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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The operator $ id/dx,$ on $ C[0,1],$ is $ C\sp 1$-scalar


Author: Ralph deLaubenfels
Journal: Proc. Amer. Math. Soc. 103 (1988), 215-221
MSC: Primary 47B40; Secondary 47E05
MathSciNet review: 938671
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Abstract: We consider closed, unbounded linear operators $ A$, on a Banach space, $ X$, with discrete real spectrum, with corresponding eigenvectors, $ \{ {x_k}\} _{ - \infty }^\infty $, such that $ A{x_k} = {a_k}{x_k},{a_k} \leq {a_{k + 1}}$, for all $ k$, and $ {\lim _{k \to \pm \infty }}\left\vert {{a_k}} \right\vert = \infty $. For arbitrary $ n$, we present necessary and sufficient conditions for $ A$ to be $ {C^n}$-scalar.

Letting $ F(s)$ be the projection whose range is the $ {\text{span}}\{ {x_k}\vert{a_k}\;{\text{is between }}0{\text{ and }}s\} $, null-space is the closure of $ {\text{span}}\{ {x_k}\vert{a_k}\;{\text{is not between }}0{\text{ and }}s\} $, we show that $ A$ is $ {C^1}$-scalar if and only if the series

$\displaystyle \sum\limits_{{a_k} > 0} {\varphi (F({a_k})x)\left( {\frac{1}{{{a_... ...arphi (F({a_k})x)\left( {\frac{1}{{{a_{k - 1}}}} - \frac{1}{{{a_k}}}} \right)} $

both converge absolutely, for all $ \varphi $ in $ {X^ * },x$ in $ X$. As a corollary, we get that $ id/dx$, on $ \{ f{\text{ in }}C[0,1]\vert f(0) = f(1)\} $, is $ {C^1}$-scalar. Also, $ id/dx$, on $ {L^1}[0,1]$, is $ {C^1}$-scalar.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1988-0938671-8
PII: S 0002-9939(1988)0938671-8
Article copyright: © Copyright 1988 American Mathematical Society