26. CJM 2009 (vol 62 pp. 242)
27. CJM 2009 (vol 62 pp. 133)
 Makarov, Konstantin A.; Skripka, Anna

Some Applications of the Perturbation Determinant in Finite von Neumann Algebras
In the finite von Neumann algebra setting, we introduce the concept
of a perturbation determinant associated with a pair of selfadjoint
elements $H_0$ and $H$ in the algebra and relate it to the concept of
the de la HarpeSkandalis homotopy invariant determinant associated
with piecewise $C^1$paths of operators joining $H_0$ and $H$. We
obtain an analog of Krein's formula that relates the perturbation
determinant and the spectral shift function and, based on this
relation, we derive subsequently (i) the BirmanSolomyak formula for
a general nonlinear perturbation, (ii) a universality of a spectral
averaging, and (iii) a generalization of the
DixmierFugledeKadison differentiation formula.
Keywords:perturbation determinant, trace formulae, von Neumann algebras Categories:47A55, 47C15, 47A53 

28. CJM 2009 (vol 61 pp. 1239)
 Davidson, Kenneth R.; Yang, Dilian

Periodicity in Rank 2 Graph Algebras
Kumjian and Pask introduced an aperiodicity condition
for higher rank graphs.
We present a detailed analysis of when this occurs
in certain rank 2 graphs.
When the algebra is aperiodic, we give another proof
of the simplicity of $\mathrm{C}^*(\mathbb{F}^+_{\theta})$.
The periodic $\mathrm{C}^*$algebras are characterized, and it is shown
that $\mathrm{C}^*(\mathbb{F}^+_{\theta}) \simeq
\mathrm{C}(\mathbb{T})\otimes\mathfrak{A}$
where $\mathfrak{A}$ is a simple $\mathrm{C}^*$algebra.
Keywords:higher rank graph, aperiodicity condition, simple $\mathrm{C}^*$algebra, expectation Categories:47L55, 47L30, 47L75, 46L05 

29. CJM 2009 (vol 61 pp. 282)
 Bouya, Brahim

Closed Ideals in Some Algebras of Analytic Functions
We obtain a complete description of closed ideals of the algebra
$\cD\cap \cL$, $0<\alpha\leq\frac{1}{2}$, where $\cD$ is the
Dirichlet space and $\cL$ is the algebra of analytic functions
satisfying the Lipschitz condition of order $\alpha$.
Categories:46E20, 30H05, 47A15 

30. CJM 2009 (vol 61 pp. 241)
 Azamov, N. A.; Carey, A. L.; Dodds, P. G.; Sukochev, F. A.

Operator Integrals, Spectral Shift, and Spectral Flow
We present a new and simple approach to the theory of multiple
operator integrals that applies to unbounded operators affiliated with general \vNa s.
For semifinite \vNa s we give applications
to the Fr\'echet differentiation of operator functions that sharpen existing results,
and establish the BirmanSolomyak representation of the spectral
shift function of M.\,G.\,Krein
in terms of an average of spectral measures in the type II setting.
We also exhibit a surprising connection between the spectral shift
function and spectral flow.
Categories:47A56, 47B49, 47A55, 46L51 

31. CJM 2009 (vol 61 pp. 190)
 Lu, Yufeng; Shang, Shuxia

Bounded Hankel Products on the Bergman Space of the Polydisk
We consider the problem of determining for which square integrable
functions $f$ and $g$ on the polydisk the densely defined Hankel
product $H_{f}H_g^\ast$ is bounded on the Bergman space of the
polydisk. Furthermore, we obtain similar results for the mixed
Haplitz products $H_{g}T_{\bar{f}}$ and $T_{f}H_{g}^{*}$, where $f$
and $g$ are square integrable on the polydisk and $f$ is analytic.
Keywords:Toeplitz operator, Hankel operator, Haplitz products, Bergman space, polydisk Categories:47B35, 47B47 

32. CJM 2009 (vol 61 pp. 50)
33. CJM 2008 (vol 60 pp. 1010)
 Galé, José E.; Miana, Pedro J.

$H^\infty$ Functional Calculus and MikhlinType 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 halfline, 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 Mikhlintype 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 Mikhlintype. 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 

34. CJM 2008 (vol 60 pp. 758)
 Bercovici, H.; Foias, C.; Pearcy, C.

On the Hyperinvariant Subspace Problem. IV
This paper is a continuation of three recent articles
concerning the structure of hyperinvariant
subspace lattices of operators on a (separable, infinite dimensional)
Hilbert space $\mathcal{H}$. We show herein, in particular, that
there exists a ``universal'' fixed blockdiagonal operator $B$ on
$\mathcal{H}$ such that if $\varepsilon>0$ is given and $T$ is
an arbitrary nonalgebraic operator on $\mathcal{H}$, then there exists
a compact operator $K$ of norm less than $\varepsilon$ such that
(i) $\Hlat(T)$ is isomorphic as a complete lattice to $\Hlat(B+K)$
and (ii) $B+K$ is a quasidiagonal, $C_{00}$, (BCP)operator with
spectrum and left essential spectrum the unit disc. In the last four
sections of the paper, we investigate the possible structures of the
hyperlattice of an arbitrary algebraic operator. Contrary to existing
conjectures, $\Hlat(T)$ need not be generated by the ranges and kernels
of the powers of $T$ in the nilpotent case. In fact, this lattice
can be infinite.
Category:47A15 

