as for Theorem 1 modified by § 1 above. We require estimates for the resolvent R(z) -(ze -a)" 1 near z = ± 1 when a is quasi-Hermitian. The growth conditions on a ensure that (3) holds. To get an estimate for R(z) off the real axis, we use the Laplace formula if y < 0, using (1). A similar estimate in the upper half plane come3 from using exp( -ita) in (4). Combining (5) with (3) we see that R(z) has a terminating Laurent expansion near z = + 1, so as in § 1

C\W -k-1
where k = min ([AJ, \ji\). Theorem 1" now follows from this estimate in the usual way. 6. As before, we see that Theorem 3" is the operator version of Theorem 1". 7. The equivalence of Theorem 1" and Theorem 2" may be shown as before. To deduce Theorem 2" from Theorem 1" we need to use the fact that the operator TF = iF' is quasi-Hermitian of order fi (rather than Hermitian) on the Banach space 8. It is not at present clear whether a version of Kolmogorov's theorem for functions of polynomial growth, and thus for quasi-Hermitian operators, is valid. Thus it is unclear whether a version of Theorem 4 holds for the second trio of theorems.