# An operator-valued $T1$ theory for symmetric CZOs

## TL;DR

This paper establishes a BMO-criterion for the boundedness of symmetric Calderón-Zygmund operators with operator-valued kernels, integrating classical and quantum probability methods, and extends results to commutators involving Riesz transforms.

## Contribution

It introduces a natural BMO-criterion for operator-valued Calderón-Zygmund operators with symmetric kernels, combining classical and quantum probability techniques.

## Key findings

- Established a BMO-criterion for $L_2$-boundedness of operator-valued CZOs.
- Proved $L_2$-boundedness of commutators with Riesz transforms for functions in Bourgain's BMO.
- Utilized operator-valued Haar multipliers in the analysis.

## Abstract

We provide a natural BMO-criterion for the $L_2$-boundedness of Calder\'on-Zygmund operators with operator-valued kernels satisfying a symmetric property. Our arguments involve both classical and quantum probability theory. In the appendix, we give a proof of the $L_2$-boundedness of the commutators $[R_j,b]$ whenever $b$ belongs to the Bourgain's vector-valued BMO space, where $R_j$ is the $j$-th Riesz transform. A common ingredient is the operator-valued Haar multiplier studied by Blasco and Pott.