Invertibility of Identity Transformation Plus Product of Two Continuous Linear Transformations
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Theorem
Let $\struct {X, \norm{, \cdot ,} }$ be the normed vector space.
Let $I : X \to X$ be the identity mapping.
Let $\map {CL} X := \map {CL} {X, X}$ be the continuous linear transformation space on $X$.
Suppose $I + A \circ B$ is invertible, where $\circ$ denotes the composition of mappings.
Then $I + B \circ A$ is invertible, with the inverse given by:
- $\paren {I + B \circ A}^{-1} = I - B \circ \paren {I + A \circ B}^{-1} \circ A$
Proof
\(\ds \paren {I + B \circ A} \circ \paren {I - B \circ \paren {I + A \circ B}^{-1} \circ A}\) | \(=\) | \(\ds I - B \circ \paren {I + A \circ B}^{-1} \circ A + B \circ A - B \circ A \circ B \circ \paren {I + A \circ B}^{-1} \circ A\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds I - B \circ \paren {I + A \circ B}^{-1} \circ A + B \circ A - B \circ \paren {I + A \circ B - I} \circ \paren {I + A \circ B}^{-1} \circ A\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds I - B \circ \paren {I + A \circ B}^{-1} \circ A + B \circ A - B \circ \paren {I + A \circ B} \circ \paren {I + A \circ B}^{-1} \circ A + B \circ \paren {I + A \circ B}^{-1} \circ A\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds I + B \circ A - B \circ A\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds I\) |
\(\ds \paren {I - B \circ \paren {I + A \circ B}^{-1} \circ A} \circ \paren {I + B \circ A}\) | \(=\) | \(\ds I + B \circ A - B \circ \paren {I + A \circ B}^{-1} \circ A - B \circ \paren {I + A \circ B}^{-1} \circ A \circ B \circ A\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds I + B \circ A - B \circ \paren {I + A \circ B}^{-1} \circ A - B \circ \paren {I + A \circ B}^{-1} \circ \paren{I + A \circ B - I} \circ A\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds I + B \circ A - B \circ \paren {I + A \circ B}^{-1} \circ A - B \circ \paren {I + A \circ B}^{-1} \circ \paren{I + A \circ B} \circ A + B \circ \paren {I + A \circ B}^{-1} \circ A\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds I + B \circ A - B \circ A\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds I\) |
$\blacksquare$
Sources
- 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): Chapter $\S 2.4$: Composition of continuous linear transformations