BRUSSELS — John Bolton’s appointment as national security adviser seems to many around the world to represent an uninhibited President Trump, and they are trying to puzzle out what exactly that means.

A fiercely intelligent man with deeply conservative, nationalistic and aggressive views about American foreign policy, Mr. Bolton may bring more consistency and predictability to President Trump’s foreign policy, many suggest. But others worry that his hawkish views on Iran and North Korea, among others, may goad Mr. Trump into seeking military solutions to diplomatic problems.

But some also wonder whether Mr. Bolton, who has played the outsider even when serving as a senior American diplomat in the State Department and at the United Nations, will be able to adjust to a high-pressure job where he must be less public, less strident and more of a mediator of differing policy views.

And it is an open question whether he will be able to manage his relationship with Mr. Trump, who seems to tire quickly of anyone outside his own family who tries to guide or restrict his behavior, his public statements or his instincts.