35. CJM 2008 (vol 60 pp. 520)
 Chen, ChangPao; Huang, HaoWei; Shen, ChunYen

Matrices Whose Norms Are Determined by Their Actions on Decreasing Sequences
Let $A=(a_{j,k})_{j,k \ge 1}$ be a nonnegative 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 

36. CJM 2007 (vol 59 pp. 1207)
 Bu, Shangquan; Le, Christian

$H^p$Maximal Regularity and Operator Valued Multipliers on Hardy Spaces
We consider maximal regularity in the $H^p$ sense for the Cauchy
problem $u'(t) + Au(t) = f(t)\ (t\in \R)$, where $A$ is a closed
operator on a Banach space $X$ and $f$ is an $X$valued function
defined on $\R$. We prove that if $X$ is an AUMD Banach space,
then $A$ satisfies $H^p$maximal regularity if and only if $A$ is
Rademacher sectorial of type $<\frac{\pi}{2}$. Moreover we find an
operator $A$ with $H^p$maximal regularity that does not have the
classical $L^p$maximal regularity. We prove a related Mikhlin
type theorem for operator valued Fourier multipliers on Hardy
spaces $H^p(\R;X)$, in the case when $X$ is an AUMD Banach space.
Keywords:$L^p$maximal regularity, $H^p$maximal regularity, Rademacher boundedness Categories:42B30, 47D06 

37. CJM 2007 (vol 59 pp. 966)
 Forrest, Brian E.; Runde, Volker; Spronk, Nico

Operator Amenability of the Fourier Algebra in the $\cb$Multiplier Norm
Let $G$ be a locally compact group, and let $A_{\cb}(G)$ denote the
closure of $A(G)$, the Fourier algebra of $G$, in the space of completely
bounded multipliers of $A(G)$. If $G$ is a weakly amenable, discrete group
such that $\cstar(G)$ is residually finitedimensional, we show that
$A_{\cb}(G)$ is operator amenable. In particular,
$A_{\cb}(\free_2)$ is operator amenable even though $\free_2$, the free
group in two generators, is not an amenable group. Moreover, we show that
if $G$ is a discrete group such that $A_{\cb}(G)$ is operator amenable,
a closed ideal of $A(G)$ is weakly completely complemented in $A(G)$
if and only if it has an approximate identity bounded in the $\cb$multiplier
norm.
Keywords:$\cb$multiplier norm, Fourier algebra, operator amenability, weak amenability Categories:43A22, 43A30, 46H25, 46J10, 46J40, 46L07, 47L25 

38. CJM 2007 (vol 59 pp. 614)
 Labuschagne, C. C. A.

Preduals and Nuclear Operators Associated with Bounded, $p$Convex, $p$Concave and Positive $p$Summing Operators
We use Krivine's form of the Grothendieck inequality
to renorm the space of bounded linear maps acting between Banach
lattices. We
construct preduals and describe the nuclear operators
associated with these preduals for this renormed space
of bounded operators as well as for
the spaces of $p$convex,
$p$concave and positive $p$summing operators acting
between Banach lattices and Banach spaces.
The nuclear operators obtained are described in
terms of factorizations through
classical Banach spaces via positive operators.
Keywords:$p$convex operator, $p$concave operator, $p$summing operator, Banach space, Banach lattice, nuclear operator, sequence space Categories:46B28, 47B10, 46B42, 46B45 

39. CJM 2007 (vol 59 pp. 638)
 MacDonald, Gordon W.

Distance from Idempotents to Nilpotents
We give bounds on the distance from a nonzero idempotent to the
set of nilpotents in the set of $n\times n$ matrices in terms of
the norm of the idempotent. We construct explicit idempotents and
nilpotents which achieve these distances, and determine exact
distances in some special cases.
Keywords:operator, matrix, nilpotent, idempotent, projection Categories:47A15, 47D03, 15A30 

40. CJM 2007 (vol 59 pp. 393)
 Servat, E.

Le splitting pour l'opÃ©rateur de KleinGordon: une approche heuristique et numÃ©rique
Dans cet article on \'etudie la diff\'erence entre les deux
premi\`eres valeurs propres, le splitting, d'un op\'erateur de
KleinGordon semiclassique unidimensionnel, dans le cas d'un
potentiel sym\'etrique pr\'esentant un double puits. Dans le cas d'une
petite barri\`ere de potentiel, B. Helffer et B. Parisse ont obtenu
des r\'esultats analogues \`a ceux existant pour l'op\'erateur de
Schr\"odinger. Dans le cas d'une grande barri\`ere de potentiel, on
obtient ici des estimations des tranform\'ees de Fourier des fonctions
propres qui conduisent \`a une conjecture du splitting. Des calculs
num\'eriques viennent appuyer cette conjecture.
Categories:35P05, 34L16, 34E05, 47A10, 47A70 

