Expand all Collapse all  Results 51  71 of 71 
51. CJM 2003 (vol 55 pp. 379)
Generalized Factorization in Hardy Spaces and the Commutant of Toeplitz Operators Every classical inner function $\varphi$ in the unit disk gives rise to
a certain factorization of functions in Hardy spaces. This factorization,
which we call the generalized Riesz factorization, coincides with the
classical Riesz factorization when $\varphi(z)=z$. In this paper we prove
several results about the generalized Riesz factorization, and we apply
this factorization theory to obtain a new description of the commutant
of analytic Toeplitz operators with inner symbols on a Hardy space. We
also discuss several related issues in the context of the Bergman space.
Categories:47B35, 30D55, 47A15 
52. CJM 2002 (vol 54 pp. 1142)
Form Domains and Eigenfunction Expansions for Differential Equations with Eigenparameter Dependent Boundary Conditions 
Form Domains and Eigenfunction Expansions for Differential Equations with Eigenparameter Dependent Boundary Conditions Form domains are characterized for regular $2n$th order differential
equations subject to general selfadjoint boundary conditions
depending affinely on the eigenparameter. Corresponding modes of
convergence for eigenfunction expansions are studied, including
uniform convergence of the first $n1$ derivatives.
Categories:47E05, 34B09, 47B50, 47B25, 34L10 
53. CJM 2002 (vol 54 pp. 998)
Resonances for Slowly Varying Perturbations of a Periodic SchrÃ¶dinger Operator We study the resonances of the operator $P(h) = \Delta_x + V(x) +
\varphi(hx)$. Here $V$ is a periodic potential, $\varphi$ a
decreasing perturbation and $h$ a small positive constant. We prove
the existence of shape resonances near the edges of the spectral bands
of $P_0 = \Delta_x + V(x)$, and we give its asymptotic expansions in
powers of $h^{\frac12}$.
Categories:35P99, 47A60, 47A40 
54. CJM 2001 (vol 53 pp. 1031)
The Complete $(L^p,L^p)$ Mapping Properties of Some Oscillatory Integrals in Several Dimensions We prove that the operators $\int_{\mathbb{R}_+^2} e^{ix^a \cdot
y^b} \varphi (x,y) f(y)\, dy$ map $L^p(\mathbb{R}^2)$ into itself
for $p \in J =\bigl[\frac{a_l+b_l}{a_l+(\frac{b_l r}{2})},\frac{a_l+b_l}
{a_l(1\frac{r}{2})}\bigr]$ if $a_l,b_l\ge 1$ and $\varphi(x,y)=xy^{r}$,
$0\le r <2$, the result is sharp. Generalizations to dimensions $d>2$
are indicated.
Categories:42B20, 46B70, 47G10 
55. CJM 2001 (vol 53 pp. 756)
Correction to: Upper Bounds for the Resonance Counting Function of SchrÃ¶dinger Operators in Odd Dimensions 
Correction to: Upper Bounds for the Resonance Counting Function of SchrÃ¶dinger Operators in Odd Dimensions The proof of Lemma~3.4 in [F] relies on the incorrect equality $\mu_j
(AB) = \mu_j (BA)$ for singular values (for a counterexample, see [S,
p.~4]). Thus, Theorem~3.1 as stated has not been proven. However,
with minor changes, we can obtain a bound for the counting function in
terms of the growth of the Fourier transform of $V$.
Categories:47A10, 47A40, 81U05 
56. CJM 2001 (vol 53 pp. 506)
Isometric Dilations of NonCommuting Finite Rank $n$Tuples A contractive $n$tuple $A=(A_1,\dots,A_n)$ has a minimal joint
isometric dilation $S=\break
(S_1,\dots,S_n)$ where the $S_i$'s are
isometries with pairwise orthogonal ranges. This determines a
representation of the CuntzToeplitz algebra. When $A$ acts on a
finite dimensional space, the $\wot$closed nonselfadjoint algebra
$\fS$ generated by $S$ is completely described in terms of the
properties of $A$. This provides complete unitary invariants for the
corresponding representations. In addition, we show that the algebra
$\fS$ is always hyperreflexive. In the last section, we describe
similarity invariants. In particular, an $n$tuple $B$ of $d\times d$
matrices is similar to an irreducible $n$tuple $A$ if and only if
a certain finite set of polynomials vanish on $B$.
