Determinants and Plemelj-Smithies formulas

TL;DR

This paper develops Plemelj-Smithies formulas for determinants in various operator algebras, including operators on the torus and elliptic operators on manifolds, linking determinants to symbols and invariant structures.

Contribution

It introduces new determinant formulas for operators on the torus, pseudo-differential operators, and elliptic operators, expanding the analytical tools for operator determinants.

Findings

Formulas for determinants of periodic pseudo-differential operators in terms of symbols

Determinant formulas for invariant operators relative to Hilbert space decompositions

Explicit determinant formulas for elliptic operators on manifolds and Lie groups

Abstract

We establish Plemelj-Smithies formulas for determinants in different algebras of operators. In particular we define a Poincar\'e type determinant for operators on the torus $\Tn$ 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.