show that$$  $$ w(AB) $$ \le\frac12\left\Vert\left\Vert A\right\Vert^2+\left\Vert B\right\Vert^2\right\Vert $$

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:
\[
w(AB) \le \frac{1}{2} (\|A\|^2 + \|B\|^2)
\]

Would you like more details or have any questions?

### 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.

Learn how to prove the inequality involving the numerical range $ w(AB) $ and norms of operators $ A $ and $ B $. This problem explores concepts from operator theory, including the numerical range, norms of operators, and inequalities such as Cauchy-Schwarz and AM-GM. Suitable for advanced undergraduate students and researchers.

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