There is every good reason to think that Jesus's followers, during his lifetime, believed that he might be this coming anointed one. Two pieces of data must be seen in tandem to recognize their full force. The first is one I have already mentioned, that "Christ" (i.e., anointed one;i.e., messiah) was far and away the most common descriptive title the early Christians used for Jesus, so much so that they often called him Christ rather than Jesus (so that, despite my little joke earlier, it really did begin to function as his name). This is very surprising, given the fact that as far as we can tell, Jesus did nothing during his life to make anyone think that he was this anointed one. That is to say, he did not come on the clouds of heaven to judge the living and the dead; he was not a priest; and he never raised an army and drove the Romans out of the promised land to set up Israel as a sovereign state. So why did his followers so commonly designate him by a title that suggested that he had done one of these things?
This question relates to the second datum. Many Christians today assume that the earliest followers of Jesus concluded that he was the messiah because of his death and resurrection: if Jesus died for sins and was raised from the dead, he must be the messiah. But such thinking is precisely wrong, for reasons that you may already have inferred from what I have said to this point. Ancient Jews had no expectation-zero expectation-that the future messiah would die and rise from the dead. That was not what the messiah was supposed to do. Whatever specific idea any Jew had about the messiah (as cosmic judge, mighty priest, powerful warrior), what they all thought was that he would be a figure of grandeur and power who would be a mighty ruler of Israel. And Jesus was