I believe that Norman Bates was guilty of the murders of Marion and Arbogast. Because his mother which is a figment of his mind or imagination dose not exist for she is dead and Norman had to have made her up. The fact that his mother might have helped him develop the split personalities, by the way she treated him might be a possibility too but he manifested his own thoughts. The idea that Norman Bates mother committed the crime seems to be irrelevant she is dead and there is no way she physically did it. The idea of mother only exists in Norman's mind. For that is his problem and he created the idea of his mother in his own fantasy world.
The way he created his split personality was because he needed attention and he couldn't deal with the fact of his mother's death. So he saved her body, tried to preserve it and even tried to talk in her voice. I had and idea that Norman was the murderer, because through out the movie they never showed the mothers face. "She" was always in the shadows and you never heard the mother and Norman talking at the same time. Not to mention on the out line and sheets the teacher gave us Sick or mother was always in quotes. There is a reason for every thing they do in the movies.
When Bates saw Marion he felt affection and the side of his mother grew larger. Every time Marion mentioned his mother Norman got quite defensive and mad. It appeared that he thought she was telling him to get rid of his mother and that struck a nerve with him. Norman couldn't take it and the feeling of his other side grew to a point were he couldn't take it and his mother emerged from his mind. So mentally his mother killed Marion. It's not like Norman's mothers' mind was some how transferred into his head.
Norman Bates must have been a fine and a normal person until some drastic events occurred (Mother dying) which made him go psycho. Everyone's personalities are molded by the people that surround them and their environment. In this case his parents disrupted Norman's mental situation. So in that event it was originally Norman's parents' fault he went crazy but you are responsible for your own actions and thoughts.