Expansion of the element \(\sin a\) via exponentials for Hermitian elements \(a\) (Q1405670)

The theory of (unbounded) Hermitian elements (i.e., \(\|\exp(ita)\|=1\)) of a complex Banach algebra has received a lot of attention in the last decades. On specific algebras of entire functions, the question whether the derivation operator \(a= -id/dx\) is Hermitian leads to Bernstein type inequalities, see, for instance, \textit{A. Browder} [Am. Math. Mon. 78, 871--873 (1971; Zbl 0224.47011)]. The present note focuses on the equality \(\|\sin a\|=|\sin a|_{sp}\), for a Hermitian element \(a\) of a Banach algebra, where the latter denotes the spectral radius. That this is so, follows from the main interpolation theorem of the note, under the assumption \(| a|\leq\pi/2\). As a corollary, the following sharper Bernstein inequality for entire functions \(f\) of exponential type \(\sigma> 0\) follows: \[ | f(x+ h)- f(x- h)|\leq 2\sin(\sigma h)\| f\|_\infty,\quad x\in\mathbb{R},\;\sigma| h|\leq \pi/2. \] The main technique is based on Fourier transforms in the framework of Paley-Wiener's class.