51. CJM 2003 (vol 55 pp. 449)
 Albeverio, Sergio; Makarov, Konstantin A.; Motovilov, Alexander K.

Graph Subspaces and the Spectral Shift Function
We obtain a new representation for the solution to the operator
Sylvester equation in the form of a Stieltjes operator integral.
We also formulate new sufficient conditions for the strong
solvability of the operator Riccati equation that ensures the
existence of reducing graph subspaces for block operator matrices.
Next, we extend the concept of the LifshitsKrein spectral shift
function associated with a pair of selfadjoint operators to the
case of pairs of admissible operators that are similar to
selfadjoint operators. Based on this new concept we express the
spectral shift function arising in a perturbation problem for block
operator matrices in terms of the angular operators associated with
the corresponding perturbed and unperturbed eigenspaces.
Categories:47B44, 47A10, 47A20, 47A40 

52. CJM 2003 (vol 55 pp. 379)
 Stessin, Michael; Zhu, Kehe

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 

53. CJM 2002 (vol 54 pp. 1142)
54. CJM 2002 (vol 54 pp. 998)
 Dimassi, Mouez

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 

55. CJM 2001 (vol 53 pp. 1031)
 Sampson, G.; Szeptycki, P.

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 

56. CJM 2001 (vol 53 pp. 756)
57. CJM 2001 (vol 53 pp. 506)
 Davidson, Kenneth R.; Kribs, David W.; Shpigel, Miron E.

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 

58. CJM 2000 (vol 52 pp. 1221)
 Hopenwasser, Alan; Peters, Justin R.; Power, Stephen C.

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 

59. CJM 2000 (vol 52 pp. 849)
 Sukochev, F. A.

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 

60. CJM 2000 (vol 52 pp. 468)
61. CJM 2000 (vol 52 pp. 119)
62. CJM 2000 (vol 52 pp. 197)
 Radjavi, Heydar

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 

63. CJM 1999 (vol 51 pp. 850)
 Muhly, Paul S.; Solel, Baruch

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 

64. CJM 1999 (vol 51 pp. 566)
 Ferenczi, V.

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 

65. CJM 1998 (vol 50 pp. 673)
 Carey, Alan; Phillips, John

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 

66. CJM 1998 (vol 50 pp. 538)
67. CJM 1998 (vol 50 pp. 658)
 Symesak, Frédéric

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 

68. CJM 1998 (vol 50 pp. 290)
 Davidson, Kenneth R.; Popescu, Gelu

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 

69. CJM 1998 (vol 50 pp. 99)
 Izuchi, Keiji; Matsugu, Yasuo

$A_\phi$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 

70. CJM 1997 (vol 49 pp. 1117)
 Hu, Zhiguo

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 

71. CJM 1997 (vol 49 pp. 736)
72. CJM 1997 (vol 49 pp. 100)
 Lance, T. L.; Stessin, M. I.

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 
