Image copyright Reuters

President Trump invited Prime Minister Theresa May to the White House and has been trying to strengthen US-UK relations. Not everybody is happy about that.

President Trump loves the UK - and seems pleased with the way that Britons voted to leave the European Union (EU). As the Atlantic Council's Reginald Dale said, describing Trump's views: "He's very pro-British, and he doesn't like the EU."

For critics of the administration, Trump's invitation to May - she's the first foreign leader to meet with the new president - and his efforts to build a closer relationship with the UK are a troubling development.

These critics see Brexit, a term that's used to describe a country's process of leaving the European Union, as a disaster.

The detractors worry that Trump will try to leverage his friendship with May and her fellow Britons as a way to express support for Brexit - and in this way will encourage more countries to leave the European Union.

On Friday afternoon, Trump and May stood together in the White House's East Room, an open space with an oak floor.

The room was decorated with marble-topped fireplaces, white candles and heavy, gold curtains (a George Washington portrait, one that was first hung on a wall in the room in 1800, was not in its usual spot).

Image copyright PA

The two leaders were in a place that was steeped in tradition - as is the so-called special relationship, a friendship between the US and the UK that dates back generations. For Trump and May, though, it's all new.

They'd met only the day before in Philadelphia, where they were both attending a US congressional retreat. When White House officials first sent out a schedule for the president's meeting with May, they spelled her name wrong, inadvertently dropping the "h" in Theresa.

Still, they had good intentions - and big plans. At the White House, the two leaders discussed Russia, counterterrorism and defence issues.

They're open to the possibility of a trade deal between the US and the UK. It could only be signed once the UK leaves the European Union, a process that will take years. But on Friday they seemed eager to get started.

"Opposites attract," May has told reporters. Understated and reserved, she has a different style than Trump. Still, she said recently that she's "not afraid to speak frankly to a president of the United States".

Image copyright AFP

On Friday - at least in the East Room - they got along well. It was an auspicious beginning for a "most special relationship", as he put it. They stood at identical lecterns - six to eight feet apart. She wore a paisley scarf, while he had on a bright-red, wide tie.

For most of the 18-minute press conference, they smiled at each other.

At one point she congratulated him on his "stunning" victory. He looked at people in the audience, a group made up of presidential aides and reporters, as if he wanted to make sure that they'd heard what she said.

"We're going to have a fantastic relationship," he told them. A moment later he spoke exuberantly about Brexit. Then the mood changed, slightly.

Up until that point, she'd been all smiles. As he spoke about Brexit, though, she looked sombre. She glanced at the people in the audience, as if she were trying to gauge their reaction to his remarks.

Afterwards she spoke briefly about the way that they both value "ordinary working people". Then he picked up his notes, which had been typed out on sheets of paper, and put them into the pocket in his jacket. They left the room together in a show of solidarity.

Their alliance rests on a solid foundation. In a speech on Thursday, May spoke about the friendship between Margaret Thatcher and Ronald Reagan, two icons for conservatives in the US.

Image copyright Getty Images

Conservatives at think-tanks in Washington have been quick to bring up this friendship when talking about the relationship between the current prime minister and the president. "Instinctively, Trump - you know - really likes Britain," said Nile Gardiner of the Heritage Foundation.

So do Trump's aides. When Sean Spicer, the White House press secretary, uses the phrase, "special relationship", he's not being snarky. "I think we've always had that special relationship," he told a reporter at a White House briefing on 23 January. "But we can always be closer."

Things were different with the Obama White House officials. They liked the Britons, too, but they gently mocked the phrase "special relationship". This reflected their views about foreign policy in Europe.

As Charles Kupchan, who was a senior adviser in the Obama White House, told me the relationship between the US and the UK was important. But the president was focused on Berlin, not London. "Diplomacy," Kupchan said, "tilted towards the Continent".

This became more pronounced after UK voters expressed their desire to leave the European Union.

Image copyright Getty Images Image caption Obama focused on the special relationship, but not to the exclusion of Europe

Trump administration officials supported the decision of UK voters to leave the European Union, and in the aftermath they've been hoping for a deeper friendship between Americans and Britons and their leaders.

The perception of a newfound closeness of the two leaders has rattled some Europeans.

Guy Verhofstadt, a former Belgium prime minister, was visiting Washington this week to promote his book, Europe's Last Chance: Why the European States Must Form a More Perfect Union.

While speaking with reporters on Thursday, he described ways that Trump has tried to undermine the European alliance. Verhofstadt said he believes that Trump is "hoping" more countries will leave - and "disintegrate the European Union".

Gardiner knows that not everyone is happy about the renewed relationship between the US and the UK or about the developments in Europe. But he wishes they'd embrace the new order.

"Brexit is about sovereignty, self determination and freedom," Gardiner told me. "These are all great things." Anyway there's not much the detractors can do about it, he said, adding: "The winds of change are blowing through Europe."

Follow @Tara_Mckelvey on Twitter.