Timeline for Uniqueness of Borel functional calculus for unbounded self-adjoint operators
Current License: CC BY-SA 4.0
4 events
| when toggle format | what | by | license | comment | |
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| Mar 27, 2023 at 18:28 | comment | added | terceira | Using the strict topology to prove the spectral theorem for unbounded s.a. operators is discussed in the monograph "Saks Spaces and Applications in Functional Analysis". | |
| Mar 27, 2023 at 17:27 | comment | added | MathMath | @terceira thanks for your comment. I have seen two different approaches to uniqueness of the Borel functional calculus: using that $C_{0}$ is dense in $C_{b}$ as you mentioned and using that every characteristic function $1_{E}$ of a Borel set $E$ is the limit of $C_{0}$ functions. However, I have found no source of proof of any of these statements. It seems that everybody knows it but nobody actually proves it explicitly. I really wanted to work out the details. Any reference is welcome! | |
| Mar 27, 2023 at 17:14 | comment | added | terceira | One way to see this is to note that $C_0$ is dense in $C_b$ (on any locally compact space). We regard the latter not with the norm but with Buck's strict topology, i.e., the finest l.c. topology which agrees with compact convergence on the unit ball. It is a standard fact that the functional calculus is continuous for this topology. | |
| Mar 27, 2023 at 16:48 | history | asked | MathMath | CC BY-SA 4.0 |