Self Adjoint Operator

Overview


A self adjoint operator is an operator where the adjoint of the operator {% T^* %} is equal to itself. That is
{% T^* = T %}

Properties


  • If {% T_1 %} and {% T_2 %} are self-adjoint, then {% aT_1 + bT_2 %} is self-adjoint
  • For {% T \in B(\mathcal{H}) %}, {% T^*T %} and {% T T^* %} are self-adjoint
  • For self adjoint operator {% T \in B(\mathcal{H}) %}, the numerical range of {% T %} is a subset of {% \mathbb{R} %}. This implies
    • all of its eigenvalues are real
    • if {% Tx \ lambda_1 x %} and {% Ty \ lambda_2 y %} and {% \lambda_1 \neq lamba_2 %} then {% \langle x, y \rangle = 0 %}