FREDHOLM OPERATORS AND THE GENERALIZED INDEX JOSEPH BREEN CONTENTS 1. Introduction 1 1.1. Index Theory in Finite Dimensions 2 2. The Space of Fredholm Operators 3 3. Vector Bundles and K-Theory 6 3.1. Vector Bundles 6 3.2. K-Theory 8 4. The Atiyah-Janich¨ Theorem 9 4.1. The Generalized Index Map 10 5. Index Theory Examples 12 5.1. Toeplitz Operators and the Toeplitz Index Theorem 12 5.2. The Atiyah-Singer Index Theorem 14 6. Conclusion 16 References 17 1. INTRODUCTION One of the most fundamental problems in mathematics is to solve linear equations of the form T f = g, where T is a linear transformation, g is known, and f is some unknown quantity. The simplest example of this comes from elementary linear algebra, which deals with solutions to matrix-vector equations of the form Ax = b. More generally, if V; W are vector spaces (or, in particular, Hilbert or Banach spaces), we might be interested in solving equations of the form T v = w where v 2 V , w 2 W , and T is a linear map from V ! W . A more explicit source of examples is that of differential equations. For instance, one can seek solutions to a system of differential equations of the form Ax(t) = x0(t), where, again, A is a matrix. From physics, we get important partial differential equations such as Poisson’s equation: ∆f = g, Pn @2 n where ∆ = i=1 2 and f and g are functions on R . In general, we can take certain partial @xi differential equations and repackage them in the form Df = g for some suitable differential operator D between Banach spaces. In each of these examples, given some sort of operator T and an equation T f = g, we wish to understand the existence and uniqueness of solutions. Unfortunately, understanding existence and uniqueness in isolation (i.e., as separate entities) often proves to be difficult. The goal of “index theory” is to do the next best thing: assign some kind of an index to T which encapsulates information about both existence and uniqueness, simultaneously. 1 2 JOSEPH BREEN Vague Example 1.1. Suppose that T is a linear map from X to Y . We want to understand the space of solutions to the equation T f = g. Suppose that we have a magic number N which measures the uniqueness of solutions, and another magic number M which measures the existence of solutions. Then we can define the index of T to be: (1.1) ind (T ) := N − M Thus, ind (T ) contains information about the existence and uniqueness of solutions, packaged into one number. Computing the index of T might not be as insightful as computing N and M separately, but it might be the case that ind (T ) is easier to calculate. Though this example is devoid of any actual meaning, it is the standard model for the concept of an index. 1.1. Index Theory in Finite Dimensions. To warm us up, let’s do something a little more concrete. Let V and W be finite dimensional vector spaces, and let T : V ! W be a linear transformation. We want to understand solutions to equations of the form T v = w. We make two observations. (i) Understanding uniqueness of solutions to T v = w corresponds to understanding injectiv- ity of T . Explicitly, if T is injective, then any solution to T v = w is unique. One way to measure the injectivity of T is to consider the dimension of the kernel of T , defined as ker T := f v 2 V : T v = 0 g, since T is injective exactly when dim ker T = 0. The larger the dimension of ker T is, the less injective T is. Hence, we have the following correspondence: ( ) ( ) uniqueness of solutions injectivity of =∼ −! dim ker T to T v = w T (ii) Similarly, we can understand the existence of solutions to T v = w by understanding the sur- jectivity of T . Explicitly, if T is surjective (i.e. if im T = W ) then existence of solutions to T v = w is guaranteed for any w 2 W . Define the cokernel of T to be coker T := W= im T , the quotient of the target space by the image of T . If quotienting makes you nervous, you can think of the cokernel as the orthogonal complement of im T in W . Hence, we can measure the surjectivity of T by considering the dimension of coker T . If dim coker T = 0, then im T is all of W , and so T is surjective. The larger dim coker T is, the less surjective T is. Thus, we have the following correspondence: ( ) ( ) existence of