Expand all Collapse all | Results 1 - 11 of 11 |
1. CJM 2009 (vol 61 pp. 282)
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 |
2. CJM 2008 (vol 60 pp. 758)
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 block-diagonal 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 |
3. CJM 2007 (vol 59 pp. 638)
Distance from Idempotents to Nilpotents We give bounds on the distance from a non-zero 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 |
4. CJM 2006 (vol 58 pp. 859)
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 |
5. CJM 2005 (vol 57 pp. 61)
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 |
6. CJM 2004 (vol 56 pp. 742)
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 |
7. CJM 2003 (vol 55 pp. 1264)
Admissible Majorants for Model Subspaces of $H^2$, Part II: Fast Winding of the Generating Inner Function |
Admissible Majorants for Model Subspaces of $H^2$, Part II: Fast Winding of the Generating Inner Function This paper is a continuation of \cite{HM02I}. 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
\cite{HM02I}, 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, {\it 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 \cite{HM02I}),
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 |
8. CJM 2003 (vol 55 pp. 1231)
Admissible Majorants for Model Subspaces of $H^2$, Part I: Slow Winding of the Generating Inner Function |
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 {\it 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 |
9. 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 |
10. 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
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 |
11. CJM 1998 (vol 50 pp. 99)
$A_\phi$-invariant subspaces on the torus Generalizing the notion of invariant subspaces on
the 2-dimensional 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 |