Algebraic And\^{o} Dilation

TL;DR

This paper addresses the Andf3 dilation problem for linear maps, demonstrating that any pair of commuting linear maps on a vector space can be extended to commuting injective linear maps, advancing the understanding of operator dilations.

Contribution

It provides a solution to the Andf3 dilation problem for linear maps, showing that commuting linear maps can be dilated to commuting injective maps, which was previously unresolved.

Findings

Any commuting linear maps can be dilated to commuting injective maps.

The result extends the understanding of dilation theory in linear algebra.

Addresses a problem posed by Krishna and Johnson in 2022.

Abstract

We solve the And\^{o} dilation problem for linear maps on vector space asked by Krishna and Johnson in \textit{[Oper. Matrices, 2022]}. More precisely, we show that any commuting linear maps on vector space can be dilated to commuting injective linear maps.