Fredholm Operator - Examples

Examples

Let H be a Hilbert space with an orthonormal basis {en} indexed by the non negative integers. The (right) shift operator S on H is defined by

This operator S is injective (actually, isometric) and has a closed range of codimension 1, hence S is Fredholm with ind(S) = −1. The powers Sk, k ≥ 0, are Fredholm with index −k. The adjoint S∗ is the left shift,

The left shift S∗ is Fredholm with index 1.

If H is the classical Hardy space H2(T) on the unit circle T in the complex plane, then the shift operator with respect to the orthonormal basis of complex exponentials

e_n : \mathrm{e}^{\mathrm{i} t} \in \mathbf{T} \rightarrow \mathrm{e}^{\mathrm{i} n t}, \quad n \ge 0, \,

is the multiplication operator Mφ with the function φ = e1. More generally, let φ be a complex continuous function on T that does not vanish on T, and let Tφ denote the Toeplitz operator with symbol φ, equal to multiplication by φ followed by the orthogonal projection P from L2(T) onto H2(T):

Then Tφ is a Fredholm operator on H2(T), with index related to the winding number around 0 of the closed path t ∈ → φ(e i t ) : the index of Tφ, as defined in this article, is the opposite of this winding number.

Read more about this topic:  Fredholm Operator

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