can you trespass in a public place? i know this is probably one for another thread, but if the librarian said to you, "i don't want him in here, he's trouble", surely you have no power to 'eject' him?
A public place can still be owned by someone (who can tell you to get lost). A shop is a good example - the public have access by payment or otherwise (so it's a "public place") but it's owned by a person or a company and they (or their agents, for example members of staff) can tell you to leave (or indeed close and keep everyone out).
A street is rather different - there is probably all sorts of common law that preserves your right to walk down the street but ultimately there isn't really a landlord as such to tell you to clear off.
A library would lie somewhere between the two. It's more complicated because there is specific legislation that applies to libraries that enables bylaws to be made in respect of their premises and you would expect local authorities to provide bylaws that allow librarians to require members of the public to leave. However regardless the library premises would be occupied by a library authority and if their employees asked you to leave then I'm guessing the position wouldn't be that dissimilar to a member of staff in a shop kicking you out.
In terms of constables assisting the removal of a trespasser, the situation may well be different in libraries because of such local bylaws (i.e. you might have a duty/power to remove someone rather than acting in a dubious 'agent of the landlord' capacity).