91Ó°ÊÓ

First, we need to understand the concepts of positive operators and commutativity. A linear operator \(\theta\) is said to be positive if \(\langle \theta x, x \rangle \geq 0\) for all \(x \in H\), where \(H\) is a Hilbert space and \(\langle \cdot, \cdot \rangle\) is the inner product defined on \(H\). Two operators, say \(\sigma\) and \(\theta\), are said to commute if \(\sigma \theta = \theta \sigma\). Given that \(\sigma \leq \tau\), it implies \(\tau - \sigma \geq 0\), which means \((\tau - \sigma)\) is a positive operator.

Now, we will use the commutativity properties of the given operators. Since \(\theta\) commutes with both \(\sigma\) and \(\tau\), we can rewrite the desired inequality \(\sigma \theta \leq \tau \theta\) as: \(\tau \theta - \sigma \theta \geq 0\) Using the commutativity property, we can rewrite this inequality as: \(\theta \tau - \theta \sigma \geq 0\) We will prove this inequality by constructing another positive operator using the given positive operators and the commutativity property.

Let's define a new operator \(X = \sqrt{\theta} (\tau - \sigma) \sqrt{\theta}\). This operator is positive since the product of positive operators is also positive. This allows us to write: \(\langle X x, x \rangle \geq 0\) for all \(x \in H\). Now, we will replace \(X\) with the expression \(\sqrt{\theta} (\tau - \sigma) \sqrt{\theta}\) and rewrite the inequality: \(\langle \sqrt{\theta} (\tau - \sigma) \sqrt{\theta} x, x \rangle \geq 0\) for all \(x \in H\).

As we know, the operators \(\sigma\) and \(\tau\) commute with \(\theta\). Therefore, the following commutativity holds: \(\theta \tau = \tau \theta \) and \(\theta \sigma = \sigma \theta\) Using these relations, we can rewrite the inequality from the previous step as: \(\langle (\theta \tau - \theta \sigma) x, x \rangle \geq 0\) for all \(x \in H\). Since this inequality holds for all \(x \in H\), it means that the operator \(\theta \tau - \theta \sigma\) is positive, that is: \(\theta \tau - \theta \sigma \geq 0\) This result is equivalent to the desired inequality: \(\sigma \theta \leq \tau \theta\). So, we have proved that if \(\sigma \leq \tau\) and \(\theta\) is a positive operator that commutes with both \(\sigma\) and \(\tau\), then \(\sigma \theta \leq \tau \theta\).