spectral inequality
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2020 ◽  
Vol 26 ◽  
pp. 54 ◽  
Author(s):  
Ivica Nakić ◽  
Matthias Täufer ◽  
Martin Tautenhahn ◽  
Ivan Veselić

We prove new bounds on the control cost for the abstract heat equation, assuming a spectral inequality or uncertainty relation for spectral projectors. In particular, we specify quantitatively how upper bounds on the control cost depend on the constants in the spectral inequality. This is then applied to the heat flow on bounded and unbounded domains modeled by a Schrödinger semigroup. This means that the heat evolution generator is allowed to contain a potential term. The observability/control set is assumed to obey an equidistribution or a thickness condition, depending on the context. Complementary lower bounds and examples show that our control cost estimates are sharp in certain asymptotic regimes. One of these is dubbed homogenization regime and corresponds to the situation where the control set becomes more and more evenly distributed throughout the domain while its density remains constant.


2018 ◽  
Vol 356 (11-12) ◽  
pp. 1131-1155
Author(s):  
Rémi Buffe ◽  
Kim Dang Phung

2009 ◽  
Vol 02 (03) ◽  
pp. 359-365
Author(s):  
Maria Apazoglou

We examine Ptak's inequality in a real Banach *-algebra which is hermitian and skew hermitian, and give conditions for a real Banach *-algebra to be isomorphic to a C* algebra.


1978 ◽  
Vol 26 (1) ◽  
pp. 86-88 ◽  
Author(s):  
Kong-Ming Chong

AbstractIn this note, a theorem of Rado which characterizes the convex hull of the set of all rearrangements of a given real n-tuple in terms of the Hardy—Littlewood—Pólya spectral order relation < is shown to be a consequence of a result of Hardy—Littlewood—Pólya and a strong spectral inequality.Subject classification (Amer. Math. Soc. (MOS) 1970): 52 A 40, 52 A 20.


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