Category:47L80 
57. CJM 2000 (vol 52 pp. 1221)
Nest Representations of TAF Algebras A nest representation of a strongly maximal TAF algebra $A$ with
diagonal $D$ is a representation $\pi$ for which $\lat \pi(A)$ is
totally ordered. We prove that $\ker \pi$ is a meet irreducible ideal
if the spectrum of $A$ is totally ordered or if (after an appropriate
similarity) the von Neumann algebra $\pi(D)''$ contains an atom.
Keywords:nest representation, meet irreducible ideal, strongly maximal TAF algebra Categories:47L40, 47L35 
58. CJM 2000 (vol 52 pp. 849)
Operator Estimates for Fredholm Modules We study estimates of the type
$$
\Vert \phi(D)  \phi(D_0) \Vert_{\emt} \leq C \cdot \Vert D  D_0
\Vert^{\alpha}, \quad \alpha = \frac12, 1
$$
where $\phi(t) = t(1 + t^2)^{1/2}$, $D_0 = D_0^*$ is an unbounded
linear operator affiliated with a semifinite von Neumann algebra
$\calM$, $D  D_0$ is a bounded selfadjoint linear operator from
$\calM$ and $(1 + D_0^2)^{1/2} \in \emt$, where $\emt$ is a symmetric
operator space associated with $\calM$. In particular, we prove that
$\phi(D)  \phi(D_0)$ belongs to the noncommutative $L_p$space for
some $p \in (1,\infty)$, provided $(1 + D_0^2)^{1/2}$ belongs to the
noncommutative weak $L_r$space for some $r \in [1,p)$. In the case
$\calM = \calB (\calH)$ and $1 \leq p \leq 2$, we show that this
result continues to hold under the weaker assumption $(1 +
D_0^2)^{1/2} \in \calC_p$. This may be regarded as an odd
counterpart of A.~Connes' result for the case of even Fredholm
modules.
Categories:46L50, 46E30, 46L87, 47A55, 58B15 
59. CJM 2000 (vol 52 pp. 468)
TwoWeight Estimates For Singular Integrals Defined On Spaces Of Homogeneous Type Twoweight inequalities of strong and weak type are obtained in the
context of spaces of homogeneous type. Various applications are
given, in particular to Cauchy singular integrals on regular curves.
Categories:47B38, 26D10 
60. CJM 2000 (vol 52 pp. 119)
Corrigendum to ``Spectral Theory for the Neumann Laplacian on Planar Domains with HornLike Ends'' Errors to a previous paper (Canad. J. Math. (2) {\bf 49}(1997),
232262) are corrected. A nonstandard regularisation of the
auxiliary operator $A$ appearing in Mourre theory is used.
Categories:35P25, 58G25, 47F05 
61. CJM 2000 (vol 52 pp. 197)
Sublinearity and Other Spectral Conditions on a Semigroup Subadditivity, sublinearity, submultiplicativity, and other
conditions are considered for spectra of pairs of operators on a
Hilbert space. Sublinearity, for example, is a weakening of the
wellknown property~$L$ and means $\sigma(A+\lambda B) \subseteq
\sigma(A) + \lambda \sigma(B)$ for all scalars $\lambda$. The
effect of these conditions is examined on commutativity,
reducibility, and triangularizability of multiplicative semigroups
of operators. A sample result is that sublinearity of spectra
implies simultaneous triangularizability for a semigroup of compact
operators.
