1
The Winding Number in Functional Analysis Johar M. Ashfaque Topological ideas such as the winding number are ubiquitous in modern mathematics often showing up in entirely unexpected contexts. For example, the winding number will arise, as will shown later, as the solution to a problem in functional analysis. Roughly speaking this problem is about counting the number of arbitrary constants that appear in the general solution of a certain integral equation. This process of relating topology to counting the number of solutions of an integral equations lies at the heart of the index theorem of Atiyah and Singer. I.
THE FREDHOLM INDEX
Let V be a vector space and U a subspace of V . Recall that the dimension of U is the number of elements in a basis for U . That is the dimensions of U is the smallest n for which we can find u1 , ..., un ∈ U such that every u ∈ U can be written u=
n X
λi ui
i=1
for some scalars λi ∈ C. Also recall that the codimension of the subspace U of V is the dimension of an algebraic complement to U , that is, the smallest n for which there exist v1 , ..., vn ∈ V such that for every v∈V v=
n X
µi vi + u,
µi ∈ C, u ∈ U.
i=1
When everything is finite-dimensional, the codimension of U in V is just dim V − dim U . But the codimension may be finite even when the V and U are both infinite-dimensional. Now let T : V → W be a linear map between vector spaces. Recall the following definitions • The kernel and image of T are defined by ker T = {v ∈ V : T v = 0}, im T = {w ∈ W : ∃v ∈ V, T v = w}. They are vector subspaces of V and W respectively. • The nullity of T is the dimension of kerT and the rank of T is the dimension of im T . • The corank of T is the codimension of im T in W . Similarly, the conullity of T is the codimension of ker T in V . One of the basic results in linear algebra is the rank-nullity theorem. One version of its statement is the following. Theorem I.1 For a linear transformation T : V → W between finite-dimensional vector spaces Nullity T − Corank T = dim V − dim W. An invertible linear map has zero nullity and zero corank. Definition I.2 Let V and W be Hilbert spaces and let T : V → W be a bounded linear operator. We say that T is a Fredholm operator if • the kernel of T has finite dimension, • the range of T has finite codimension. Proposition I.3 The kernel and the range of a Fredholm operator on Hilbert space are closed subspaces.
2 Proposition I.4 The index Ind T of a Fredholm operator T is the difference of dimensions Nullity T − Corank T. The rank-nullity theorem can be restated as follows: if V and W are finite-dimensional and T : V → W is a linear map then Ind T = dim V − dim W . In other words, the index does not depend on T at all. This statement is not true for maps from an infinite-dimensional space to itself. Example. Let V = W = l2 , the Hilbert space of square-summable sequences. Let T : V → W be the linear operator defined by T (a0 , a1 , a2 , a3 , ...) = (0, a0 , a1 , a2 , ...) called the unilateral shift. Clearly, Nullity T = 0 while Corank T = 1 so T is the Fredholm operator of index -1. The adjoint operator T ∗ called the unilateral backward shift defined by T ∗ (a0 , a1 , a2 , a3 , ...) = (a1 , a2 , a3 , a4 , ...) has index +1. The following theorem captures the key properties of the Fredholm index. Theorem I.5 Let H be a Hilbert space. Then the space Fred(H) of Fredholm operators H → H is an open subset of B(H) (the collection of all bounded linear operators on a Hilbert space H). The index function Ind : Fred(H) → Z is constant on the path components of Fred(H) and two Fredholm operators belong to the same path component of Fred(H) if and only if they have the same index. If H is infinite-dimensional, all integers can be obtained as the indices of appropriate Fredholm operators.
II.
ATKINSON’S THEOREM A.
Operators: A Review
Let H and K be Hilbert spaces and let T : H → K be a linear map. One says that T is bounded if the quantity ||T || = sup{||T x|| : ||x|| ≤ 1} is finite. In that case ||T || is called the norm of T . The bounded linear maps are exactly those which are continuous when we consider H and K as metric spaces. The collection of all bounded linear maps form H to K denoted B(H; K) then becomes a normed vector space and the completeness of K also implies that B(H; K) is also complete. A bounded linear map is also called a linear operator. Note. If H = K then we simply write B(H). Linear maps can be compose as well as added that is to say B(H) is not only a normed vector space but is also a ring. Notice the inequality relating the norm to the composition of linear maps ||ST || ≤ ||S|| ||T || which follows from the definition of the norm of an operator. Suppose that T : H → K is a bounded linear map. Then for each fixed y ∈ K, the map from H to C defined by x 7→ hT x, yi is bounded and linear. Proposition II.1 The map T ∗ : K → H so defined is bounded linear operator with norm ||T ∗ || = ||T ||. The operator T ∗ is called the adjoint.
