11.5: Operadores positivos
- Page ID
- 115001
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Recordemos que los operadores autounidos son el operador analógico para números reales. Definamos ahora el operador analógico para números reales positivos (o, más precisamente, no negativos).
Definición 11.5.1. Un operador\(T\in \mathcal{L}(V)\) se llama positivo (denotado\(T\ge 0\)) si\(T=T^*\) y\(\inner{Tv}{v} \ge 0\) para todos\(v\in V\).
Si la\(V\) is a complex vector space, then the condition of autounión se desprende de la condición\(\inner{Tv}{v} \ge 0\) y por lo tanto se puede dejar caer.
Ejemplo 11.5.2. Tenga en cuenta que, para todos\(T \in \mathcal{L}(V)\), tenemos\(T^*T\ge 0\) desde entonces\(T^*T\) es autocolindante y\(\inner{T^*Tv}{v}=\inner{Tv}{Tv} \ge 0\).
Ejemplo 11.5.3. Let\(U\subset V\) be a subspace of \(V\) and \(P_U\) be the orthogonal projection onto \(U\).
Entonces\(P_U\ge 0\). Para ver esto, escribir\(V=U \oplus U^\bot\) y\(v=u_v+u_v^\bot\) para cada uno\(v\in V\), dónde\(u_v \in U\) y\(u_v^\bot \in U^\bot\). \(\inner{P_U v}{w} = \inner{u_v}{u_w+u_w^\bot} = \inner{u_v}{u_w} = \inner{u_v+u_v^\bot}{u_w} = \inner{v}{P_U w}\)Entonces para eso\(P_U^*=P_U\). También, fijando\(v=w\) en la cadena de ecuaciones anterior, obtenemos\(\inner{P_U v}{v}=\inner{u_v}{u_v} \ge 0\), para todos\(v\in V\). De ahí,\(P_U\ge 0\).
Si\(\lambda\) es un valor propio de un operador positivo\(T\) y\(v\in V\) es un vector propio asociado, entonces\(\inner{Tv}{v} = \inner{\lambda v}{v} = \lambda \inner{v}{v} \ge 0\). Ya que\(\inner{v}{v}\ge 0\) para todos los vectores\(v\in V\), se deduce que\(\lambda\ge 0\). Este hecho se puede utilizar para definir\(\sqrt{T}\) mediante la configuración
\ begin {ecuación*}
\ sqrt {T} e_i =\ sqrt {\ lambda_i} e_i,
\ end {ecuación*}
donde\(\lambda_i\) están los valores propios de\(T\) con respecto a la base ortonormal\(e=(e_1,\ldots,e_n)\). Sabemos que estos existen por el Teorema Espectral.