Categories:47A15, 47D03, 15A30, 20A20, 47A10, 47B10 
62. CJM 1999 (vol 51 pp. 850)
Tensor Algebras, Induced Representations, and the Wold Decomposition Our objective in this sequel to \cite{MSp96a} is to develop extensions,
to representations of tensor algebras over $C^{*}$correspondences, of
two fundamental facts about isometries on Hilbert space: The Wold
decomposition theorem and Beurling's theorem, and to apply these to
the analysis of the invariant subspace structure of certain subalgebras
of CuntzKrieger algebras.
Keywords:tensor algebras, correspondence, induced representation, Wold decomposition, Beurling's theorem Categories:46L05, 46L40, 46L89, 47D15, 47D25, 46M10, 46M99, 47A20, 47A45, 47B35 
63. CJM 1999 (vol 51 pp. 566)
Quotient Hereditarily Indecomposable Banach Spaces A Banach space $X$ is said to be {\it quotient hereditarily
indecomposable\/} if no infinite dimensional quotient of a subspace
of $X$ is decomposable. We provide an example of a quotient
hereditarily indecomposable space, namely the space $X_{\GM}$
constructed by W.~T.~Gowers and B.~Maurey in \cite{GM}. Then we
provide an example of a reflexive hereditarily indecomposable space
$\hat{X}$ whose dual is not hereditarily indecomposable; so
$\hat{X}$ is not quotient hereditarily indecomposable. We also
show that every operator on $\hat{X}^*$ is a strictly singular
perturbation of an homothetic map.
Categories:46B20, 47B99 
64. 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
selfadjoint 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 selfadjoint operator. The path
$$
D_t^u:=(1t) D+tuDu^\ast=D+tB
$$
is a ``continuous'' path of unbounded selfadjoint ``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 normcontinuous path of (bounded) selfadjoint 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 halfinteger:
$$
\alpha (X)={1\over {\tilde {C}_m}}\Tr \Bigl(X\bigl(1+(D+B)^2\bigr)^{m}\Bigr)
$$
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 righthand
side. We also prove a bounded finitely summable version of the form:
$$
\sf(\{F_t\})={1\over C_n}\int_0^1\Tr\Bigl({d\over dt}(F_t)(1F_t^2)^n\Bigr)\,dt
$$
for $n\geq{{p1}\over 2}$ 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 
65. CJM 1998 (vol 50 pp. 538)
Upper bounds for the resonance counting function of SchrÃ¶dinger operators in odd dimensions The purpose of this note is to provide a simple proof of the sharp
polynomial upper bound for the resonance counting function of
a Schr\"odinger operator in odd dimensions. At the same time
we generalize the result to the class of superexponentially
decreasing potentials.
Categories:47A10, 47A40, 81U05 
66. CJM 1998 (vol 50 pp. 658)
Hankel operators on pseudoconvex domains of finite type in ${\Bbb C}^2$ The aim of this paper is to study small Hankel operators $h$ on the
Hardy space or on weighted Bergman spaces, where $\Omega$ is a
finite type domain in ${\Bbbvii C}^2$ or a strictly pseudoconvex
domain in ${\Bbbvii C}^n$. We give a sufficient condition on the
symbol $f$ so that $h$ belongs to the Schatten class ${\cal S}_p$,
$1\le p<+\infty$.
Categories:32A37, 47B35, 47B10, 46E22 
67. CJM 1998 (vol 50 pp. 290)
Noncommutative disc algebras for semigroups We study noncommutative disc algebras associated to the free
product of discrete subsemigroups of $\bbR^+$. These algebras are
associated to generalized Cuntz algebras, which are shown to be
simple and purely infinite. The nonselfadjoint subalgebras
determine the semigroup up to isomorphism. Moreover, we establish
a dilation theorem for contractive representations of these
semigroups which yields a variant of the von Neumann inequality.
These methods are applied to establish a solution to the truncated
moment problem in this context.
