Inverse mapping

(inverse operator) of a single-valued onto mapping (operator)

A single-valued mapping such that

(1)

(2)

where , and are any sets.

If satisfies only condition (1), then it is called a left-inverse mapping of , and if it satisfies only condition (2), it is a right-inverse mapping of . The inverse mapping exists if and only if for each the complete inverse image consists of a single element . If has an inverse mapping , then the equation

(3)

has a unique solution for each . If only a right inverse exists, then a solution of (3) exists, but its uniqueness is an open question. If only a left inverse exists, then the solution is unique, assuming that it exists. If and are vector spaces and if is a linear operator from into , then is also linear, if it exists. In general, if and are endowed with some kind of structure, it may happen that certain properties of are also inherited by , assuming that it exists. E.g., if and are Banach spaces and is a closed operator, then is also closed; if is a Hilbert space and is self-adjoint, then is also self-adjoint; if is an odd function, then is also odd, etc. The continuity of does not always imply the continuity of for many important classes of linear operators, for example for completely-continuous operators. The following are important tests for the continuity of the inverse of a linear operator.

Let be a finite-dimensional vector space, with a certain basis, and let be given by the matrix with respect to this basis. Then exists if and only if (in this case and are automatically continuous).

Let and be Banach spaces, and let be a continuous linear operator from into .

1) If , where , then exists and is continuous.

2) If , , then exists, is continuous and

where the series on the right-hand side converges in the norm of the space .

3) The operator exists and is continuous on all of if and only if the conjugate has an inverse which is defined and continuous on . Here .

4) If exists, is continuous and if , then also exists, is continuous and

Thus, the set of invertible operators is open in in the uniform topology of this space.

5) Banach's open mapping theorem: If is a one-to-one mapping of onto , 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 onto a separated barrelled space 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. E.g., if is self-adjoint and is not real, then 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)