Operator monotone function

Operator monotone functions are like magic wands for making sure that things stay the same or get better when you do certain things to them. Imagine you have a big bag of toys, each one is different size and color. You have a special rule: if a bigger toy can fit inside a smaller one, that means it's even better and you want it more. That's how operator monotone functions work too.

But instead of toys, we are talking about mathematical things called operators. These operators can do things like stretch, squeeze, and twist numbers or other mathematical things. When you use an operator monotone function on an operator, it makes sure that if one operator is bigger than another, then the bigger one is always better. Just like how bigger toys are always better if they can fit inside smaller ones according to your rule.

This is really useful in many areas of math, like calculus, linear algebra, and even quantum mechanics. These fields deal with different mathematical objects that can be manipulated with operators. By using operator monotone functions, mathematicians can make sure that the things they do to these operators don't make them worse, but always make them better or keep them the same. This helps them solve problems and discover new things in the world of math.