Expand all Collapse all | Results 1 - 4 of 4 |
1. CJM 2010 (vol 62 pp. 1419)
BMO-Estimates for Maximal Operators via Approximations of the Identity with Non-Doubling Measures
Let $\mu$ be a nonnegative Radon measure
on $\mathbb{R}^d$ that satisfies the growth condition that there exist
constants $C_0>0$ and $n\in(0,d]$ such that for all $x\in\mathbb{R}^d$ and
$r>0$, ${\mu(B(x,\,r))\le C_0r^n}$, where $B(x,r)$ is the open ball
centered at $x$ and having radius $r$. In this paper, the authors prove
that if $f$ belongs to the $\textrm {BMO}$-type space $\textrm{RBMO}(\mu)$ of Tolsa, then
the homogeneous maximal function $\dot{\mathcal{M}}_S(f)$ (when $\mathbb{R}^d$ is not an
initial cube) and the inhomogeneous maximal function
$\mathcal{M}_S(f)$ (when $\mathbb{R}^d$ is an initial cube)
associated with a given approximation of the identity $S$ of Tolsa are
either infinite everywhere or finite almost everywhere,
and in the latter case, $\dot{\mathcal{M}}_S$ and $\mathcal{M}_S$ are bounded from
$\textrm{RBMO}(\mu)$ to the $\textrm {BLO}$-type
space $\textrm{RBLO}(\mu)$. The authors also prove that the inhomogeneous
maximal operator $\mathcal{M}_S$ is bounded from the local
$\textrm {BMO}$-type space $\textrm{rbmo}(\mu)$
to the local $\textrm {BLO}$-type space $\textrm{rblo}(\mu)$.
Keywords:Non-doubling measure, maximal operator, approximation of the identity, RBMO(mu), RBLO(mu), rbmo(mu), rblo(mu) Categories:42B25, 42B30, 47A30, 43A99 |
2. CJM 2008 (vol 60 pp. 520)
Matrices Whose Norms Are Determined by Their Actions on Decreasing Sequences Let $A=(a_{j,k})_{j,k \ge 1}$ be a non-negative matrix. In this
paper, we characterize those $A$ for which $\|A\|_{E, F}$ are
determined by their actions on decreasing sequences, where $E$ and
$F$ are suitable normed Riesz spaces of sequences. In particular,
our results can apply to the following spaces: $\ell_p$, $d(w,p)$,
and $\ell_p(w)$. The results established here generalize
ones given by Bennett; Chen, Luor, and Ou; Jameson; and
Jameson and Lashkaripour.
Keywords:norms of matrices, normed Riesz spaces, weighted mean matrices, NÃ¶rlund mean matrices, summability matrices, matrices with row decreasing Categories:15A60, 40G05, 47A30, 47B37, 46B42 |
3. CJM 2004 (vol 56 pp. 134)
Linear Operators on Matrix Algebras that Preserve the Numerical Range, Numerical Radius or the States |
Linear Operators on Matrix Algebras that Preserve the Numerical Range, Numerical Radius or the States Every norm $\nu$ on $\mathbf{C}^n$ induces two norm numerical
ranges on the algebra $M_n$ of all $n\times n$ complex matrices,
the spatial numerical range
$$
W(A)= \{x^*Ay : x, y \in \mathbf{C}^n,\nu^D(x) = \nu(y) = x^*y = 1\},
$$
where $\nu^D$ is the norm dual to $\nu$, and the algebra numerical range
$$
V(A) = \{ f(A) : f \in \mathcal{S} \},
$$
where $\mathcal{S}$ is the set of states on the normed algebra
$M_n$ under the operator norm induced by $\nu$. For a symmetric
norm $\nu$, we identify all linear maps on $M_n$ that preserve
either one of the two norm numerical ranges or the set of states or
vector states. We also identify the numerical radius isometries,
{\it i.e.}, linear maps that preserve the (one) numerical radius
induced by either numerical range. In particular, it is shown that
if $\nu$ is not the $\ell_1$, $\ell_2$, or $\ell_\infty$ norms,
then the linear maps that preserve either numerical range or either
set of states are ``inner'', {\it i.e.}, of the form $A\mapsto
Q^*AQ$, where $Q$ is a product of a diagonal unitary matrix and a
permutation matrix and the numerical radius isometries are
unimodular scalar multiples of such inner maps. For the $\ell_1$
and the $\ell_\infty$ norms, the results are quite different.
