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Operator semigroups in Banach space theory

Pietro Aiena, Manuel González, Antonio Martínez-Abejón (2001)

Bollettino dell'Unione Matematica Italiana

In questo lavoro, motivati dalla teoria di Fredholm in spazi di Banach e dalla cosiddetta teoria degli ideali di operatori nel senso di Pietsch, viene definito un nuovo concetto di semigruppo di operatori. Questa nuova definizione include quella di molte classi di operatori già studiate in letteratura, come la classe degli operatori di semi-Fredholm, quella degli operatori tauberiani ed altre ancora. Inoltre permette un nuovo ed unificante approccio ad una serie di problemi in teoria degli operatori...

Operator theoretic properties of semigroups in terms of their generators

S. Blunck, L. Weis (2001)

Studia Mathematica

Let ( T t ) be a C₀ semigroup with generator A on a Banach space X and let be an operator ideal, e.g. the class of compact, Hilbert-Schmidt or trace class operators. We show that the resolvent R(λ,A) of A belongs to if and only if the integrated semigroup S t : = ∫ 0 t T s d s belongs to . For analytic semigroups, S t ∈ implies T t ∈ , and in this case we give precise estimates for the growth of the -norm of T t (e.g. the trace of T t ) in terms of the resolvent growth and the imbedding D(A) ↪ X.

Optimal time and space regularity for solutions of degenerate differential equations

Alberto Favaron (2009)

Open Mathematics

We derive optimal regularity, in both time and space, for solutions of the Cauchy problem related to a degenerate differential equation in a Banach space X. Our results exhibit a sort of prevalence for space regularity, in the sense that the higher is the order of regularity with respect to space, the lower is the corresponding order of regularity with respect to time.

Optimization problems for structural acoustic models with thermoelasticity and smart materials

Irena Lasiecka (2000)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

Optimization problem for a structural acoustic model with controls governed by unbounded operators on the state space is considered. This type of controls arises naturally in the context of "smart material technology". The main result of the paper provides an optimal synthesis and solvability of associated nonstandard Riccati equations. It is shown that in spite of the unboundedness of control operators, the resulting gain operators (feedbacks) are bounded on the state space. This allows to provide...

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