I mean, I'm no space expert (which is evidenced by the fact that I don't even know what a space expert is called), but I'm pretty sure outer space is huge, and therefore not easily patrolled. To make things more difficult, it also extends in all directions.It's actually relatively simple. The militarily-important satellites the US government uses daily are in predictable orbits around the Earth, which (all things considered) limits the space we're talking about considerably. A satellite in orbit 20,000 kilometers up is rather high up there, but it's not that far, all things considered.
There are only a few ways for America to conceivably "dominate space", if we mean the denial of space to non-friendly nations - either with ground-based weaponry designed to shoot down other countries' assets, or (more speculatively) with actual orbiting weaponry. The orbital weaponry - despite the expense - at least as the option of possibly being deniable: a satellite that "mysteriously" blows up while orbiting the Pacific might not be credited to the Americans. Ground based weaponry by defintion leaves a trail.
The problem for the US is that it is shockingly easy to destroy an orbiting satellite in retaliation. Their orbits are predictable, and most are easily tracked. (Almost by definition, the dangerous satellites are the ones whose orbits are above you, where they can see you and you can see them.) A modestly determined opponent could put together an anti-satellite weapon, as both the US and USSR did during the Cold War. Think what was doable for the Soviets 30 years ago is beyond the Chinese, the Indians, or for that matter the Brazilians today?
(There are obvious exceptions - GPS satellites are much too high above the Earth for the kind of near-term land-based ASAT weapon that a middle power is likely to build. A high-power laser, on the other hand...)
The point is, if America is seriously going to start threatening other people's space assets, they need to be prepared for others to do the same in short order. But unless America is willing to shoot down other people's satellites, or otherwise render them inoperable, it's difficult to think of what Washington could conceivably do.
More questionable is the entire idea that America has some kind of incomparable, irreplaceable advantage in space. America's space assets are highly impressive, but these are technologies that can be, and are, applied in other areas such as aircraft. (Witness the rise of unmanned aerial vehicles, with arguably higher battlefield value than satellites.) The actual role of space as a dimension of warfare - such as land, air, and sea conventionally - is so far uncertain. (Not saying there isn't a real oneThe biggest - and most important - exception is the role of global telecommunications in the military (low-frequency communications satellites are used to call submarines, etc.) But it's not news to say that America is opposed to people interfering with it's communications.
None of this negates the fact that, in a hypothetical war with the US over Taiwan, I imagine China would start trying to plink US satellites. Of course, China would also be trying to sink US carrier groups, which any sensible leader would see as a far more important threat.