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Suppose that $H$ and $K$ are Hilbert spaces and $T\in ℬ(H,K)$. The adjoint of $T$ is the unique operator $T^{\ast }\in ℬ(K,H)$ satisfying $$⟨Th,k⟩=⟨h,T^{\ast }k⟩$$ for every $h\in H$ and $k\in K$.

Suppose that $H$ is a Hilbert space. An operator $T\in ℬ\left(H\right)$ is self-adjoint if $T=T^{\ast }$.

Suppose that $H$ is a Hilbert space. An operator $T\in ℬ\left(H\right)$ is normal if $T{T}^{\ast }=T^{\ast }T$.

Suppose that $H$ and $K$ are Hilbert spaces. A surjective linear map $U:H\to K$ is unitary if $$⟨Uh,Ug⟩=⟨h,g⟩$$ for every $h,g\in H$.

Suppose that $H$ is a Hilbert space. An operator $P\in ℬ\left(H\right)$ is a projection if $P^2=P$. It is an orthogonal projection if also $P=P^{\ast }$.

Suppose that $T\in ℬ\left(H\right)$. The spectrum of $T$ is $$\sigma \left(T\right):=\{\lambda \in ℂ:T-\lambda I\text{ is not invertible}\}.$$