Electronic Resource
Springer
Integral equations and operator theory
26 (1996), S. 136-187
ISSN:
1420-8989
Keywords:
47B53
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract A general theory of tracestr D A and determinantsdet D (I+A) in normed algebrasD of operators acting in Banach spacesB is proposed. In this approach trace and determinant are defined as continuous extensions of the corresponding functionals from finite dimensional operators. We characterize the algebras for which such extensions exist and describe sets of possible values of traces and determinants for the same operator in different algebras. In spite of the fact that the extended traces and determinants may differ in different algebrasD, operatorI+A (withA ∈D) is invertible inB if and only ifdet D (I+A) does not vanish. Cramer's rule and formulas for the resolvent are obtained and they are expressed in different algebras by the same formulas viadet D (I+A) andtr D (A). A large set of examples and illustrations are also presented.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01191855
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