Definition
Let H be a Hilbert space and G the set of bounded invertible operators on H of the form I + T, where T is a trace-class operator. G is a group because
It has a natural metric given by d(X, Y) = ||X - Y||1, where || ยท ||1 is the trace-class norm.
If H is a Hilbert space with inner product, then so too is the kth exterior power with inner product
In particular
gives an orthonormal basis of if (ei) is an orthonormal basis of H. If A is a bounded operator on H, then A functorially defines a bounded operator on by
If A is trace-class, then (A) is also trace-class with
This shows that the definition of the Fredholm determinant given by
makes sense.
Read more about this topic: Fredholm Determinant
Famous quotes containing the word definition:
“The man who knows governments most completely is he who troubles himself least about a definition which shall give their essence. Enjoying an intimate acquaintance with all their particularities in turn, he would naturally regard an abstract conception in which these were unified as a thing more misleading than enlightening.”
—William James (1842–1910)
“Mothers often are too easily intimidated by their children’s negative reactions...When the child cries or is unhappy, the mother reads this as meaning that she is a failure. This is why it is so important for a mother to know...that the process of growing up involves by definition things that her child is not going to like. Her job is not to create a bed of roses, but to help him learn how to pick his way through the thorns.”
—Elaine Heffner (20th century)
“No man, not even a doctor, ever gives any other definition of what a nurse should be than this—”devoted and obedient.” This definition would do just as well for a porter. It might even do for a horse. It would not do for a policeman.”
—Florence Nightingale (1820–1910)