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A two-sided sequence with values in a complex unital Banach algebra is a cosine sequence if it satisfies for any n,m ∈ ℤ with c₀ equal to the unity of the algebra. A cosine sequence is bounded if . A (bounded) group decomposition for a cosine sequence is a representation of c as for every n ∈ ℤ, where b is an invertible element of the algebra (satisfying , respectively). It is known that every bounded cosine sequence possesses a universally defined group decomposition, the so-called...
Si studiano esistenza, unicità e regolarità delle soluzioni strette, classiche e forti dell'equazione di evoluzione non autonoma con il dato iniziale , in uno spazio di Banach . Gli operatori sono generatori infinitesimali di semi-gruppi analitici ed hanno dominio indipendente da e non necessariamente denso in . Si danno condizioni necessarie e sufficienti per l'esistenza e la regolarità hölderiana della soluzione e della sua derivata.
Let A denote a complex unital Banach algebra. We characterize properties such as boundedness, relative compactness, and convergence of the sequence for an arbitrary x ∈ A, using σ(x) and resolvent conditions. Under these circumstances, we investigate elements in the peripheral spectrum, and give further conclusions, also involving the behaviour of and .
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