On τ-essentially Invertibility of τ-measurable Operators

Abstract

Let $$ \mathcal{M} $$ be a von Neumann algebra of operators on a Hilbert space and τ be a faithful normal semifinite trace on $$ \mathcal{M} $$ . Let I be the unit of the algebra $$ \mathcal{M} $$ . A τ-measurable operator A is said to be τ-essentially right (or left) invertible if there exists a τ-measurable operator B such that the operator I − AB (or I − BA) is τ-compact. A necessary and sufficient condition for an operator A to be τ-essentially left invertible is that A∗A (or, equivalently, $$ \sqrt{A^{\ast }A} $$ ) is τ-essentially invertible. We present a sufficient condition that a τ-measurable operator A not be τ-essentially left invertible. For τ-measurable operators A and P = P2 the following conditions are equivalent: 1. A is τ-essential right inverse for P; 2. A is τ-essential left inverse for P; 3. I − A,I − P are τ-compact; 4. PA is τ-essential left inverse for P. For τ-measurable operators A = A3, B = B3 the following conditions are equivalent: 1. B is τ-essential right inverse for A; 2. B is τ-essential left inverse for A. Pairs of faithful normal semifinite traces on $$ \mathcal{M} $$ are considered.