Compact operator on Hilbert space

Okay, let's imagine that you have a big toy box (which is called a Hilbert space). In this toy box, you have a bunch of toys (which are called vectors). Now, imagine that you have a special toy (which is called a compact operator) that can take any toy in the box and shrink it down to a smaller size.

Let's say you have a toy truck (which is a vector) in your toy box, and you want to shrink it down. You can use this special toy (compact operator) to do that. The compact operator takes the toy truck and squeezes it down to a smaller size, so that it takes up less space in the toy box.

This compact operator can do this for any toy in the box, not just the toy truck. It can shrink down any toy to a smaller size. And the cool thing about this operator is that it can keep shrinking and shrinking the toys infinitely, so they become really, really small.

So, a compact operator on a Hilbert space is like a toy in a toy box that can take any other toy in the box and shrink it down to a smaller size. It's called "compact" because it can keep shrinking things down indefinitely, making them more and more "condensed" in the space.