"dear professor" is correct and usable in written English.
You can use it when you are addressing a professor, either formally or informally. For example: Dear Professor Smith, I hope you are doing well. I am writing to ask for your advice on getting involved in research projects at the university. Thank you for your time, [Your name].
"My dear Professor," Schnitzler wrote in 1906, in a translation by UCL's Judith Beniston, "Even if you have almost forgotten who I am, allow me nonetheless to add my congratulations to the many that you will be receiving today.
"Dear Professor Dodd, I thought you were dead, but now that I know that you're alive, I'd like to come study with you," Mr. Buffett said he wrote in his letter.
It must be said here that Ms. Bruni's good taste was influenced by her good friend Marianne Faithfull (another Jagger alumna), or her "dear professor," as Ms. Bruni has referred to her more than once.
When I first came to the UK and wrote emails, I'd be really respectful and say "dear professor", now I just call them by their first names, which I couldn't imagine doing in France.
While a student at Radcliffe in the late eighteen-nineties, Stein, faced with an examination in William James's philosophy course for which she has not studied, writes on the examination paper, "Dear Professor James, I am so sorry but really I do not feel a bit like an examination paper in philosophy to-day," and leaves the examination room.
Please be sure to include "Dear Professor Investor" in the subject line.
"My dear old professor, Alan Westin, would be shaking his head if he read this article," the mayor said with a laugh, and he was right.
"Oh dear," the Professor said, "another bloody dead turtle boy in the garden.
SAMETZ--Arnold W. The members of the NYU Leonard N. Stern School of Business community mourn the passing of our dear friend and professor emeritus, Arnie Sametz.
"I want to completely better myself, I want to be worthy of you, you must help me to transform, you will be my little professor, dear, and I will blindly listen to you like a master that I adore," she wrote in January 1952.
