Linear Operator Theory in Banach Algebras
Summary
Linear Operator Theory in Banach Algebras studies bounded linear mappings within complete normed algebras and their extensions across both algebraic and analytic frameworks. Banach algebras provide a unified environment in which algebraic operations and norm topology coexist, enabling a detailed analysis of spectra, resolvents and functional calculi. Central topics include the classification of operators by their spectrum and approximate point spectrum, the study of semi‐Fredholm and quasinilpotent elements, and the behaviour of derivations and automorphisms. Perturbation theory examines stability of spectral properties under small changes, while noncommutative functional calculi allow analytic functions to be applied to operators. These developments underpin applications in differential equations, signal processing, control theory and quantum mechanics, and inspire advances in noncommutative geometry, spectral approximation algorithms and the theory of operator semigroups.
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Linear Operator Theory in Banach Algebras publication trend
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Technical terms
Banach algebra: A complete normed algebra over the real or complex numbers in which the norm satisfies ∥ab∥≤∥a∥∥b∥.
Linear operator: A bounded linear map between normed vector spaces, studied via its algebraic and topological properties.
Spectrum: The set of complex scalars λ for which an operator T−λI fails to be invertible in a Banach algebra.
Semi‐Weyl operator: An operator that is semi‐Fredholm with infinite ascent or descent under specific compact perturbations.
Linear pencil: A parameterised family λx+y of elements in an algebra, whose invertibility is studied as λ varies.
Ascent and descent: The smallest non‐negative integers at which the nullity (ascent) or deficiency (descent) of powers of T−λI stabilises.
Jordan triple product: The bilinear map {a,b,c}=ab*c+cb*a that generalises Jordan algebra structures to operator contexts.
References
- Linear Maps Preserving the Set of Semi-Weyl Operators. Mathematics (2023).
- Some spectral domain in approximate point-spectrum-preserving maps on B(X). Journal of Inequalities and Applications (2023).
- Banach algebra mappings preserving the invertibility of linear pencils. Linear Algebra and its Applications (2024).
- Maps preserving ascent or descent of triple Jordan product. Advances in Operator Theory (2024).
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