Timeline for Is there a reasonable notion of spectral theorem on a pre-Hilbert space?

7 events

when toggle format	what		by	license	comment
Aug 10, 2021 at 19:04	vote	accept	Sanchayan Dutta
Dec 28, 2019 at 23:40	answer	added	Nik Weaver		timeline score: 8
Dec 28, 2019 at 18:09	answer	added	Vadim Alekseev		timeline score: 3
Dec 28, 2019 at 17:18	comment	added	Nate Eldredge		One example to consider is the Laplacian $\Delta$ on the pre-Hilbert space $E = C_c^\infty(\Omega)$, where $\Omega$ is some nice domain (e.g. a ball). Then $\Delta$ is symmetric, everywhere defined, and negative definite, so you would hope any "reasonable" version of the spectral theorem would apply to it. Now you would also hope any "reasonable" version of the spectral theorem would let you define the semigroup $e^{t\Delta}$ on $E$, but this is impossible since solutions of the heat equation do not stay compactly supported.
Dec 28, 2019 at 16:09	history	edited	Sanchayan Dutta		edited tags
Dec 28, 2019 at 7:50	history	edited	Sanchayan Dutta	CC BY-SA 4.0	edited title
Dec 28, 2019 at 7:32	history	asked	Sanchayan Dutta	CC BY-SA 4.0