On commutators of quadratic operators | BlMSA-Tsinghua Quantum Symmetry Seminar

Abstract

A bounded linear operator A is said to be quadratic if there is a polynomial p of degree 2 suchthat p(A)=0. Square zero operators, involutions, and idempotents are all typical quadraticoperators. We will give characterizations of matrices could be expressed as commutators of twosquare zero matrices, and explain some related results about limits of commutators of two squarezero operators acting on a complex, separable Hilbert space H. We will also study the norm-closure of the set Ce of bounded linear operators acting on H which may be expressed as thecommutator of two idempotent operators. In particular, biquasitriangular operators belong to thenorm-closure of Ce are fully charateriezed. This talk is based on joint papers with Laurent Marcouxand Heydar Radjavi.