To show that $w(AB) \le \frac{1}{2} \left( \|A\|^2 + \|B\|^2 \right)$, we need to understand the numerical range $w(T)$ of an operator $T$ and use properties of norms and inner products.

Definitions and Preliminaries

1. Numerical Range: The numerical range $w(T)$ of an operator $T$ is defined as: $w(T) = \sup_{\|x\| = 1} |\langle Tx, x \rangle|$

2. Norm of an Operator: The norm of an operator $T$ is defined as: $\|T\| = \sup_{\|x\| = 1} \|Tx\|$

Steps to Prove the Inequality

Step 1: Express $w(AB)$

By definition, we have: $w(AB) = \sup_{\|x\| = 1} |\langle ABx, x \rangle|$

Step 2: Apply the Cauchy-Schwarz Inequality

Using the Cauchy-Schwarz inequality for the inner product, we get: $|\langle ABx, x \rangle| \le \|ABx\| \|x\|$ Since $\|x\| = 1$, this simplifies to: $|\langle ABx, x \rangle| \le \|ABx\|$

Step 3: Use the Sub-multiplicative Property of the Norm

The norm of a product of operators satisfies the sub-multiplicative property: $\|ABx\| \le \|A\| \|Bx\|$

Step 4: Apply the Norm Inequality to $B$

Now, we apply the operator norm to $B$: $\|Bx\| \le \|B\| \|x\|$ Since $\|x\| = 1$, we have: $\|Bx\| \le \|B\|$

Step 5: Combine the Inequalities

Combining the results from Steps 3 and 4, we get: $\|ABx\| \le \|A\| \|B\|$ Thus, $|\langle ABx, x \rangle| \le \|A\| \|B\|$ Taking the supremum over all $\|x\| = 1$, we obtain: $w(AB) \le \|A\| \|B\|$

Step 6: Use the Arithmetic Mean-Geometric Mean Inequality

Finally, we use the Arithmetic Mean-Geometric Mean (AM-GM) inequality to relate $\|A\|\|B\|$ to $\frac{1}{2}(\|A\|^2 + \|B\|^2)$. The AM-GM inequality states: $\|A\| \|B\| \le \frac{1}{2} (\|A\|^2 + \|B\|^2)$

Conclusion

Combining all the steps, we get: $w(AB) \le \|A\| \|B\| \le \frac{1}{2} (\|A\|^2 + \|B\|^2)$ Thus, we have shown that: $w(AB) \le \frac{1}{2} (\|A\|^2 + \|B\|^2)$

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Relative Questions

1. What is the definition of the numerical range $w(T)$ of an operator $T$?
2. How do we define the norm of an operator $T$?
3. What is the Cauchy-Schwarz inequality in the context of inner products?
4. What does the sub-multiplicative property of norms state?
5. How does the norm inequality apply to an operator $B$ acting on a vector $x$?
6. What is the Arithmetic Mean-Geometric Mean (AM-GM) inequality?
7. How do we use the AM-GM inequality to relate $\|A\|\|B\|$ to $\frac{1}{2}(\|A\|^2 + \|B\|^2)$?
8. Can we extend this result to other types of operator products or norms?

Tip

When dealing with inequalities in linear algebra, using properties like the Cauchy-Schwarz inequality, sub-multiplicative property, and AM-GM inequality can provide powerful tools to simplify and prove relationships.