
# 11.5 Positive operators

Recall that self-adjoint operators are the operator analog for real numbers. Let us now define the operator analog for positive (or, more precisely, nonnegative) real numbers.

Definition 11.5.1. An operator $$T\in \mathcal{L}(V)$$ is called positive (denoted $$T\ge 0$$) if $$T=T^*$$ and $$\inner{Tv}{v} \ge 0$$ for all $$v\in V$$.

If $$V$$ is a complex vector space, then the condition of self-adjointness follows from the condition $$\inner{Tv}{v} \ge 0$$ and hence can be dropped.

Example 11.5.2.  Note that, for all $$T \in \mathcal{L}(V)$$, we have $$T^*T\ge 0$$ since $$T^*T$$ is self-adjoint and $$\inner{T^*Tv}{v}=\inner{Tv}{Tv} \ge 0$$.

Example 11.5.3.  Let $$U\subset V$$ be a subspace of $$V$$ and $$P_U$$ be the orthogonal projection onto $$U$$.

Then $$P_U\ge 0$$. To see this, write $$V=U \oplus U^\bot$$ and $$v=u_v+u_v^\bot$$ for each $$v\in V$$, where $$u_v \in U$$ and $$u_v^\bot \in U^\bot$$. Then $$\inner{P_U v}{w} = \inner{u_v}{u_w+u_w^\bot} = \inner{u_v}{u_w} = \inner{u_v+u_v^\bot}{u_w} = \inner{v}{P_U w}$$ so that $$P_U^*=P_U$$. Also, setting $$v=w$$ in the above string of equations, we obtain $$\inner{P_U v}{v}=\inner{u_v}{u_v} \ge 0$$, for all $$v\in V$$. Hence, $$P_U\ge 0$$.

If $$\lambda$$ is an eigenvalue of a positive operator $$T$$ and $$v\in V$$ is an associated eigenvector, then $$\inner{Tv}{v} = \inner{\lambda v}{v} = \lambda \inner{v}{v} \ge 0$$. Since $$\inner{v}{v}\ge 0$$ for all vectors $$v\in V$$, it follows that $$\lambda\ge 0$$. This fact can be used to define $$\sqrt{T}$$ by setting

\begin{equation*}
\sqrt{T} e_i = \sqrt{\lambda_i} e_i,
\end{equation*}

where $$\lambda_i$$ are the eigenvalues of $$T$$ with respect to the orthonormal basis $$e=(e_1,\ldots,e_n)$$. We know that these exist by the Spectral Theorem.

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