Expand all Collapse all | Results 1 - 7 of 7 |
1. CJM 2013 (vol 66 pp. 1143)
Maps Preserving Complementarity of Closed Subspaces of a Hilbert Space Let $\mathcal{H}$ and $\mathcal{K}$ be infinite-dimensional separable
Hilbert spaces and ${\rm Lat}\,\mathcal{H}$ the lattice of all closed subspaces oh $\mathcal{H}$.
We describe the general form of pairs of bijective maps $\phi , \psi :
{\rm Lat}\,\mathcal{H} \to {\rm Lat}\,\mathcal{K}$ having the property that for every pair
$U,V \in {\rm Lat}\,\mathcal{H}$ we have $\mathcal{H} = U \oplus V \iff \mathcal{K} = \phi (U) \oplus \psi (V)$. Then we reformulate this theorem as a description
of bijective image equality and kernel equality preserving maps acting on bounded linear idempotent operators. Several known
structural results for maps on idempotents are easy consequences.
Keywords:Hilbert space, lattice of closed subspaces, complemented subspaces, adjacent subspaces, idempotents Categories:46B20, 47B49 |
2. CJM 2013 (vol 66 pp. 387)
Composition of Inner Functions We study the image of the model subspace $K_\theta$ under the
composition operator $C_\varphi$, where $\varphi$ and $\theta$ are
inner functions, and find the smallest model subspace which contains
the linear manifold $C_\varphi K_\theta$. Then we characterize the
case when $C_\varphi$ maps $K_\theta$ into itself. This case leads to
the study of the inner functions $\varphi$ and $\psi$ such that the
composition $\psi\circ\varphi$ is a divisor of $\psi$ in the family of
inner functions.
Keywords:composition operators, inner functions, Blaschke products, model subspaces Categories:30D55, 30D05, 47B33 |
3. CJM 2012 (vol 64 pp. 254)
Corrigendum to ``On $\mathbb{Z}$-modules of Algebraic Integers'' We fix a mistake in the proof of Theorem 1.6 in the paper in the title.
Keywords:Pisot numbers, algebraic integers, number rings, Schmidt subspace theorem Categories:11R04, 11R06 |
4. CJM 2009 (vol 61 pp. 503)
Subspaces of de~Branges Spaces Generated by Majorants For a given de~Branges space $\mc H(E)$ we investigate
de~Branges subspaces defined in terms of majorants
on the real axis. If $\omega$ is a nonnegative function on $\mathbb R$,
we consider the subspace
\[
\mc R_\omega(E)=\clos_{\mc H(E)} \big\{F\in\mc H(E):
\text{ there exists } C>0:
|E^{-1} F|\leq C\omega \mbox{ on }{\mathbb R}\big\}
.
\]
We show that $\mc R_\omega(E)$ is a de~Branges subspace and
describe all subspaces of this form. Moreover,
we give a criterion for the existence of positive minimal majorants.
Keywords:de~Branges subspace, majorant, Beurling-Malliavin Theorem Categories:46E20, 30D15, 46E22 |
5. CJM 2007 (vol 59 pp. 63)
Some Results on the Schroeder--Bernstein Property for Separable Banach Spaces We construct a continuum of mutually
non-isomorphic
separable Banach spaces which are complemented in each other.
Consequently, the Schroeder--Bernstein Index of any of these spaces is
$2^{\aleph_0}$. Our
construction is based on a Banach space introduced by W. T. Gowers
and
B. Maurey in 1997.
We also use classical descriptive set theory methods, as in some
work of the first author and C. Rosendal, to improve some results
of P. G. Casazza and
of N. J. Kalton on the
Schroeder--Bernstein Property for
spaces with an unconditional finite-dimensional Schauder
decomposition.
Keywords:complemented subspaces,, Schroeder-Bernstein property Categories:46B03, 46B20 |
6. 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 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 |
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 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 |