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Search: MSC category 47A15 ( Invariant subspaces [See also 47A46] )

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1. CJM 2009 (vol 61 pp. 282)

Bouya, Brahim
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)

Bercovici, H.; Foias, C.; Pearcy, C.
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)

MacDonald, Gordon W.
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)

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

5. 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

6. 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

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

8. 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

9. 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

10. 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

11. CJM 1998 (vol 50 pp. 99)

Izuchi, Keiji; Matsugu, Yasuo
$A_\phi$-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

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