Police said on Wednesday that they had made around 370 arrests for illegal assembly as well as other offences, with 10 involving violations of the new law.
Police will use a new purple flag to warn protesters if they display banners or shout slogans that may constitute a crime under the law.
A man and a woman holding Hong Kong independence signs were arrested in two separate incidents hours after the new law took effect.
When Hong Kong media tycoon Jimmy Lai arrived in Hong Kong at age 12, the city felt like heaven to him, brimming with hope and freedom.
Three other women were detained for possessing items advocating independence.
"We should stand with the Hong Kong people and support those staying in Hong Kong", he said.
Trump "wants to ensure, with a handful of exceptions, that Hong Kong is treated just like mainland China because that's the way that General Secretary Xi has chosen to treat that place as well", Pompeo told reporters.
This measure violates paragraphs 3 (3) and (5) of the Joint Declaration, and directly threatens the rights contained in the United Nations International Covenant on Civil and Political Rights which under the Joint Declaration are to be protected in Hong Kong.
The legislation has been widely criticized by opposition lawmakers in Hong Kong, human rights groups and politicians worldwide, with many saying it will cement Beijing's direct control over the semi-autonomous city.
"Hong Kong Police express the strongest condemnation against such violent act".
But as the territory wakes up Wednesday to mark the 23rd anniversary of the transfer, it does so under a contentious new national security law that ensures the semiautonomous city is firmly under China's control and that makes clear Beijing has no qualms about ignoring global criticism.
Hong Kong was under United Kingdom jurisdiction until Britain handed it to China in 1997 with a guarantee that Beijing would preserve the city's judicial and legislative autonomy for 50 years. At the time, he said the law marked the start of a "bloody cultural revolution".
We will grant BN (O) s five years limited leave to remain, with the right to work or study. This means that if the law conflicts with any existing laws in Hong Kong, the new national security law will take priority.
In Beijing, the executive deputy director of the Cabinet's Hong Kong affairs office, Zhang Xiaoming, said Hong Kong people are allowed to criticize the ruling Communist Party but can not turn those complaints "into actions".
The bill targets police units that have cracked down on Hong Kong protesters, as well as Chinese Communist Party officials responsible for imposing a strict "national security" law on Hong Kong, which is considered a special administrative region within China and maintains its own governing and economic systems.
It has responded by announcing plans to allow millions of Hong Kongers with British National Overseas status to relocate with their families and eventually apply for citizenship.