nLab Fredholm group

Definition

Denote by

• $\mathscr{H}$ the countably-infinite-dimensional separable Hilbert space,

• $\mathcal{B}(\mathscr{H})$ the norm topology-space of bounded operators,

• $\mathcal{K}(\mathscr{H}) \subset \mathcal{B}(\mathscr{H})$ the subset of compact operators,

• $GL(\mathscr{H}) \subset \mathcal{B}(\mathscr{H})$ the general linear group,

• $\mathrm{U}(\mathscr{H}) \subset GL(\mathscr{H})$ the unitary group (cf. at U(ℋ)).

Definition

The Fredholm group of $\mathscr{H}$ is the subgroup of the general linear group of those elements which differ from the identity operator by a compact operator:

$$GL^c(\mathscr{H}) \coloneqq \big\{ \phi \in GL(\mathscr{H}) \,\big\vert\, \phi - id \in \mathcal{K}(\mathscr{H}) \big\} \,.$$

(cf. Atiyah & Singer 1969 pp 10, Swanson 1978 p 191, Mukherjea 1970 p. 653)

Analogously, the unitary Fredholm group is the subgroup of U(ℋ) of those elements which differ from the identity operator by a compact operator:

$$U^c(\mathscr{H}) \coloneqq \big\{ U \in \mathrm{U}(\mathscr{H}) \,\big\vert\, U - id \in \mathcal{K}(\mathscr{H}) \big\} \,.$$

(cf. Atiyah & Singer 1969 pp 10, Andruchow & Larotonda 2010, DSBW 2023 p. 87, 247)

References

The Fredholm group is named in variation of Fredholm operators after Ivar Fredholm.

Mentioning of the Fredholm group:

• Michael Atiyah, Isadore Singer, pp 10 of: Index theory for skew-adjoint Fredholm operators, Publications Mathématiques de l’Institut des Hautes Scientifiques 37 1 (1969) 5-26 [doi:10.1007/BF02684885, pdf]

  (not calling these groups by any name other than a symbol)

• Kalyan K. Mukherjea: The Homotopy Type of Fredholm Manifolds, Transactions of the AMS 149 (1970) 653-663 [doi:10.1090/S0002-9947-1970-0259954-4, pdf]

• R. C Swanson: Fredholm intersection theory and elliptic boundary deformation problems, I, Journal of Differential Equations 28 2 (1978) 189-201 [doi:10.1016/0022-0396(78)90066-9]

Mentioning of the unitary Fredholm group:

• Michael Atiyah, Isadore Singer: Index theory for skew-adjoint Fredholm operators, Publications Mathématiques de l’Institut des Hautes Scientifiques 37 1 (1969) 5-26 [doi:10.1007/BF02684885, pdf]

• Esteban Andruchow, Gabriel Larotonda: The rectifiable distance in the unitary Fredholm group, Studia Mathematica Volume 196 2 (2010) 151-178 [arXiv:0812.4475, eudml:285458]

• Nora Doll, Hermann Schulz-Baldes, Nils Waterstraat, Sections 3.7 and 8.1 in: Spectral Flow — A Functional Analytic and Index-Theoretic Approach, Studies in Mathematics 94, De Gruyter (2023) [doi:10.1515/9783111172477, hdl:20.500.12657/63798, ch8:pdf]