Von Neumann's theorem
In mathematics, von Neumann's theorem is a result in the operator theory of linear operators on Hilbert spaces.
Statement of the theorem
Let G and H be Hilbert spaces, and let T : dom(T) ⊆ G → H be an unbounded operator from G into H. Suppose that T is a closed operator and that T is densely defined, i.e. dom(T) is dense in G. Let T∗ : dom(T∗) ⊆ H → G denote the adjoint of T. Then T∗T is also densely defined, and it is self-adjoint. That is,
and the operators on the right- and left-hand sides have the same dense domain in G.
gollark: It kind of makes sense. Causing the immune system to bother more/less or something.
gollark: You mean the other eleventh?
gollark: The NHS does actually have quite good web design, in my opinion.
gollark: This is actually quite impressive-looking, though. If only public health entities had this sort of thing, but correct.
gollark: You can make nicely designed websites for *anything*.
References
This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.