A few remarks about operator theory, topology, and analysis on metric spaces

Some basic facts about Fredholm indices are briefly reviewed, often used in connection with Toeplitz and pseudodifferential operators, and which may be relevant for operators associated to fractals. Let H be a complex infinite-dimensional separable Hilbert space, which is of course unique up to isomorphism. Let B(H) be the Banach algebra of bounded linear operators on H, and let C(H) be the closed ideal in B(H) consisting of compact operators on H. The quotient A(H) = B(H)/C(H) is a C algebra known as the Calkin algebra. A bounded linear operator T on H is said to be Fredholm if the kernel of T is finite-dimensional and T maps H onto a closed linear subspace of H of finite codimension. Sometimes it is convenient to consider unbounded linear operators too, such as differential operators. It may also be helpful to include operators between different Hilbert spaces. One way to deal with unbounded linear operators is to treat them as bounded linear operators between different Hilbert spaces. If T ∈ B(H) is Fredholm, then there is an R ∈ B(H) such that RT − I and T R − I are finite rank operators, where I is the identity operator on H. Conversely, T ∈ B(H) is Fredholm if there is an R ∈ B(H) such that RT − I and T R− I are compact operators. Equivalently, T ∈ B(H) is Fredholm if and only if the corresponding element T̂ of A(H) is invertible. The index of a Fredholm operator T ∈ B(H) is defined to be the difference between the dimension of the kernel of T and the codimension of T (H) in H. If T, T ′ ∈ B(H), T is Fredholm, and T ′ − T is compact, then T ′ is a Fredholm operator with the same index as T . If T, T ′ ∈ B(H), T is Fredholm, and T ′ is sufficiently close to T in the operator norm topology, then T ′ is again Fredholm and has the same index as T . The composition of two Fredholm operators is Fredholm, and the index of the composition is equal to the sum of the indices. Many questions in analysis are concerned with invertability of linear operators. One might start by showing that an operator is Fredholm, and then try to analyze the kernel and cokernel. These may be considered as nuisances to be minimized. By contrast, from the perspective of topology, nontrivial indices are an opportunity.