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1. CJM Online first

Chen, Yanni; Hadwin, Don; Liu, Zhe; Nordgren, Eric
 A Beurling Theorem for Generalized Hardy Spaces on a Multiply Connected Domain The object of this paper is to prove a version of the Beurling-Helson-Lowdenslager invariant subspace theorem for operators on certain Banach spaces of functions on a multiply connected domain in $\mathbb{C}$. The norms for these spaces are either the usual Lebesgue and Hardy space norms or certain continuous gauge norms. In the Hardy space case the expected corollaries include the characterization of the cyclic vectors as the outer functions in this context, a demonstration that the set of analytic multiplication operators is maximal abelian and reflexive, and a determination of the closed operators that commute with all analytic multiplication operators. Keywords:Beurling theorem, invariant subspace, generalized Hardy space, gauge norm, multiply connected domain, Forelli projection, inner-outer factorization, affiliated operatorCategories:47L10, 30H10

2. CJM 2013 (vol 66 pp. 1143)

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, idempotentsCategories:46B20, 47B49

3. 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 subspacesCategories:30D55, 30D05, 47B33

4. 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 theoremCategories:11R04, 11R06

5. 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 TheoremCategories:46E20, 30D15, 46E22

6. 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 propertyCategories:46B03, 46B20

7. CJM 2003 (vol 55 pp. 1231)

 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 majorantCategories:30D55, 47A15
 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 majorantCategories:30D55, 47A15