Boundedness properties of fermionic operators

TL;DR

This paper investigates the boundedness of fermionic second quantization operators, establishing conditions relating operator classes to bounds involving the number operator, and explores implications for quadratic creation and annihilation operators.

Contribution

It provides new bounds for fermionic operators based on Schatten class conditions and links these bounds to properties of the number operator, with applications to quadratic operators.

Findings

Boundedness of $d ext{Gamma}(B)$ by $N^{s/2}$ under Schatten class conditions

Inverse implications from number operator estimates to Schatten class conditions

Extension of results to quadratic creation and annihilation operators

Abstract

The fermionic second quantization operator $d\Gamma(B)$ is shown to be bounded by a power $N^{s/2}$ of the number operator $N$ given that the operator $B$ belongs to the $r$-th von Neumann-Schatten class, $s=2(r-1)/r$. Conversely, number operator estimates for $d\Gamma(B)$ imply von Neumann-Schatten conditions on $B$. Quadratic creation and annihilation operators are treated as well.