First of all create a custom attribute called : "updated_by" from magento backend and don't forget to drag that into default attribute set otherwise it will not work!!

Add events.xml file to vendor/module/etc/adminhtml/

<?xml version="1.0"?>
<config xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:noNamespaceSchemaLocation="urn:magento:framework:Event/etc/events.xsd">
    <event name="catalog_product_save_after">
        <observer name="observer_name" instance="vendor\module\Observer\ProductSaveAfter" />
    </event>
</config>
Add ProductSaveAfter.php at vendor/module/Observer/

<?php
namespace vendor\module\Observer;
use Magento\Framework\Event\ObserverInterface;
class ProductSaveAfter implements ObserverInterface
{
    public function execute(\Magento\Framework\Event\Observer $observer)
    {
        $instance = \Magento\Framework\App\ObjectManager::getInstance();
        $productId = (int) $observer->getProduct()->getId();
        $user = $instance->get('Magento\Backend\Model\Auth\Session')->getUser()->getUsername();
        $action = $instance->create('Magento\Catalog\Model\Product\Action');
        $action->updateAttributes([$productId], array('updated_by' => $user), 0);
    }
}