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Search: All articles in the CJM digital archive with keyword subspace

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1. CJM Online first

Plevnik, Lucijan; Šemrl, Peter
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)

Mashreghi, J.; Shabankhah, M.
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)

Bell, Jason P.; Hare, Kevin G.
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)

Baranov, Anton; Woracek, Harald
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)

Ferenczi, Valentin; Galego, Elói Medina
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)

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 {\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

7. 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 \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

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