# On Multivariate Matsaev's Conjecture

## TL;DR

This paper investigates multivariate extensions of Matsaev's conjecture in both commutative and non-commutative L^p-spaces, ultimately showing the conjecture's failure in the multivariate case and presenting dilation results.

## Contribution

It demonstrates the falsehood of the multivariate Matsaev's conjecture for all 1<p<∞ and provides joint dilation results in non-commutative L^p-spaces.

## Key findings

- Multivariate Matsaev's conjecture is false for all 1<p<∞.
- Established joint dilation results in non-commutative L^p-spaces.
- Extended understanding of operator behavior in multivariate non-commutative settings.

## Abstract

In this article, we study multivariate generalizations of the Matsaev's conjecture in commutative and non-commutative $L^p$-spaces.We prove that the multivariate analogue of Matsaev's conjecture is eventually false for all $1<p<\infty.$ We exhibit various joint dilation results on non-commutative $L^p$-spaces.