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Search: All articles in the CJM digital archive with keyword semigroup

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1. CJM 2009 (vol 62 pp. 74)

Ducrot, Arnaud; Liu, Zhihua; Magal, Pierre
Projectors on the Generalized Eigenspaces for Neutral Functional Differential Equations in $L^{p}$ Spaces
We present the explicit formulas for the projectors on the generalized eigenspaces associated with some eigenvalues for linear neutral functional differential equations (NFDE) in $L^{p}$ spaces by using integrated semigroup theory. The analysis is based on the main result established elsewhere by the authors and results by Magal and Ruan on non-densely defined Cauchy problem. We formulate the NFDE as a non-densely defined Cauchy problem and obtain some spectral properties from which we then derive explicit formulas for the projectors on the generalized eigenspaces associated with some eigenvalues. Such explicit formulas are important in studying bifurcations in some semi-linear problems.

Keywords:neutral functional differential equations, semi-linear problem, integrated semigroup, spectrum, projectors
Categories:34K05, 35K57, 47A56, 47H20

2. CJM 2009 (vol 61 pp. 534)

Chen, Chuan-Zhong; Sun, Wei
Girsanov Transformations for Non-Symmetric Diffusions
Let $X$ be a diffusion process, which is assumed to be associated with a (non-symmetric) strongly local Dirichlet form $(\mathcal{E},\mathcal{D}(\mathcal{E}))$ on $L^2(E;m)$. For $u\in{\mathcal{D}}({\mathcal{E}})_e$, the extended Dirichlet space, we investigate some properties of the Girsanov transformed process $Y$ of $X$. First, let $\widehat{X}$ be the dual process of $X$ and $\widehat{Y}$ the Girsanov transformed process of $\widehat{X}$. We give a necessary and sufficient condition for $(Y,\widehat{Y})$ to be in duality with respect to the measure $e^{2u}m$. We also construct a counterexample, which shows that this condition may not be satisfied and hence $(Y,\widehat{Y})$ may not be dual processes. Then we present a sufficient condition under which $Y$ is associated with a semi-Dirichlet form. Moreover, we give an explicit representation of the semi-Dirichlet form.

Keywords:Diffusion, non-symmetric Dirichlet form, Girsanov transformation, $h$-transformation, perturbation of Dirichlet form, generalized Feynman-Kac semigroup
Categories:60J45, 31C25, 60J57

3. CJM 2008 (vol 60 pp. 1010)

Galé, José E.; Miana, Pedro J.
$H^\infty$ Functional Calculus and Mikhlin-Type Multiplier Conditions
Let $T$ be a sectorial operator. It is known that the existence of a bounded (suitably scaled) $H^\infty$ calculus for $T$, on every sector containing the positive half-line, is equivalent to the existence of a bounded functional calculus on the Besov algebra $\Lambda_{\infty,1}^\alpha(\R^+)$. Such an algebra includes functions defined by Mikhlin-type conditions and so the Besov calculus can be seen as a result on multipliers for $T$. In this paper, we use fractional derivation to analyse in detail the relationship between $\Lambda_{\infty,1}^\alpha$ and Banach algebras of Mikhlin-type. As a result, we obtain a new version of the quoted equivalence.

Keywords:functional calculus, fractional calculus, Mikhlin multipliers, analytic semigroups, unbounded operators, quasimultipliers
Categories:47A60, 47D03, 46J15, 26A33, 47L60, 47B48, 43A22

4. CJM 2006 (vol 58 pp. 859)

Read, C. J.
Nonstandard Ideals from Nonstandard Dual Pairs for $L^1(\omega)$ and $l^1(\omega)$
The Banach convolution algebras $l^1(\omega)$ and their continuous counterparts $L^1(\bR^+,\omega)$ are much studied, because (when the submultiplicative weight function $\omega$ is radical) they are pretty much the prototypic examples of commutative radical Banach algebras. In cases of ``nice'' weights $\omega$, the only closed ideals they have are the obvious, or ``standard'', ideals. But in the general case, a brilliant but very difficult paper of Marc Thomas shows that nonstandard ideals exist in $l^1(\omega)$. His proof was successfully exported to the continuous case $L^1(\bR^+,\omega)$ by Dales and McClure, but remained difficult. In this paper we first present a small improvement: a new and easier proof of the existence of nonstandard ideals in $l^1(\omega)$ and $L^1(\bR^+,\omega)$. The new proof is based on the idea of a ``nonstandard dual pair'' which we introduce. We are then able to make a much larger improvement: we find nonstandard ideals in $L^1(\bR^+,\omega)$ containing functions whose supports extend all the way down to zero in $\bR^+$, thereby solving what has become a notorious problem in the area.

Keywords:Banach algebra, radical, ideal, standard ideal, semigroup
Categories:46J45, 46J20, 47A15

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