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Operators with Index and Homotopy Theory Part 1 Operators with Index and Homotopy Theory ‘‘Ifwedonotsucceedinsolvinga mathematicalproblem,thereason frequentlyisourfailuretorecognize themoregeneralstandpointfrom whichtheproblembeforeusappears onlyasasinglelinkinachainof relatedproblems.’’ (D.Hilbert,1900) CHAPTER 1 Fredholm Operators 1. Hierarchy of Mathematical Objects “In the hierarchy of branches of mathematics, certain points are recognizable where there is a definite transition from one level of abstraction to a higher level. The first level of mathematical abstraction leads us to the concept of the individ- ual numbers, as indicated for example by the Arabic numerals, without as yet any undetermined symbol representing some unspecified number. This is the stage of elementary arithmetic; in algebra we use undetermined literal symbols, but con- sider only individual specified combinations of these symbols. The next stage is that of analysis, and its fundamental notion is that of the arbitrary dependence of one number on another or of several others — the function. Still more sophisti- cated is that branch of mathematics in which the elementary concept is that of the transformation of one function into another, or, as it is also known, the operator.” Thus Norbert Wiener characterized the “hierarchy” of mathematical objects [Wie33,p.1]. Very roughly we can say: Classical questions of analysis are aimed mainly at investigations within the third or fourth level. This is true for real and complex analysis, as well as the functional analysis of differential operators with its focus on existence and uniqueness theorems, regularity of solutions, asymp- totic or boundary behavior which are of particular interest here. Thereby research progresses naturally to operators of more complex composition and greater gener- ality without usually changing the concerns in principle; the work remains directed mainly towards qualitative results. In contrast it was topologists, as Michael Atiyah variously noted, who turned systematically towards quantitative questions in their topological investigations of algebraic manifolds, their determination of quantitative measures of qualitative behavior, the definition of global topological invariants, the computation of inter- section numbers and dimensions. In this way, they again broadly broke through the rigid separation of the “hierarchical levels” and specifically investigated rela- tions between these levels, mainly of the second and third level (algebraic surface — set of zeros of an algebraic function) with the first, but also of the fourth level (Laplace operators on Riemannian manifolds, Cauchy-Riemann operators, Hodge theory) with the first. This last direction, starting with the work of William V. D. Hodge, continuing with Kunihiko Kodaira and Donald Spencer, with Henri Cartan and Jean Pierre Serre, with Friedrich Hirzebruch, Michael Atiyah and others, can perhaps be best described with the key word differential topology or analysis on manifolds.Both its relation with and distinction from analysis proper is that (from [Ati68b, p.57]) 2 2. THE CONCEPT OF FREDHOLM OPERATOR 3 “Roughly speaking we might say that the analysts were dealing with complicated operators and simple spaces (or were only ask- ing simple questions), while the algebraic geometers and topolo- gists were only dealing with simple operators but were studying rather general manifolds and asking more refined questions.” We can read, e.g. in [Bri74, p.278-283] and the literature given there, to what degree the contrast between quantitative and qualitative questions and methods must be considered a driving force in the development of mathematics beyond the realm sketched above. Actually, in the 1920’s already, mathematicians such as Fritz Noether and TorstenCarlemanhaddevelopedthepurelyfunctional analytic concept of the index of an operator in connection with integral equations, and had determined its essen- tial properties. But “although its (the theory of Fredholm operators) construction did not require the development of significantly different means, it developed very slowlyandrequiredtheefforts of very many mathematicians” [GK57, p.185].And although Soviet mathematicians such as Ilja V. Vekua had hit upon the index of elliptic differential equations at the beginning of the 1950’s, we find no reference to these applications in the quoted principal work on Fredholm operators. In 1960 Israel M. Gelfand published a programmatic article asking for a systematic study of elliptic differential equations from this quantitative point of view. He took as a starting point the theory of Fredholm operators with its theorem on the homotopy invariance of the index (see below). Only after the subsequent work of Michail S. Agranovich, Alexander S. Dynin, Aisik I. Volpert, and finally of Michael Atiyah, Raoul Bott, Klaus Jänich and Isadore M. Singer, did it become clear that the theory of Fredholm operators is indeed fundamental for numerous quantitative computa- tions, and a genuine link connecting the higher “hierarchical levels” with the lowest one, the numbers. 