41. 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 

42. CJM 2006 (vol 58 pp. 548)
 Galanopoulos, P.; Papadimitrakis, M.

Hausdorff and QuasiHausdorff Matrices on Spaces of Analytic Functions
We consider Hausdorff and quasiHausdorff matrices as operators
on classical spaces of analytic functions such as the Hardy and
the Bergman spaces, the Dirichlet space, the Bloch spaces and $\BMOA$. When the generating
sequence of the matrix is the moment sequence of a measure $\mu$,
we find the conditions on $\mu$ which are equivalent to the boundedness
of the matrix on the various spaces.
Categories:47B38, 46E15, 40G05, 42A20 

43. CJM 2005 (vol 57 pp. 1249)
44. CJM 2005 (vol 57 pp. 771)
 Schrohe, E.; Seiler, J.

The Resolvent of Closed Extensions of Cone Differential Operators
We study closed extensions $\underline A$ of
an elliptic differential operator $A$ on a manifold with conical
singularities, acting as an unbounded operator on a weighted $L_p$space.
Under suitable conditions we show that the resolvent
$(\lambda\underline A)^{1}$ exists
in a sector of the complex plane and decays like $1/\lambda$ as
$\lambda\to\infty$. Moreover, we determine the structure of the resolvent
with enough precision to guarantee existence and boundedness of imaginary
powers of $\underline A$.
As an application we treat the LaplaceBeltrami operator for a metric with
straight conical degeneracy and describe domains yielding
maximal regularity for the Cauchy problem $\dot{u}\Delta u=f$, $u(0)=0$.
Keywords:Manifolds with conical singularities, resolvent, maximal regularity Categories:35J70, 47A10, 58J40 

45. CJM 2005 (vol 57 pp. 506)
 Gross, Leonard; Grothaus, Martin

Reverse Hypercontractivity for Subharmonic Functions
Contractivity and hypercontractivity properties of semigroups
are now well understood when the generator, $A$, is a Dirichlet form
operator.
It has been shown that in some holomorphic function spaces the
semigroup operators, $e^{tA}$, can be bounded {\it below} from
$L^p$ to $L^q$ when $p,q$ and $t$ are suitably related.
We will show that such lower boundedness occurs also in spaces
of subharmonic functions.
Keywords:Reverse hypercontractivity, subharmonic Categories:58J35, 47D03, 47D07, 32Q99, 60J35 

46. CJM 2005 (vol 57 pp. 225)
 BoossBavnbek, Bernhelm; Lesch, Matthias; Phillips, John

Unbounded Fredholm Operators and Spectral Flow
We study the gap (= ``projection norm'' = ``graph distance'') topology
of the space of all (not necessarily bounded) selfadjoint Fredholm
operators in a separable Hilbert space by the Cayley transform and
direct methods. In particular, we show the surprising result that
this space is connected in contrast to the bounded case. Moreover, we
present a rigorous definition of spectral flow of a path of such
operators (actually alternative but mutually equivalent definitions)
and prove the homotopy invariance. As an example, we discuss operator
curves on manifolds with boundary.
Categories:58J30, 47A53, 19K56, 58J32 

47. CJM 2005 (vol 57 pp. 61)
 Binding, Paul; Strauss, Vladimir

On Operators with Spectral Square but without Resolvent Points
Decompositions of spectral type are
obtained for closed Hilbert space operators with empty resolvent
set, but whose square has closure which is spectral. Krein space
situations are also discussed.
Keywords:unbounded operators, closed operators,, spectral resolution, indefinite metric Categories:47A05, 47A15, 47B40, 47B50, 46C20 

48. CJM 2004 (vol 56 pp. 742)
 Jiang, Chunlan

Similarity Classification of CowenDouglas Operators
Let $\cal H$ be a complex separable Hilbert space
and ${\cal L}({\cal H})$ denote the collection of
bounded linear operators on ${\cal H}$.
An operator $A$ in ${\cal L}({\cal H})$
is said to be strongly irreducible, if
${\cal A}^{\prime}(T)$, the commutant of $A$, has no nontrivial idempotent.
An operator $A$ in ${\cal L}({\cal H})$ is said to a CowenDouglas
operator, if there exists $\Omega$, a connected open subset of
$C$, and $n$, a positive integer, such that
(a) ${\Omega}{\subset}{\sigma}(A)=\{z{\in}C; Az {\text {not invertible}}\};$
(b) $\ran(Az)={\cal H}$, for $z$ in $\Omega$;
(c) $\bigvee_{z{\in}{\Omega}}$\ker$(Az)={\cal H}$ and
(d) $\dim \ker(Az)=n$ for $z$ in $\Omega$.
In the paper, we give a similarity classification of strongly
irreducible CowenDouglas operators by using the $K_0$group of
the commutant algebra as an invariant.
Categories:47A15, 47C15, 13E05, 13F05 

49. CJM 2004 (vol 56 pp. 277)
50. CJM 2004 (vol 56 pp. 134)
 Li, ChiKwong; Sourour, Ahmed Ramzi

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 
