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Borichev-Tomilov Theorem

The Borichev-Tomilov theorem characterizes polynomial decay of a bounded strongly continuous semigroup on a Hilbert space. Such decay is a form of semigroup stability. Let (T(t))_(t>=0) have infinitesimal generator A, suppose that the imaginary axis is contained in the resolvent set of A, and let alpha>0. Then

||(isI-A)^(-1)||=O(|s|^alpha) (|s|->infty) <==> ||T(t)A^(-1)||=O(t^(-1/alpha)) (t->infty).

Here I is the identity operator, and (isI-A)^(-1) is the resolvent operator at is. Thus a quantitative frequency-domain estimate gives the corresponding optimal polynomial rate in the time domain.

See also

Big-O Notation, Gearhart-Prüss-Huang Theorem, Infinitesimal Generator, Resolvent Operator, Resolvent Set, Semigroup Stability, Strongly Continuous Semigroup

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References

Borichev, A. and Tomilov, Y. "Optimal Polynomial Decay of Functions and Operator Semigroups." Math. Ann. 347, 455-478, 2010. https://doi.org/10.1007/s00208-009-0439-0.

Cite this as:

Weisstein, Eric W. "Borichev-Tomilov Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Borichev-TomilovTheorem.html

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