solutions surjectivity of =∼ −! dim coker T to T v = w T In the spirit of the vague example from above, N = dim ker T is a number which measures the uniqueness of solutions / injectivity of T , and M = dim coker T is a number which measures the existence of solutions / surjectivity of T . Modeling equation (1.1), we make the following definition. Definition 1.2. Let V and W be finite dimensional vector spaces, and let T : V ! W be linear. The index of T is ind (T ) := dim ker T − dim coker T FREDHOLM OPERATORS AND THE GENERALIZED INDEX 3 Note that if T is invertible, then dim ker T = dim coker T = 0, and so ind (T ) = 0. This suggests that the index of T measures how far T is from being invertible. Note, however, that ind (T ) = 0 does not imply that T is invertible; all we can say in this case is that dim ker T = dim coker T . Another thing to note is that the Rank-Nullity theorem renders index theory in finite dimensions practically useless, as demonstrated by the next proposition. Proposition 1.3. Let V and W be finite dimensional vector spaces, and let T : V ! W be linear. Then ind (T ) = dim V − dim W Proof. Since coker T = W= im T , we have dim coker T = dim W= im T = dim W − dim im T . Hence, ind (T ) = dim ker T − dim coker T = dim ker T − (dim W − dim im T ) = (dim ker T + dim im T ) − dim W By the Rank-Nullity theorem, dim ker T + dim im T = dim V , and we are done. In other words, the index of a map between finite dimensional spaces doesn’t actually depend on map itself; it only depends on the dimensions of the domain and codomain. Hence, the concept of an index only becomes useful when we move into the world of infinite dimensional vector spaces. 2. THE SPACE OF FREDHOLM OPERATORS Instead of considering a linear map T between finite dimensional vector spaces V and W , we will study linear operators T : H!H on infinite dimensional complex Hilbert spaces. As in the finite dimensional case, we want to understand the space of solutions to equations of the form T f = g. Also as before, this is equivalent to understanding injectivity and surjectivity of T . We might be tempted to define the index of an operator in the same way as before: ind (T ) := dim ker T − dim coker T But now that we’re working in infinite dimensions, we run into the possibility that one or both of dim ker T and dim coker T are infinite. For this reason, ind (T ) is not well-defined for all operators (for example, the operator 0 : H!H which sends everything to 0 has an infinite-dimensional kernel and an infinite-dimensional cokernel.). To remedy this, we will define a new class of operators for which the index is well-defined. Definition 2.1. An operator T : H!H is Fredholm if dim ker T and dim coker T are both finite. For a Fredholm operator T , its Fredholm index is: ind (T ) := dim ker T − dim coker T We denote the set of Fredholm operators on H by F(H). We can think about these Fredholm operators as being “almost-invertible” in the sense that the kernel and cokernel are small enough to measure. As in the finite dimensional case, the Fredholm index of an operator gives a measurement for how defective (i.e. not invertible) such an operator is. Example 2.2. An important class of Fredholm operators is the shift operators. Let fe1; e2;::: g be an orthonormal basis for H, and let U : H!H be the operator defined by Uej = ej+1. The image of U is: im U = Spanfe2; e3;::: g 4 JOSEPH BREEN Hence, coker U = H= im U = Spanfe1g, and so dim coker U = 1. Since U is injective, dim ker U = 0, and we have: ind (U) = dim ker U − dim coker U = 0 − 1 = −1 We can generalize this and consider the n-shift operator, which is the nth power of U: n U ej = ej+n n n Then im U = Spanfen+1; en+2;::: g. Repeating the computation above gives ind (U ) = −n. We can also consider the backwards-shift operator, which is the adjoint1 of U: ∗ U ej = ej−1 ∗ ∗ In this case, U is surjective, and ker U = Spanfe1g. Hence, ind (U ∗) = 1 − 0 = 1 The backwards n-shift operator (U ∗)n has index n. This example suggests the following general properties of the index. Proposition 2.3. (i) The map ind : F(H) ! Z is surjective; (ii) If T 2 F(H), then T ∗ 2 F(H), and ind (T ∗) = −ind (T ); (iii) If T;S 2 F(H), then TS 2 F(H), and ind (TS) = ind (T ) + ind (S).