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Mathematics LibreTexts

11.5: Positive operators

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    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

    \sqrt{T} e_i = \sqrt{\lambda_i} e_i,

    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.