Keywords:Numerical range, numerical radius, state, isometry Categories:15A60, 15A04, 47A12, 47A30 |
4. CJM 1998 (vol 50 pp. 673)
Fredholm modules and spectral flow An {\it odd unbounded\/} (respectively, $p$-{\it summable})
{\it Fredholm module\/} for a unital Banach $\ast$-algebra, $A$, is a pair $(H,D)$
where $A$ is represented on the Hilbert space, $H$, and $D$ is an unbounded
self-adjoint operator on $H$ satisfying:
\item{(1)} $(1+D^2)^{-1}$ is compact (respectively, $\Trace\bigl((1+D^2)^{-(p/2)}\bigr)
<\infty$), and
\item{(2)} $\{a\in A\mid [D,a]$ is bounded$\}$ is a dense
$\ast-$subalgebra of $A$.
If $u$ is a unitary in the dense $\ast-$subalgebra mentioned in (2) then
$$
uDu^\ast=D+u[D,u^{\ast}]=D+B
$$
where $B$ is a bounded self-adjoint operator. The path
$$
D_t^u:=(1-t) D+tuDu^\ast=D+tB
$$
is a ``continuous'' path of unbounded self-adjoint ``Fredholm'' operators.
More precisely, we show that
$$
F_t^u:=D_t^u \bigl(1+(D_t^u)^2\bigr)^{-{1\over 2}}
$$
is a norm-continuous path of (bounded) self-adjoint Fredholm
operators. The {\it spectral flow\/} of this path $\{F_t^u\}$ (or $\{
D_t^u\}$) is roughly speaking the net number of eigenvalues that pass
through $0$ in the positive direction as $t$ runs from $0$ to $1$.
This integer,
$$
\sf(\{D_t^u\}):=\sf(\{F_t^u\}),
$$
recovers the pairing of the $K$-homology class $[D]$ with the $K$-theory
class [$u$].
We use I.~M.~Singer's idea (as did E.~Getzler in the $\theta$-summable
case) to consider the operator $B$ as a parameter in the Banach manifold,
$B_{\sa}(H)$, so that spectral flow can be exhibited as the integral
of a closed $1$-form on this manifold. Now, for $B$ in our manifold,
any $X\in T_B(B_{\sa}(H))$ is given by an $X$ in $B_{\sa}(H)$ as the
derivative at $B$ along the curve $t\mapsto B+tX$ in the manifold.
Then we show that for $m$ a sufficiently large half-integer:
$$
\alpha (X)={1\over {\tilde {C}_m}}\Tr \Bigl(X\bigl(1+(D+B)^2\bigr)^{-m}\Bigr)
$$
is a closed $1$-form. For any piecewise smooth path $\{D_t=D+B_t\}$ with
$D_0$ and $D_1$ unitarily equivalent we show that
$$
\sf(\{D_t\})={1\over {\tilde {C}_m}} \int_0^1\Tr \Bigl({d\over {dt}}
(D_t)(1+D_t^2)^{-m}\Bigr)\,dt
$$
the integral of the $1$-form $\alpha$. If $D_0$ and $D_1$ are not unitarily
equivalent, we must add a pair of correction terms to the right-hand
side. We also prove a bounded finitely summable version of the form:
$$
\sf(\{F_t\})={1\over C_n}\int_0^1\Tr\Bigl({d\over dt}(F_t)(1-F_t^2)^n\Bigr)\,dt
$$
for $n\geq{{p-1}\over 2}$ an integer. The unbounded case is proved by
reducing to the bounded case via the map $D\mapsto F=D(1+D^2
)^{-{1\over 2}}$. We prove simultaneously a type II version of our
results.
Categories:46L80, 19K33, 47A30, 47A55 |