Question

Prove that 𝑇1 and 𝑇2 are self adjoint operators on a Hilbert space

H, prove that 𝑇1 𝑇2 +𝑇2 𝑇1 is self adjoint

Answer 1

the adjoint of T is operator T^* for which

$\langle Tv,w\rangle=\langle v,T^*w\rangle$

T is self-adjoint if T=T^*

if 𝑇₁ and 𝑇₂ are self adjoint operators then:

$\langle (𝑇_1 𝑇_2 +𝑇_2 𝑇_1)v,w\rangle=\langle v,(𝑇_1 𝑇_2 +𝑇_2 𝑇_1)^*w\rangle$

$(𝑇_1 𝑇_2 +𝑇_2 𝑇_1)^*=(𝑇_1 𝑇_2)^*+(𝑇_2 𝑇_1)^*=T_2^*T_1^*+T_1^*T_2^*=𝑇_1 𝑇_2 +𝑇_2 𝑇_1$

so, $𝑇_1 𝑇_2 +𝑇_2 𝑇_1$ is self-adjoint