Category:47D25 
68. CJM 1998 (vol 50 pp. 99)
$A_\phi$invariant subspaces on the torus Generalizing the notion of invariant subspaces on
the 2dimensional torus $T^2$, we study the structure
of $A_\phi$invariant subspaces of $L^2(T^2)$. A
complete description is given of $A_\phi$invariant
subspaces that satisfy conditions similar to those
studied by Mandrekar, Nakazi, and Takahashi.
Categories:32A35, 47A15 
69. CJM 1997 (vol 49 pp. 1117)
The von Neumann algebra $\VN(G)$ of a locally compact group and quotients of its subspaces Let $\VN(G)$ be the von Neumann algebra of a locally
compact group $G$. We denote by $\mu$ the initial
ordinal with $\abs{\mu}$ equal to the smallest cardinality
of an open basis at the unit of $G$ and $X= \{\alpha;
\alpha < \mu \}$. We show that if $G$ is nondiscrete
then there exist an isometric $*$isomorphism $\kappa$
of $l^{\infty}(X)$ into $\VN(G)$ and a positive linear
mapping $\pi$ of $\VN(G)$ onto $l^{\infty}(X)$ such that
$\pi\circ\kappa = \id_{l^{\infty}(X)}$ and $\kappa$ and
$\pi$ have certain additional properties. Let $\UCB
(\hat{G})$ be the $C^{*}$algebra generated by
operators in $\VN(G)$ with compact support and
$F(\hat{G})$ the space of all $T \in \VN(G)$ such that
all topologically invariant means on $\VN(G)$ attain the
same value at $T$. The construction of the mapping $\pi$
leads to the conclusion that the quotient space $\UCB
(\hat{G})/F(\hat{G})\cap \UCB(\hat{G})$ has
$l^{\infty}(X)$ as a continuous linear image if $G$ is
nondiscrete. When $G$ is further assumed to be
nonmetrizable, it is shown that $\UCB(\hat{G})/F
(\hat{G})\cap \UCB(\hat{G})$ contains a linear
isomorphic copy of $l^{\infty}(X)$. Similar results are
also obtained for other quotient spaces.
Categories:22D25, 43A22, 43A30, 22D15, 43A07, 47D35 
70. CJM 1997 (vol 49 pp. 736)
Dilations of one parameter Semigroups of positive Contractions on $L^{\lowercase {p}}$ spaces It is proved in this note, that a strongly continuous semigroup of
(sub)positive contractions acting on an $L^p$space, for $1

71. CJM 1997 (vol 49 pp. 100)
Multiplication Invariant Subspaces of Hardy Spaces This paper studies closed subspaces $L$
of the Hardy spaces $H^p$ which are $g$invariant ({\it i.e.},
$g\cdot L \subseteq L)$ where $g$ is inner, $g\neq 1$. If
$p=2$, the Wold decomposition theorem implies that there is
a countable ``$g$basis'' $f_1, f_2,\ldots$ of
$L$ in the sense that $L$ is a direct sum of spaces
$f_j\cdot H^2[g]$ where $H^2[g] = \{f\circ g \mid f\in H^2\}$.
The basis elements $f_j$ satisfy the
additional property that $\int_T f_j^2 g^k=0$,
$k=1,2,\ldots\,.$ We call such functions $g$$2$inner.
It also
follows that any $f\in H^2$ can be factored $f=h_{f,2}\cdot
(F_2\circ g)$ where $h_{f,2}$ is $g$$2$inner and $F$ is
outer, generalizing the classical Riesz factorization.
Using $L^p$ estimates for the canonical decomposition of
$H^2$, we find a factorization $f=h_{f,p} \cdot (F_p \circ
g)$ for $f\in H^p$. If $p\geq 1$ and $g$ is a finite
Blaschke product we obtain, for any $g$invariant
$L\subseteq H^p$, a finite $g$basis of $g$$p$inner
functions.
Categories:30H05, 46E15, 47B38 