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51. 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

52. CJM 2006 (vol 58 pp. 548)

Galanopoulos, P.; Papadimitrakis, M.
Hausdorff and Quasi-Hausdorff Matrices on Spaces of Analytic Functions
We consider Hausdorff and quasi-Hausdorff 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

53. CJM 2005 (vol 57 pp. 1249)

Lindström, Mikael; Saksman, Eero; Tylli, Hans-Olav
Strictly Singular and Cosingular Multiplications
Let $L(X)$ be the space of bounded linear operators on the Banach space $X$. We study the strict singularity andcosingularity of the two-sided multiplication operators $S \mapsto ASB$ on $L(X)$, where $A,B \in L(X)$ are fixed bounded operators and $X$ is a classical Banach space. Let $1
Categories:47B47, 46B28

54. 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 Laplace--Beltrami 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

55. 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

56. CJM 2005 (vol 57 pp. 225)

Booss-Bavnbek, 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) self-adjoint 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

57. 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

58. CJM 2004 (vol 56 pp. 742)

Jiang, Chunlan
Similarity Classification of Cowen-Douglas 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 non-trivial idempotent. An operator $A$ in ${\cal L}({\cal H})$ is said to a Cowen-Douglas operator, if there exists $\Omega$, a connected open subset of $C$, and $n$, a positive integer, such that (a) ${\Omega}{\subset}{\sigma}(A)=\{z{\in}C; A-z {\text {not invertible}}\};$ (b) $\ran(A-z)={\cal H}$, for $z$ in $\Omega$; (c) $\bigvee_{z{\in}{\Omega}}$\ker$(A-z)={\cal H}$ and (d) $\dim \ker(A-z)=n$ for $z$ in $\Omega$. In the paper, we give a similarity classification of strongly irreducible Cowen-Douglas operators by using the $K_0$-group of the commutant algebra as an invariant.

Categories:47A15, 47C15, 13E05, 13F05

59. CJM 2004 (vol 56 pp. 277)

Dostanić, Milutin R.
Spectral Properties of the Commutator of Bergman's Projection and the Operator of Multiplication by an Analytic Function
It is shown that the singular values of the operator $aP-Pa$, where $P$ is Bergman's projection over a bounded domain $\Omega$ and $a$ is a function analytic on $\bar{\Omega}$, detect the length of the boundary of $a(\Omega)$. Also we point out the relation of that operator and the spectral asymptotics of a Hankel operator with an anti-analytic symbol.


60. CJM 2004 (vol 56 pp. 134)

Li, Chi-Kwong; 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

61. CJM 2003 (vol 55 pp. 1264)

Havin, Victor; Mashreghi, Javad
Admissible Majorants for Model Subspaces of $H^2$, Part II: Fast Winding of the Generating Inner Function
This paper is a continuation of Part I [6]. We consider the model subspaces $K_\Theta=H^2\ominus\Theta H^2$ of the Hardy space $H^2$ generated by an inner function $\Theta$ in the upper half plane. Our main object is the class of admissible majorants for $K_\Theta$, denoted by Adm $\Theta$ and consisting of all functions $\omega$ defined on $\mathbb{R}$ such that there exists an $f \ne 0$, $f \in K_\Theta$ satisfying $|f(x)|\leq\omega(x)$ almost everywhere on $\mathbb{R}$. Firstly, using some simple Hilbert transform techniques, we obtain a general multiplier theorem applicable to any $K_\Theta$ generated by a meromorphic inner function. In contrast with [6], we consider the generating functions $\Theta$ such that the unit vector $\Theta(x)$ winds up fast as $x$ grows from $-\infty$ to $\infty$. In particular, we consider $\Theta=B$ where $B$ is a Blaschke product with ``horizontal'' zeros, i.e., almost uniformly distributed in a strip parallel to and separated from $\mathbb{R}$. It is shown, among other things, that for any such $B$, any even $\omega$ decreasing on $(0,\infty)$ with a finite logarithmic integral is in Adm $B$ (unlike the ``vertical'' case treated in [6]), thus generalizing (with a new proof) a classical result related to Adm $\exp(i\sigma z)$, $\sigma>0$. Some oscillating $\omega$'s in Adm $B$ are also described. Our theme is related to the Beurling-Malliavin multiplier theorem devoted to Adm $\exp(i\sigma z)$, $\sigma>0$, and to de Branges' space $\mathcal{H}(E)$.

