In all of the Star Wars movies, a Jedi's main weapon is the lightsaber. We can see them with many different colors.
So my question is, can the owner of a lightsaber be killed by it?
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The most obvious example I can think of is in Revenge of the Sith when Anakin kills Count Dooku using his own lightsaber and Dooku's to chop his head off as mentioned in Shreedhar's answer.
However, if you want an example of a Jedi killing themselves with their own lightsaber, and so not someone else killing them with it, we have a Legends example. In the story "The Duty" from the comic Star Wars Tales 12 Gira commits suicide with his own lightsaber after being bested by Darth Vader.
A lightsaber is simply a weapon thus it can be used to kill even its owner. The one instance that I remember happening on-screen is the murder of Count Dooku by Anakin.
In Revenge of the Sith, Anakin overpowers Dooku severing his hands and when Palpatine then tells Anakin to execute Dooku on the spot, he does so without the slightest of hesitation. There, Anakin uses two lightsabers: his and Dooku's. You can watch it here.
Another instance, off-screen (specifically, from a deleted scene) is the death of the Jedi master Shaak Ti. In Revenge of the Sith, Shaak Ti is captured by General Grievous and killed in front of Obi-Wan and Anakin (video here, thanks @Valorum). However, I am not sure if the lightsaber that Grievous used belonged to Shaak Ti.