3 Definition II.2 An operator T ∈ B(H) has finite rank if im T is finite-dimensional. An operator T is compact if there exists a sequence of finite rank operators Tn that converges to T in the norm. The space of compact operators on H is denoted κ(H). It is easy to see that the sum of two finite rank operators is finite rank ad that the composition of a finite rank operator and a bounded operator is finite rank. For compact operators we have the following. Proposition II.3 κ(H) forms a closed, two-sided ideal in B(H). This is to say that the sum of two compact operators is compact, the product of a compact operator with a bounded operator is compact and the limit of a convergent sequence of compact operators is compact. Lemma II.4 (Closed Graph Lemma) An algebraically invertible operator on Hilbert space is topologically invertible. That is, if T : V → W is a bijective bounded linear map between Hilbert spaces then the linear map T −1 : W → V is also bounded.
B.
The Theorem of Atkinson
Theorem II.5 Let T be a bounded operator on a Hilbert space H. Then the following conditions are equivalent: • T is Fredholm. • T is invertible modulo finite rank operators: there is a bounded operator S such that I− and I − T S are of finite rank. • ST T is invertible modulo compact operators: there is a bounded operator S such that I − ST and I − T S are compact operators. Atkinson’s theorem can also be expressed as follows: an operator T is Fredholm if and only if its image in the quotient algebra B(H) κ(H) is invertible. This quotient which is known as the Calkin algebra is an important object in operator theory. Corollary II.6 Let T be a Fredholm operator and K a compact operator. Then T + K is Fredholm and has the same index as T . Proposition II.7 If T1 , T2 are Fredholm operators on a Hilbert space H then so is their composition T1 T2 with Ind T1 T2 = Ind T1 + Ind T2 .
III.
TOEPLITZ INDEX THEOREM
Definition III.1 The Hardy space H 2 (S 1 ) is the closed subspace of L2 (S 1 ) comprised of those functions f ∈ L2 (S 1 ) all of whose negative Fourier coefficients are zero. In other words, those for which hf, en i = 0 for n < 0 where en (t) = eint ,
t ∈ [0, 2π],
n ∈ Z.
The Hardy projection P is the orthogonal projection onto Hardy space. Thus P f has the same Fourier coefficients as f for n ≥ 0 and zero Fourier coefficients for n < 0. Remark III.2 If we think of f as an L2 function of z = eit defined on the unit circle in the complex plane, then the functions in the Hardy space are precisely those that involve only non-negative powers of z and therefore extend to holomorphic functions defined on the unit disc.
4 Definition III.3 Let g be a continuous function on the circle. The Toeplitz operator with symbol g is the operator on the Hardy space defined by Tg = P Mg where the multiplication operator Mg on H is defined as Mg (f ) = gf,
∀f ∈ H = L2 (S 1 ).
In other words, to compute Tg (f ) we first have to multiply f by g and then project the result back into the Hardy space. Proposition III.4 If the function g is nowhere vanishing, then the Toeplitz operator Tg if a Fredholm operator. Theorem III.5 (Toeplitz Index Theorem) Let g : S 1 → C/{0} be a nowhere-vanishing function. Then Ind Tg = −wn(g, 0) where we consider g as a loop that does not pass through 0. Example. Suppose that g(t) = eit . Then for each of the basis elements e0 , e1 , e2 , ... of the Hardy space we have Tg en = en+1 . Thus Tg is in fact a unilateral shift and has index -1. On the other hand, the path g described is simply the unit circle traversed once in the positive direction so wn(g, 0) = +1. Theorem III.6 Let ψ : S 1 → GL(n, C) be a continuous, matrix-valued symbol and let Tψ be the corresponding matrix Toeplitz operator Then Tψ is Fredholm and its index is given by Ind Tψ = −wn(det ψ, 0) where det ψ is the path in C/{0} given by the determinant of ψ.