Bounded functional calculi for unbounded operators

Conference item

This article summarises the theory of several bounded functional calculi for unbounded operators that have recently been discovered. They extend the Hille–Phillips calculus for (negative) generators A of certain bounded C0-semigroups, in particular for bounded semigroups on Hilbert spaces and bounded holomorphic semigroups on Banach spaces. They include functions outside the Hille-Phillips class, and they generally give sharper bounds for the norms of the resulting operators f(A). The calculi are mostly based on appropriate reproducing formulas for the relevant classes of functions, and they rely on significant and interesting developments of function theory. They are compatible with standard functional calculi and they admit appropriate convergence lemmas and spectral mapping theorems. They can also be used to derive several well-known operator norm-estimates, provide generalisations of some of them, and extend the general theory of operator semigroups. Our aim is to help readers to make use of these calculi without having to understand the details of their construction.

Authors: Batty, C; Gomilko, A; Tomilov, Y

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Birkhäuser
• Host title: Operators, Semigroups, Algebras and Function Theory: Volume from IWOTA Lancaster 2021
• Pages: 27-60
• Series: Operator Theory: Advances and Applications
• Series number: 292
• Place of publication: Cham, Switzerland
• Publication date: 2023-12-07
• Acceptance date: 2022-03-30
• Event title: International Workshop on Operator Theory and its Applications (IWOTA 2021)
• Event location: Lancaster, UK
• Event website: https://www.lancaster.ac.uk/maths/iwotauk2021/
• Event start date: 2021-08-16
• Event end date: 2021-08-20
• DOI: 10.1007/978-3-031-38020-4_2
• EISSN: 2296-4878
• ISSN: 0255-0156
• EISBN: 9783031380204
• ISBN: 9783031380198
• Language: English