Definition:Invertible Bounded Linear Transformation

Definition
Let $\HH$ and $\KK$ be Hilbert spaces.

Let $\map \BB {\HH, \KK}$ be the set of bounded linear transformations from $\HH$ to $\KK$.

Let $A \in \map \BB {\HH, \KK}$ be a bounded linear transformation.

An inverse for $A$ is a bounded linear transformation $A^{-1} \in \map \BB {\KK, \HH}$ satisfying:


 * $AA^{-1} = I_\KK$
 * $A^{-1}A = I_\HH$

where $I_\KK$ and $I_\HH$ denote the identity operators on $\KK$ and $\HH$ respectively.

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

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

Invertible Bounded Linear Operator
When $\HH = \KK$, the notation simplifies considerably, and $A$ is said to be a invertible (bounded) linear operator.

Also see

 * Definition:Inverse Element, of which this is an instantiation.