Why $ \Vert A \Vert \leq \frac{1}{2} ( \Vert A + A^* \Vert + \Vert A - A^* \Vert )?$

Question

Let $E$ be a complex Hilbert space.

If $A\in \mathcal{L}(E)$, why we have $$ \Vert A \Vert \leq \frac{1}{2} ( \Vert A + A^* \Vert + \Vert A - A^* \Vert )?$$

Answer 1

Notice the equality $$ A = \frac12 (A + A^*) + \frac12 (A-A^*).$$