Decomposition of Hyperbolic Semigroups

In this article, the Banach space is decomposed into the direct sum of two closed subspaces such that the semigroup becomes forward exponentially stable on one subspace and backward exponentially stable on another subspace. Hyperbolic semigroup is characterised in terms of the spectrum of its cogenerator. Further, we study the rescaled hyperbolic semigroup to analyse its spectrum.


Introduction
The theory of Hyperbolicity is one of the basic sources in the study of partial differential equations. Hyperbolic semigroups are studied in the models of beams and waves as well as the transport equation and networks of nonhomogeneous transmission lines. If (T(ζ)) ζ≥0 is a hyperbolic semigroup, then every operator T(ζ) in the semigroup must itself be hyperbolic [1]. The literature on hyperbolic semigroups is very rich [2].
The classical results are presented in the books (5; 8; 14). Over the years, the notion of hyperbolicity was broadened (non-uniform hyperbolicity) [3] and relaxed (partial hyperbolicity) [4] to encompass a much larger class of systems. In [5], Herbert and Daniel obtain a complete characterization of Fredholm spectrum of the semigroup generated by sharp energy estimates [6]. The existence and uniqueness of conformal measures are briefly explained by [7]. [8] proved that the rate of convergence of the semigroup to the point one is exponential. Quasi hyperbolic semigroups [9] and hyperbolic dynamical systems [10] gives a path way to study the variations of hyperbolic semigroups. In contrast to the known result on this topic, this paper is motivated to study the decomposition of hyperbolic semigroups [11][12][13][14][15]. The aim of the present study is to bridge the gap explored from the above literature. Our results enable us to efficiently analyse the hyperbolic semigroup in terms of its spectrum of its cogenerator.

Preliminaries
We remark that if A be the generator of a bounded semigroup of T(ζ) then σ(A) ⊂ {z: Rez ≤ 0} but not conversely.

Theorem 2.3
For C 0 -semigroup T(ζ) on a Banach space X the following assertions are equivalent: on a Banach space X with generator A satisfies the spectral mapping theorem if σ(T(ζ))/ {0} = e ζσ(A) for every ζ ≥ 0.
µ and which shows

Conclusion
The Banach space is decomposed into the direct sum of two closed subspaces. This leads to characterise the hyperbolic semigroup in terms of the spectrum of its cogenerator. Also rescaled hyperbolic semigroup is developed to analyse its spectrum. By analyzing the spectrum, one can study the equality of spectral and growth bound for strongly continuous semigroup which have extensive applications in quantum theory and stochastic processes.