Definition:Invertible Bounded Linear Transformation

Definition
Let $H, K$ be Hilbert spaces.

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

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


 * $AA^{-1} = I_K$
 * $A^{-1}A = I_H$

where $I_K, I_H$ denote the identity operators on $K, H$, respectively.

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

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

Invertible Bounded Linear Operator
When $H = K$, the notation simplifies considerably, and $A$ is said to be a invertible linear operator.