Functional Analysis by Peter D. Lax

Author:Peter D. Lax
Language: eng
Format: epub
Publisher: Wiley

27

INDEX THEORY

We start with stating the main results of index theory in chapter 2. Let U, V be linear spaces, in general, infinite dimensional. A linear map T : U → V is said to have finite index if it has these properties:

(i) The nullspace NT of T, is a finite-dimensional subspace of U.

(ii) The quotient space V/RT, RT the range of T, is finite dimensional.

For such an operator we define the index as

(1)

A map G from one linear space into another is called degenerate if its range is finite dimensional. We recall from chapter 2 the following results:

Theorem A. A linear map T : U → V has finite index iff T has a pseudoinverse, that is, a linear map S : V → U such that

(2)

where I denotes the identity in U and V respectively, and G, H are degenerate maps.

Theorem B. Let T : U → V and R : V → W be linear maps with finite index. Then their product RT has finite index, and

(3)

Theorem C. Let T : U → V be a linear map with finite index, and G : U → V a degenerate linear map. Then T + G has finite index, and

(4)

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