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Let $\omega$ be a weight on a right cancellative semigroup $S$. Let$\|\cdot\|_{\omega}$ be the weighted norm on the weighted discrete semigroupalgebra $\ell^1(S, \omega)$. In this paper, we prove that the weight $\omega$satisfies F-property if and only if the operator norm $\| \cdot \|_{\omega op}$of $\| \cdot \|_{\omega}$ is exactly equal to another weighted norm $\| \cdot\|_{\widetilde{\omega}_1}$ [Theorem 2.5 ($iii$)]. Though its proof iselementary, the result is unexpectedly surprising. In particular, $\| \cdot\|_{1 op}$ is same as $\| \cdot \|_1$ on $\ell^1(S)$. Moreover, variousexamples are discussed to understand the relating among $\| \cdot \|_{\omegaop}$, $\| \cdot \|_{\omega}$, and $\ell^1(S, \omega)$.

The Operator Norm on Weighted Discrete Semigroup Algebras $\ell^1(S, \omega)$