Do the compressors and turbine rotate at the same speed? Since the turbine and the compressors are connected to each other via a common shaft, do they both rotate at the same speed. Because I've read that in some jet engines (single spool engines) they don't rotate at the same speed:


  Single spool jet engine. These are those jet engines who have one shaft inside them. The compressors and turbines rotate on a same shaft. The speed of compressors and turbines decreases half way down the compressor stages, that is why they can’t match the local airflow. The various turbines and compressors is forced to rotate at same speed which lead to their inefficiencies.


(quora.com)

Is this true?
That's one badly-worded quote:


  Single spool jet engine. These are those jet engines who have one shaft inside them. The compressors and turbines rotate on a same shaft. 


Correct. In a single-spool engine, all of the compressor and turbine stages rotate at the same speed because they're attached rigidly to the shaft. 


  The speed of compressors and turbines decreases half way down the compressor stages, that is why they can’t match the local airflow. 


No, the speed of the compressors and turbines doesn't change. It's the airflow that changes, and causes a mismatch.