The inclusion of the Schur algebra in B(l^2) is not inverse-closed

TL;DR

This paper disproves a conjecture by Q. Sun by showing that the Schur algebra, consisting of operators bounded on l^1 and l^{ty}, is not inverse-closed, as an invertible operator in l^2 has an inverse not bounded on l^1 or l^{ty}.

Contribution

The paper provides a counterexample demonstrating that the Schur algebra is not inverse-closed, challenging previous conjectures in operator algebra theory.

Findings

Counterexample of an operator in the Schur algebra with an inverse not in the algebra.

Disproof of Q. Sun's conjecture on inverse-closedness.

Clarification of the limitations of the Schur algebra's inverse properties.

Abstract

The Schur algebra is the algebra of operators which are bounded on l^1 and on l^{\infty}. Q. Sun conjectured that the Schur algebra is inverse-closed. In this note, we disprove this conjecture. Precisely, we exhibit an operator in the Schur algebra, invertible in l^2, whose inverse is not bounded on l^1 nor on l^{\infty}.