Definition:Inverse (Bounded Linear Operator)

Definition
Let $H$ be a Hilbert space.

Let $A \in B \left({H}\right)$ be a bounded linear operator.

An inverse for $A$ is a bounded linear operator $A^{-1} \in B \left({H}\right)$ satisfying:


 * $AA^{-1} = A^{-1}A = I$

where $I$ denotes the identity operator on $H$.

If such a $A^{-1}$ exists, $A$ is said to be an invertible linear operator.

The operation assigning $A^{-1}$ to $A$ is referred to as inverting.