Inverse mapping

(inverse operator) of a single-valued onto mapping (operator) $ f: M \to f[M] $

A single-valued mapping $ g $ such that \begin{align} g \circ f & = \mathsf{Id}_{X} \quad \text{on} ~ M, \qquad (1) \\ f \circ g & = \mathsf{Id}_{Y} \quad \text{on} ~ f[M], \qquad (2) \end{align} where $ M \subseteq X $, $ f[M] \subseteq Y $, and $ X $ and $ Y $ are any sets.

If $ g $ satisfies only Condition (1), then it is called a left-inverse mapping of $ f $, and if it satisfies only Condition (2), then it is called a right-inverse mapping of $ f $. The inverse mapping $ f^{-1} $ exists if and only if for each $ y \in f[M] $, the inverse image $ {f^{\leftarrow}}[\{ y \}] $ consists of just a single element $ x \in M $. If $ f $ has an inverse mapping $ f^{-1} $, then the equation $$ f(x) = y \qquad (3) $$ has a unique solution for each $ y \in f[M] $. If only a right inverse $ f_{R}^{-1} $ exists, then a solution of (3) exists, but its uniqueness is an open question. If only a left inverse $ f_{L}^{-1} $ exists, then any solution is unique, assuming that it exists.

If $ X $ and $ Y $ are vector spaces, and if $ A $ is a linear operator from $ X $ into $ Y $, then $ A^{-1} $ is also linear, if it exists. In general, if $ X $ and $ Y $ are endowed with some kind of structure, it may happen that certain properties of $ A $ are also inherited by $ A^{-1} $, assuming that it exists. For example:

• If $ X $ and $ Y $ are Banach spaces, and if $ A: X \to Y $ is a closed operator, then $ A^{-1} $ is also closed.
• If $ \mathcal{H} $ is a Hilbert space and $ A: \mathcal{H} \to \mathcal{H} $ is self-adjoint, then $ A^{-1} $ is also self-adjoint.
• If $ f: \mathbf{R} \to \mathbf{R} $ is an odd function, then $ f^{-1} $ is also odd.

The continuity of $ A $ does not always imply the continuity of $ A^{-1} $ for many important classes of linear operators, e.g., completely-continuous operators. The following are important tests for the continuity of the inverse of a linear operator.

Let $ X $ be a finite-dimensional vector space, with a certain basis, and let $ A: X \to X $ be given by the matrix $ [a_{ij}] $ with respect to this basis. Then $ A^{-1} $ exists if and only if $ \det([a_{ij}]) \neq 0 $ (in this case, $ A $ and $ A^{-1} $ are automatically continuous).

Let $ X $ and $ Y $ be Banach spaces, and let $ A $ be a continuous linear operator from $ X $ into $ Y $.

1. If $ \| A(x) \| \geq m \| x \| $, where $ m > 0 $, then $ A^{-1} $ exists and is continuous.
2. If $ X = Y $ and $ \| A \| < 1 $, then $ (\mathsf{Id} - A)^{-1} $ exists, is continuous, and $$ (\mathsf{Id} - A)^{-1} = \sum_{n = 0}^{\infty} A^{n}, $$ where the series on the right-hand side converges in the norm of the space $ \mathcal{L}(X) $.
3. The operator $ A^{-1} $ exists and is continuous on all of $ Y $ if and only if the conjugate $ A^{*} $ has an inverse that is defined and continuous on $ X^{*} $. Here, $ (A^{-1})^{*} = (A^{*})^{-1} $.
4. If $ A^{-1} $ exists and is continuous, and if $ \| A - B \| < \| A^{-1} \|^{-1} $, then $ B^{-1} $ also exists, is continuous, and $$ B^{-1} = A^{-1} \sum_{n = 0}^{\infty} [(A - B) A^{-1}]^{n}. $$ Therefore, the set of invertible operators is open in $ \mathcal{L}(X,Y) $ in the uniform topology of this space.
5. Banach’s Open-Mapping Theorem. If $ A $ is a one-to-one mapping of $ X $ onto $ Y $, then the inverse mapping, which exists, is continuous. This theorem has the following generalization: A one-to-one continuous linear mapping of a fully-complete space $ X $ onto a separated barrelled space $ Y $ is a topological isomorphism.

The spectral theory of linear operators on a Hilbert space contains a number of results on the existence and continuity of the inverse of a continuous linear operator. For example, if $ A $ is self-adjoint and $ \lambda $ is not real, then $ (A - \lambda \cdot \mathsf{Id})^{-1} $ exists and is continuous.

References

[1]	N. Dunford, J.T. Schwartz, “Linear operators. General theory”, 1, Interscience (1958).
[2]	L.V. Kantorovich, G.P. Akilov, “Functional analysis”, Pergamon (1982). (Translated from Russian)
[3]	W. Rudin, “Functional analysis”, McGraw-Hill (1979).
[4]	A.P. Robertson, W.S. Robertson, “Topological vector spaces”, Cambridge Univ. Press (1964).