As jjcale mentions in a comment, the index of a Fredholm operator is very important in physics. One way to define the Chern number of a topological insulator is in terms of the index of a Fredholm operator, as explained in [1].
There is also the concept of an index of a pair of projections. This is seen a lot recently in physics papers, for example in [2]. That paper uses as a reference on the index of certain pairs of projections a paper in the Journal of Functional Analysis [3].
For physics published this year, see [4]. That paper discusses trace class operators and cites a text in functional analysis.
[1] Bellissard, Jean, Andreas van Elst, and Hermann Schulz‐Baldes. "The noncommutative geometry of the quantum Hall effect." Journal of Mathematical Physics 35.10 (1994): 5373-5451.
[2] Akagi, Yutaka, Hosho Katsura, and Tohru Koma. "A New Numerical Method for Topological Insulators with Strong Disorder." Journal of the Physical Society of Japan 86.12 (2017): 123710.
[3] Avron, J., Ruth Seiler, and Barry Simon. "The index of a pair of projections." Journal of Functional Analysis 120.1 (1994): 220-237.
[4] Zhi Li and Roger S. K. Mong, "A Local formula for the Z_2 invariant of topological insulators" Phys. Rev. B 100, 205101 – Published 4 November 2019.