Definition:Pseudoinverse of Bounded Linear Transformation

Definition

Let $\struct {X, \norm \cdot_X}$ and $\struct {Y, \norm \cdot_Y}$ be normed vector spaces.

Let $S: X \to Y$ be a bounded linear transformation.

Let $T: Y \to X$ be a bounded linear transformation.

$S$ and $T$ are pseudoinverse to each other if and only if:

$T \circ S - I_X$ is compact

and:

$S \circ T - I_Y$ is compact

where:

$\circ$ denotes the composition

$I_X$ denotes the identity mapping of $X$

$I_Y$ denotes the identity mapping of $Y$

This article, or a section of it, needs explaining. In particular: $T \circ S - I_X$ and $S \circ T - I_Y$ You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code.

This article is complete as far as it goes, but it could do with expansion. In particular: Establish the conceptual connection between compactness and degeneracy, as the definition is different according to whether L.T. is bounded or not You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding this information. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Expand}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Also see

• Definition:Pseudoinverse of Linear Transformation

Sources

• 2002: Peter D. Lax: Functional Analysis: $27.1$: The Noether Index