Can proteases be degraded by themselves?

Yes, they can. Proteases can be cleaved by other protease molecules, either another unit of the same protease or a different protease, as long as they contain a specific cleavage site for them to recognize. This process is called autolysis, which acts as a method of regulating protease activity. Some proteases are less active after autolysis (e.g. TEV protease), while others are more active (e.g. trypsinogen).