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I'm a postdoc in mathematics in the US.
My question is exactly in the title. Sometimes you get an email from someone that you know, usually with a link to their job ad, asking you to apply and saying that you'd be a good fit, or something along those lines.
Faculty members, how often do you do this, and how does this change the likelihood of someone getting a job (or even, getting an interview)?
When you get an email inviting you to apply for a position, it can mean many things.
It may mean nothing special. At one extreme (and this doesn't sound like it applies in your case), faculty send out a batch email to many of the other academics they know informing them of the position. This is typically not personalised. Often there is also a hope with such emails that people will share the job advertisement around with colleagues.
It may mean that they think you would be a good fit, and that they would like you to apply. Within this range, there would be degrees of support. At the very least, the academic is indicating that they'd like you to apply. In some cases, they may be really hoping that you get the position. They may want you to get the position for various reasons: (a) they like you and want to support your career, (b) they think you'd be best suited for the job and the department, (c) they would rather have you as a colleague than someone else, perhaps because they like your area of research, they get along well with you, and so on. In other cases, they just want a strong applicant pool; so you would contribute to the strength of the applicant pool. From this perspective, their goal is to ensure that the department hires someone strong, and they see you as a strong candidate, but they would be even happier if a stronger candidate also applied and got the position.
Academics also vary in the degree to which they are involved in the decision of who to hire (e.g., no involvement, input of academic considered by selection panel, on the formal selection panel, person who makes final decision, etc.). If you are receiving an email from someone who is more involved with making the decision, then the email is likely to be more indicative of your chances of getting the position were you to apply.
More generally, if you are considering applying, receiving such an email is an invitation to have a chat with the sender to discuss the position and learn more about what the job entails and what your chances are of getting the position, were you to apply.
Update:
How do they know that you'd be a strong candidate? You write:
Would it be fair to say that if you get multiple of these emails from decent institutions, you are a strong candidate? It surprises me that people can make these deduction before seeing the application package, though.
PostDocs have a reputation. This is formed by many factors: (a) publications, particularly main or first author publications that give an insight into your capabilities (b) where you did your PhD, (c) where you are doing your PostDoc, (d) presentations and interactions at conferences, (e) your reputation within your academic social network, (f) your online activities, and so on. Some of this information is a publicly available. In other cases, it will depend on whether the academic sending you the email knows you personally or knows people who know you (and presumably are recommending you).
But yes, in general, I would think that if you are getting regular personalized emails from academics encouraging you to apply for faculty positions, that sounds like a good sign for your career. That said, you'd probably want to form the assessment of how your career is travelling based on other information. I imagine follow-up contact with these academics would help to clarify your chances of getting such positions. And presumably, you are starting to be able to independently assess (perhaps with advice from mentors) how your track record is going in terms of skills, experience, publications, and other academic outputs.
