Positive operators in \Lop

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Section 10.2 Positive operators in \(\Lop\)

We’ll start with a definition similar to the matrix case.

Definition 10.2.1.

An operator \(T \in \Lop\) is called positive and we write \(T \geq 0\) if for all \(u \in \hilbert\text{,}\) we have \(\ip{Tu}{u} \geq 0\text{.}\)

Theorem 10.2.2.

For \(T \in \Lop\text{,}\) the following are equivalent.

\(\displaystyle T \geq 0.\)
\(T\ad = T\) and \(\sigma(A) \subseteq \{x \in \R: x \geq 0\}.\)
\(T = X\ad X\) for some Hilbert space \(\mathcal{K}\) and \(X \in \mathcal{L}(\hilbert, \mathcal{K})\text{.}\)
\(T = X^2\) for some \(X \in \Lop\text{.}\)

The proof of this theorem can be found, for example, in Conway. In finite dimensions, this theorem is proved in Axler as Theorem 7.39. While the \(X\) so that \(X^2 = T\) isn’t unique, there is a unique positive square root, for which we say \(A = \sqrt{T}\) if \(A \geq 0\) and \(A^2 = T\text{.}\)

The next part of this discussion follows Agler, McCarthy, and Young’s Operator Analysis. The theorem to follow is called by them the fundamental fact, as it will allow us to change frame back and forth from analysis estimates to the alebraic property of positivity.

Theorem 10.2.3.

If \(\hilbert, \K\) are Hilbert spaces, then \(\norm{T} \leq 1\) if and only if \(1 - T\ad T \geq 0\text{.}\)

Proof.

\begin{align*} \norm{T} \leq 1 \amp\iff \norm{x}^2 \geq \norm{Tx}^2 \,\, \forall x \in \hilbert\\ \amp\iff \ip{x}{x} \geq \ip{Tx}{Tx} \,\, \forall x \in \hilbert\\ \amp\iff \ip{(1 - T\ad T)x}{x} \geq 0 \,\, \forall x \in \hilbert\\ \amp\iff 1- T\ad T \geq 0 \end{align*}

Lemma 10.2.4.

Let \(T \in \Lop\text{.}\) Then if \(\ip{Tx}{x} = 0\) for all \(x \in \hilbert\text{,}\) then \(A = 0\text{.}\)

Proof.

Exercise.

We also get a nice characterization of isometries.

Definition 10.2.5.

An operator \(T \in \Lop\) is called an isometry if \(\norm{Tx} = \norm{x}\) for all \(x \in \hilbert\text{.}\)

Lemma 10.2.6.

Let \(T \in \Lop\text{.}\) Then \(T\) is an isometry if and only if \(1 - T\ad T = 0\)

Proof.

Modify the proof of Theorem 10.2.3 and invoke Lemma 10.2.4 at the appropriate moment.