Fredholm Δ-filtrations of Fredholm groups

In this short note, we show that some well-known filtrations of infinite dimensional groups of Fredholm operators associated to certain perturbation classes are, in fact, Fredholm Δ-filtrations (see Definition 5 below.) For example, let E be a separable infinite dimensional real Hilbert space. The group GL K ( E ) of all invertible operators on E which are compact perturbations of the identity is the structure group for Hilbert Fredholm manifolds and bundles modelled on E [1] , [2] , [3] . Using an orthonormal basis, there are canonical inclusions of general linear groups: GL ( 1 ) ⊂ ⋯ ⊂ GL ( n ) ⊂ GL ( n + 1 ) ⊂ ⋯ ⊂ GL ( ∞ ) = lim → GL ( n ) ⊂ GL K ( E ) . We show this is a Fredholm Δ-filtration of the Fredholm manifold GL K ( E ) with dimension sequence Δ ( n ) = dim ⁡ ( GL ( n ) ) = n 2 , which was not discussed in the classical Fredholm manifold literature because of the rigid constraint that the dimensions of a filtration increase only by one.