Keywords:Hardy space, inner function, shift operator, model, subspace, Hilbert transform, admissible majorant
Categories:30D55, 47A15

62. CJM 2003 (vol 55 pp. 1231)

Havin, Victor; Mashreghi, Javad
Admissible Majorants for Model Subspaces of $H^2$, Part I: Slow Winding of the Generating Inner Function
A model subspace $K_\Theta$ of the Hardy space $H^2 = H^2 (\mathbb{C}_+)$ for the upper half plane $\mathbb{C}_+$ is $H^2(\mathbb{C}_+) \ominus \Theta H^2(\mathbb{C}_+)$ where $\Theta$ is an inner function in $\mathbb{C}_+$. A function $\omega \colon \mathbb{R}\mapsto[0,\infty)$ is called an admissible majorant for $K_\Theta$ if there exists an $f \in K_\Theta$, $f \not\equiv 0$, $|f(x)|\leq \omega(x)$ almost everywhere on $\mathbb{R}$. For some (mainly meromorphic) $\Theta$'s some parts of Adm $\Theta$ (the set of all admissible majorants for $K_\Theta$) are explicitly described. These descriptions depend on the rate of growth of $\arg \Theta$ along $\mathbb{R}$. This paper is about slowly growing arguments (slower than $x$). Our results exhibit the dependence of Adm $B$ on the geometry of the zeros of the Blaschke product $B$. A complete description of Adm $B$ is obtained for $B$'s with purely imaginary (``vertical'') zeros. We show that in this case a unique minimal admissible majorant exists.

Keywords:Hardy space, inner function, shift operator, model, subspace, Hilbert transform, admissible majorant
Categories:30D55, 47A15

63. 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 Lifshits-Krein spectral shift function associated with a pair of self-adjoint operators to the case of pairs of admissible operators that are similar to self-adjoint 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

64. 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

65. CJM 2002 (vol 54 pp. 1142)

Binding, Paul; Ćurgus, Branko
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 self-adjoint boundary conditions depending affinely on the eigenparameter. Corresponding modes of convergence for eigenfunction expansions are studied, including uniform convergence of the first $n-1$ derivatives.

Categories:47E05, 34B09, 47B50, 47B25, 34L10

66. CJM 2002 (vol 54 pp. 998)

Dimassi, Mouez
Resonances for Slowly Varying Perturbations of a Periodic Schrödinger 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

67. 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 $\varphi(x,y)=|x-y|^{-r}$, $0\le r <2$, the result is sharp. Generalizations to dimensions $d>2$ are indicated.

Categories:42B20, 46B70, 47G10

68. CJM 2001 (vol 53 pp. 756)

Froese, Richard
Correction to: Upper Bounds for the Resonance Counting Function of Schrödinger 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

69. CJM 2001 (vol 53 pp. 506)

Davidson, Kenneth R.; Kribs, David W.; Shpigel, Miron E.
Isometric Dilations of Non-Commuting 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 Cuntz-Toeplitz algebra. When $A$ acts on a finite dimensional space, the $\wot$-closed nonself-adjoint 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 hyper-reflexive. 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 $B$.


70. 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

71. 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 self-adjoint 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 non-commutative $L_p$-space for some $p \in (1,\infty)$, provided $(1 + D_0^2)^{-1/2}$ belongs to the non-commutative 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

72. CJM 2000 (vol 52 pp. 468)

Edmunds, D. E.; Kokilashvili, V.; Meskhi, A.
Two-Weight Estimates For Singular Integrals Defined On Spaces Of Homogeneous Type
Two-weight 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

73. CJM 2000 (vol 52 pp. 119)

Edward, Julian
Corrigendum to ``Spectral Theory for the Neumann Laplacian on Planar Domains with Horn-Like Ends''
Errors to a previous paper (Canad. J. Math. (2) {\bf 49}(1997), 232--262) are corrected. A non-standard regularisation of the auxiliary operator $A$ appearing in Mourre theory is used.

Categories:35P25, 58G25, 47F05

74. 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 well-known 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

75. 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 Cuntz-Krieger algebras.

Keywords:tensor algebras, correspondence, induced representation, Wold decomposition, Beurling's theorem
Categories:46L05, 46L40, 46L89, 47D15, 47D25, 46M10, 46M99, 47A20, 47A45, 47B35
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