Abstract. The following conjecture of J. R. Partington, raised in [8], is proved: If $T$ is a Hermitian operator in a complex Banach space and $d(\lambda,\sigma(T))$ denotes the distance of the complex number $\lambda $ to the spectrum $\sigma(T)$ of $T,$ then $$\|(\lambda I-T)^{-1}\|\le{\pi\over 2} d(\lambda,\sigma(T))^{-1}$$ for all $\lambda\in \msbm C\setminus\sigma (T).$ For the proof we compute the infimum of the norms of all functions $f\in L^1 ({\msbm R}),$ whose inverse Fourier transform extends $${\msbm R}\setminus(-1, 1)\ni x\mapstochar\rightarrow {1\over\alpha -ix},$$ $\alpha $ being a real parameter

AMS Subject Classification (1991): 47B44, 42A38

Received July 17, 1994. (Registered under 5602/2009.)