Author:
Cardona Duván,Delgado Julio,Ruzhansky Michael
Abstract
AbstractWe establish Plemelj-Smithies formulas for determinants in different algebras of operators. In particular we define a Poincaré type determinant for operators on the torus $${{\mathbb T}^n}$$
T
n
and deduce formulas for determinants of periodic pseudo-differential operators in terms of the symbol. On the other hand, by applying a recently introduced notion of invariant operators relative to fixed decompositions of Hilbert spaces we also obtain formulae for determinants with respect to the trace class. The analysis makes use of the corresponding notion of full matrix-symbol. We also derive explicit formulas for determinants associated to elliptic operators on compact manifolds, compact Lie groups, and on homogeneous vector bundles over compact homogeneous manifolds.
Publisher
Springer Science and Business Media LLC
Reference31 articles.
1. Borodin, A., Corwin, I., Remenik, D.: Log-gamma polymer free energy fluctuations via a Fredholm determinant identity. Commun. Math. Phys. 324(1), 215–232 (2013)
2. Bothner, T., Its, A.: Asymptotics of a Fredholm determinant corresponding to the first bulk critical universality class in random matrix models. Commun. Math. Phys. 328(1), 155–202 (2014)
3. Bott, R., The index theorem for homogeneous differential operators. In: Differential and Combinatorial Topology. A Symposion in Honour of Marston Morse, pp. 167–186. (1965)
4. Bornemann, F.: On the numerical evaluation of Fredholm determinants. Math. Comput. 79(270), 871–915 (2010)
5. Delgado, J.: A trace formula for nuclear operators on $$L^p$$. In: Pseudo-Differential Operators: Complex Analysis and Partial Differential Equations Volume 205 of Oper. Theory Adv. Appl., pp. 181–193. Birkhäuser Verlag, Basel (2010)