On Two Methods of Studying the Invertibility of Operators in $ C^*$-Algebras Induced by Dynamical Systems

Antonevich, A. B.

doi:10.1070/sm1985v052n01abeh002730

Cited by 12 publications

(18 citation statements)

References 15 publications

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“…As noted in the beginning of this section, to prove (3.1) it suffices to show: 1 # _ ap (T ) implies 0 # _ ap (1 ). In fact, we show that _ ap (1 ) contains the entire imaginary axis whenever 1 # _ ap (T ).…”

Section: Spectral Mapping Theoremmentioning

confidence: 93%

“…Our proof of the Spectral Projection Theorem for LSPFs also differs from that in [48]. Here, we extend the algebraic technique used in [1,26] (for Hilbert spaces) so that it applies in the general setting of Banach spaces. One advantage to this approach is that we are able to discuss the case of`d iscrete'' time t # Z and``pointwise dichotomies'' (see Section 5; cf.…”

Section: Introductionmentioning

confidence: 93%

“…By a standard rescaling technique [36], it suffices to consider only T=T j (D) and &Tf&f &=O(1ÂN ). Using f, we construct a function g # C 0 (3, X ) such that &1g&=O(1ÂN ) &g&, hence showing that 0 # _ ap (1 ).…”

Section: Spectral Mapping Theoremmentioning

confidence: 99%

“…t along the trajectory through %. These operators were, in fact, introduced in [16] for the line setting, and in [1,8,26] for the compact setting. A byproduct of our approach is the following theorem.…”

Section: Introductionmentioning

confidence: 97%

“…These techniques build upon those used in [1] which addresses the finite dimensional setting and uses some C*-algebraic methods. These methods were also used in [26] to prove the results for a uniformly continuous cocycle on a Hilbert space.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Evolutionary Semigroups and Dichotomy of Linear Skew-Product Flows on Locally Compact Spaces with Banach Fibers

Latushkin

Montgomery‐Smith

Randolph

1996

Journal of Differential Equations

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We study evolutionary semigroups generated by a strongly continuous semicocycle over a locally compact metric space acting on Banach fibers. This setting simultaneously covers evolutionary semigroups arising from non-autonomous abstract Cauchy problems and C 0 -semigroups, and linear skew-product flows. The spectral mapping theorem for these semigroups is proved. The hyperbolicity of the semigroup is related to the exponential dichotomy of the corresponding linear skew-product flow. To this end a Banach algebra of weighted composition operators is studied. The results are applied in the study of:``roughness'' of the dichotomy, dichotomy and solutions of nonhomogeneous equations, Green's function for a linear skew-product flow,``pointwise'' dichotomy versus``global'' dichotomy, and evolutionary semigroups along trajectories of the flow. 1996Academic Press, Inc.

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Section: Spectral Mapping Theoremmentioning

confidence: 93%

Section: Introductionmentioning

confidence: 93%

Section: Spectral Mapping Theoremmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Evolutionary Semigroups and Dichotomy of Linear Skew-Product Flows on Locally Compact Spaces with Banach Fibers

Latushkin

Montgomery‐Smith

Randolph

1996

Journal of Differential Equations

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Evolution Semigroups, Translation Algebras, and Exponential Dichotomy of Cocycles

Latushkin

Schnaubelt

1999

Journal of Differential Equations

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We study the exponential dichotomy of an exponentially bounded, strongly continuous cocycle over a continuous flow on a locally compact metric space 3 acting on a Banach space X. Our main tool is the associated evolution semigroup on C 0 (3; X ). We prove that the cocycle has exponential dichotomy if and only if the evolution semigroup is hyperbolic if and only if the imaginary axis is contained in the resolvent set of the generator of the evolution semigroup. To show the latter equivalence, we establish the spectral mappingÂannular hull theorem for the evolution semigroup. In addition, dichotomy is characterized in terms of the hyperbolicity of a family of weighted shift operators defined on c 0 (Z; X ). Here we develop Banach algebra techniques and study weighted translation algebras that contain the evolution operators. These results imply that dichotomy persists under small perturbations of the cocycle and of the underlying compact metric space. Also, exponential dichotomy follows from pointwise discrete dichotomies with uniform constants. Finally, we extend to our situation the classical Perron theorem which says that dichotomy is equivalent to the existence and uniqueness of bounded, continuous, mild solutions to the inhomogeneous equation. Academic Press

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Evolutionary Semigroups and Lyapunov Theorems in Banach Spaces

Latushkin

Montgomery‐Smith

1995

Journal of Functional Analysis

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On Two Methods of Studying the Invertibility of Operators in $ C^*$-Algebras Induced by Dynamical Systems

Cited by 12 publications

References 15 publications

Evolutionary Semigroups and Dichotomy of Linear Skew-Product Flows on Locally Compact Spaces with Banach Fibers

Evolutionary Semigroups and Dichotomy of Linear Skew-Product Flows on Locally Compact Spaces with Banach Fibers

Evolution Semigroups, Translation Algebras, and Exponential Dichotomy of Cocycles

Evolutionary Semigroups and Lyapunov Theorems in Banach Spaces

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