Expand all Collapse all | Results 1 - 4 of 4 |
1. CJM 2009 (vol 62 pp. 74)
Projectors on the Generalized Eigenspaces for Neutral Functional Differential Equations in $L^{p}$ Spaces |
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)
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)
$H^\infty$ 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)
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 |