The angle along a curve and range-kernel complementarity
| Autor: | Nikos Yannakakis, Dimosthenis Drivaliaris |
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| Rok vydání: | 2019 |
| Předmět: |
Pure mathematics
Algebra and Number Theory Spectrum (functional analysis) Sigma Spectral set Bounded operator law.invention Functional Analysis (math.FA) Mathematics - Functional Analysis Mathematics - Spectral Theory Range (mathematics) Kernel (algebra) Invertible matrix law FOS: Mathematics Commutative algebra Spectral Theory (math.SP) Analysis Mathematics 47A10 47A15 |
| DOI: | 10.48550/arxiv.1908.03555 |
| Popis: | We define the angle of a bounded linear operator A along a curve emanating from the origin and use it to characterize range-kernel complementarity. In particular we show that if $$\sigma (A)$$ does not separate 0 from $$\infty $$ , then $$X=R(A)\oplus N(A)$$ if and only if R(A) is closed and some angle of A is less than $$\pi $$ . We first apply this result to invertible operators that have a spectral set that does not separate 0 from $$\infty $$ . Next we extend the notion of angle along a curve to Banach algebras and use it to prove two characterizations of elements in a semisimple and in a $$C^*$$ commutative algebra respectively, whose spectrum does not separate 0 from $$\infty $$ . |
| Databáze: | OpenAIRE |
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