Abstract
A general theory of regularized and Hilbert-Carleman determinants in normed algebras of operators acting in Banach spaces is proposed. In this approach regularized determinants are defined as continuous extensions of the corresponding determinants of finite dimensional operators. We characterize the algebras for which such extensions exist, describe the main properties of the extended determinants, obtain Cramer's rule and the formulas for the resolvent which are expressed via the extended tracestr(Ak) of iterations and regularized determinants.
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