Technically that would be a different thermodynamic cycle. For a true jet engine, you need compression, heating, and then expansion. Using the heat of compression doesn't count; it's thermodynamically equivalent to a propeller (just less efficient because not all of the work is being converted directly into kinetic energy, but rather a bunch is converted into heat).

One could however make an electric jet engine. You "just" need to replace the fuel-based heating element with an electric-based one. Then the front compressor would be electric, and you'd have no use for the turbine, since its sole purpose in conventional jet engines is to extract energy from the combustion to power the compressor.

I have this recurring daydream of an electrodynamic jet engine. The inlet ionizes the airflow, the compressor is electromagnetic, the heating element inductive. Just one fancy-shaped tube with a bunch of complicated electronics around it. That would be really interesting from a whole lot of perspectives -- for example, noise reduction. It also would open up a lot more flexibility in terms of power density (which is more or less determined by the amount of heat you can dump into the engine, which is more or less limited by the melting point of the turbine blades).

Something else that's interesting about that idea is that it doesn't depend on oxygen in the air, which means you could use it on Mars and Venus. I've never looked into the numbers for Venus, but as I remember, the local speed of sound is a significant challenge for rotorcraft on Mars, so having a rotorless, purely-electric jet engine there could open up a lot of flexibility in terms of aircraft performance.