Washington (CNN) President Donald Trump sought to project momentum into his dealings with North Korea on Wednesday, saying a summit between leader Kim Jong Un and South Korean President Moon Jae-in boded well for peace despite little indication that the two leaders had made any real progress toward denuclearization. In contrast, lawmakers and experts urged caution.

Trump struck an optimistic tone while speaking to reporters on the South Lawn of the White House just hours after Moon and Kim announced that North Korea would close a key missile test facility in the presence of "international experts" and potentially destroy a key nuclear complex if the United States agrees to corresponding measures.

"We had very good news from North Korea, South Korea. They met, and we had some great responses," he said before again touting his personal rapport with Kim.

"The relationships, I have to tell you, at least on a personal basis, they're very good. It's very much calmed down," Trump said. "We're talking. It's very calm. He's calm. I'm calm. So we'll see what happens."

But several US lawmakers made it very clear that they remain skeptical about North Korea's willingness to dismantle its nuclear weapons program, encouraging the administration to keep up its pressure campaign on Pyongyang despite the public display of cooperation from Kim and Moon.

"Surprise, surprise: North Korea wants concessions from the US for steps far short of denuclearization. Glad the admin has made no commitments. Maximum pressure campaign should proceed," Republican Rep. Ed Royce of California, chairman of the House Foreign Affairs Committee, tweeted following the summit.

Trump's Wednesday comments came as the US and its allies have ramped up efforts to monitor and document evidence of sanctions violations.

US Ambassador to the United Nations Nikki Haley slammed Russia on Monday, accusing Moscow of "cheating" and acting like a "virus" by helping North Korea evade international sanctions aimed at curbing Pyongyang's nuclear and ballistic missile programs.

Congressional lawmakers have also pushed administration officials to step up enforcement of shipping sanctions during recent hearings on Capitol Hill.

'A gesture that mimics disarmament'

As part of the agreement signed Wednesday, Pyongyang pledged to destroy both the Tongchang-ri missile engine test site and the Yongbyon nuclear site, which is believed to be used for the production of fissile material, if the United States takes reciprocal measures.

But some analysts warn that offer may amount to nothing more than "a gesture that mimics disarmament."

"North Korea had previously offered to dismantle the engine test stand at Tongchang-ri, which is no longer necessary for developing North Korea's missiles. The offer to allow observers is new, but North Korea made a similar offer with regard to the nuclear test site, and it turned out they meant journalists," according to Jeffrey Lewis, director for the East Asia Nonproliferation Program at the Middlebury Institute of International Studies at Monterey.

"The offer on Yongbyon is more interesting," he said, noting that while it is possible Kim may be willing to destroy the site, he appeared to make only a very vague commitment conditioned on US actions.

And even if Kim follows through with that pledge, Lewis said, many experts believe "North Korea built the Yongbyon enrichment plant with the intention of offering it in negotiations, while keeping Kangson and perhaps other enrichment sites."

"So these measures are gestures that resemble disarmament, but they don't actually constrain North Korea's nuclear and missile programs," he told CNN.

Sen. Graham slams Moon's visit

Kim's offer is expected to be a topic of discussion when Trump and Moon meet at the United Nations later this month, a conversation he hopes will help pave the way for a second Trump-Kim summit, one diplomatic source told CNN.

While this source said South Korea ultimately hopes for a deal predicated on North Korea's commitment to a step-for-step process, Sen. Lindsey Graham suggested this week that Moon's visit to Pyongyang may undermine US efforts to achieve denuclearization.

"I'm concerned South Korea's visit is going to undermine efforts by @SecPompeo and Ambassador @nikkihaley to impose maximum pressure on the North Korean regime. While North Korea has stopped testing missiles and nuclear devices, they have NOT moved toward denuclearization," the South Carolina Republican said in a tweet.

The rapid warming of relations between North Korea and South Korea is also raising some eyebrows at the State Department, according to Mintaro Oba, a former Korea desk officer who served under President Barack Obama.

"There's a fear that moving too quickly on inter-Korean relations will undermine US efforts to maintain pressure on North Korea to denuclearize and prematurely open the door to renewed economic engagements," he said.

"It's a dynamic that comes from a long-standing mismatch in outlook between Seoul, which prioritizes peace and stability in its immediate neighborhood, and Washington -- which has an overwhelming focus on denuclearization," Oba added.

Kim said Wednesday that he and Moon are committed to showing the world "how this divided nation is going to bring about a new future on its own."

But that future depends greatly on the outcome of talks between the US and North Korea.

And analysts say the ball is now in Washington's court.

Negotiations between the US and North Korea appear to have hit an impasse since Trump and Kim met in Singapore more than three months ago.

"What the United States needs to be looking for right now are genuine steps from the North Koreans that indicate a willingness to move the process forward. If North Korea is genuinely willing to close down Yongbyon and to allow in inspectors ... those are just partial steps, but those are genuine steps forward," said Michael Fuchs, a senior fellow at the Center for American Progress and former deputy assistant secretary of state for East Asian and Pacific affairs.

"The biggest outstanding question still remains -- what price the North Koreans want to extract from the United States," Fuchs said.

However, the White House said earlier this month that Trump is open to another summit with Kim despite national security adviser John Bolton's assertion that the US is still waiting for Pyongyang to take steps toward denuclearization.

A second Trump-Kim summit?

Trump has routinely touted his personal rapport with Kim though his failure to secure a concrete pledge on denuclearization during the Singapore summit has raised concerns that the North Korean leader is using flattery to draw out the process without making any real concessions.

Moon has also become increasingly convinced that direct talks between the two leaders are the key to any agreement, the diplomatic source told CNN, adding that Seoul is taking its cues from what Trump says, rather than comments by Bolton or Secretary of State Mike Pompeo.

Standing side by side with Kim in Pyongyang, Moon expressed hope on Wednesday that talks would resume between North Korea and the US.

"They have continuously shown their trust towards one another and I hope there will be another summit between the two countries as soon as possible," Moon said.

Moon and his top advisers have consistently said they want to make inter-Korea meetings a regular part of North-South relations and see them as a helpful step in establishing a permanent peace.

Ahead of this week's talks, it was expected that the two leaders would continue to work to formally end the Korean War, which ended in an armistice 65 years ago.

While a formal peace regime officially ending the war would need buy-in from the US and China -- the other participants in the conflict -- experts agree there is nothing to stop the two Koreas declaring an end to the war themselves or signing a bilateral peace treaty.

A big part of any negotiation to end the war would be the status of the thousands of US troops stationed in South Korea as part of the two countries' alliance. The North has long seen the US military's large footprint in South Korea as a direct threat.