Thank you for your post and welcome to the Bose Community.
Due to the Bluetooth Low Energy technology, the speaker will always be ready to connect with any paired devices. However, I do not believe that the speaker should be turning straight back on once powering down. Generally, this works if you come back into range with the speaker and then select it from your device's pairing list.
Therefore, I would recommend resetting the speaker. You can do this by holding the Power button for 10 seconds until the Bluetooth light blinks blue and a voice prompt instructs you to select a language.
Additionally, please check for/install any available firmware updates. Simply visit btu.bose.com on your computer to do this.