2. The Concept of Fredholm Operator Let H be a (separable) complex Hilbert space, and let be the Banach algebra (e.g., [Ped, p.128], [Rud, p.228], [Sche, p.201]) of boundedB linear operators T : H H with the operator norm → T := sup Tu : u 1 < , k k {k k k k ≤ } ∞ where is the norm in H induced by the inner product , . k·k h· ·i Definition 1.1. An operator T is called a Fredholm operator,if ∈ B Ker T := u H : Tu =0 and Coker T := H/Im(T ) { ∈ } are finite-dimensional. This means that the homogeneous equation Tu =0has only finitely many linearly independent solutions, and to solve Tu = v,itissufficient that v satisfy a finite number of linear conditions (e.g., see Exercise 2.1b below). We write T and define the index of T by ∈ F index T := dim Ker T dim Coker T. − The codimension of Im(T )=T (H)= Tu : u H is dim Coker T . { ∈ } 4 1. FREDHOLM OPERATORS Remark 1.2. We can analogously define Fredholm operators T : H H0, → where H and H0 are Hilbert spaces, Banach spaces, or general topological vector spaces. In this case, we use the notation (H, H0) and (H, H0), corresponding to and above. However, in order to counteractB a proliferationF of notation and symbolsB inF this section, we will deal with a single Hilbert space H and its operators as far as possible. The general case H = H0 does not require new arguments at this point. Later, however, we shall apply6 the theory of Fredholm operators to elliptic differential and pseudo-differential operators where a strict distinction between H and H0 (namely, the compact embedding of the domain H into H0,seeChapter9) becomes decisive. All results are valid for Banach spaces and a large part for Fréchet spaces also. For details see for example [PRR, p.182-318]. We will not use any of these but will be able to restrict ourselves entirely to the theory of Hilbert space whose treatment is in parts far simpler. Motivated by analysis on symmetric spaces — with transformation group G — operators have been studied whose “index” is not a number but an element of the representation ring R(G) generated by the characters of finite-dimensional represen- tations of G [AS68a, p.519f] or, still more generally, is a distribution on G [Ati74, p.9-17]. We will not treat this largely analogous theory, nor the generalization of the Fredholm theory to the discrete situation of von Neumann algebras as it has been carried out — with real-valued index in [Bre, 1968/1969]. 2 Exercise 1.3. Let L (Z+) be the space of sequences c =(c0,c1,c2,...) of complex numbers with square-summable absolute values; i.e., ∞ 2 cn < . n=0 | | ∞ X 2 L (Z+) is a Hilbert space (see Chapter A below). Show that the forward shift + shift :(c0,c1,c2,...) (0,c0,c1,c2,...) 7→ and the backward shift shift− :(c0,c1,c2,...) (c1,c2,c3,...) 7→ + are Fredholm operators with index(shift )= 1 and index(shift−)=+1. 2 − [Warning: Just as we can regard L (Z+) as the limit of the finite dimensional vector + spaces Cm (as m ), we can approximate shift by endomorphisms of Cm given, relative to the standard→∞ basis, by the m m matrix × 00 0 0 ··· .. .. .. 1 . . 0 .. .. .. 0 . . . .. .. .. 0 0 010 ··· Note that the kernel and cokernel of this endomorphism are one-dimensional, whence the index is zero. (See Exercise 1.4 below.). 3. ALGEBRAIC PROPERTIES. OPERATORS OF FINITE RANK 5 We have yet another situation, when we consider the Hilbert space L2(Z) of se- quences c =(...,c 2,c 1,c0,c1,c2,...) with − − 2 ∞ 2 2 c0 + cn + c n < . − | | n=1 | | | | ∞ X ³ ´ The corresponding shift operators are now bijective, and hence have index zero.] 3. Algebraic Properties. Operators of Finite Rank Exercise 1.4. For finite-dimensional vector spaces the notion of Fredholm operator is empty, since then every linear map is a Fredholm operator. Moreover, the index no longer depends on the explicit form of the map, but only on the dimensions of the vector spaces between which it operates. More precisely, show that every linear map T : H H0 where H and H0 are finite-dimensional vector spaces has index given by → index T =dimH dim H0. − [Hint: One first recalls the vector-space isomorphism H/ Ker(T ) ∼= Im(T ) and then (since H and H0 are finite-dimensional) obtains the well-known identity from linear algebra dim H dim Ker T =dimH0 dim Coker T.] − − [Warning: If we let the dimensions of H and H0 go to , we obtain only the formula index T = . Thus, we need an additional theory,∞ to give this difference a particular value.]∞−∞ Exercise 1.5. For two Fredholm operators F : H H and G : H0 H0 consider the direct sum, → → F G : H H0 H H0. ⊕ ⊕ → ⊕ Show that F G is a Fredholm operator with ⊕ index(F G) = index F +indexG ⊕ [Hint: First verify that Ker(F G)=KerF Ker G, and the corresponding fact for Im(F G).
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