Stable invariant subspaces for operators on Hilbert space

John B. Conway; Don Hadwin

Displaying similar documents to “Stable invariant subspaces for operators on Hilbert space”

Automatic continuity of operators commuting with translations

J. Alaminos, J. Extremera, A. R. Villena (2006)

Studia Mathematica

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Let $τ_{X}$ and $τ_{Y}$ be representations of a topological group G on Banach spaces X and Y, respectively. We investigate the continuity of the linear operators Φ: X → Y with the property that $Φ \circ τ_{X} (t) = τ_{Y} (t) \circ Φ$ for each t ∈ G in terms of the invariant vectors in Y and the automatic continuity of the invariant linear functionals on X.

Stable solutions of $- Δ u = f (u)$ in $ℝ^{N}$

Louis Dupaigne, Alberto Farina (2010)

Journal of the European Mathematical Society

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Several Liouville-type theorems are presented for stable solutions of the equation $- Δ u = f (u)$ in $ℝ^{N}$ , where $f > 0$ is a general convex, nondecreasing function. Extensions to solutions which are merely stable outside a compact set are discussed.

Shift invariant operators and a saturation theorem

Karol Dziedziul (2003)

Applicationes Mathematicae

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The properties of shift invariant operators $Q_{h}$ are proved: It is shown that Q has polynomial order r iff r is the rate of convergence of $Q_{h}$ . A weak saturation theorem is given. If f is replaced by $Q_{f} h$ in the weak saturation formula the asymptotics of the expression is calculated. Moreover, bootstrap approximation is introduced.

A note on integer translates of a square integrable function on ℝ

Maciej Paluszyński (2010)

Colloquium Mathematicae

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We consider the subspace of L²(ℝ) spanned by the integer shifts of one function ψ, and formulate a condition on the family ${ψ (\cdot - n)}_{n = - \infty}^{\infty}$ , which is equivalent to the weight function $\sum_{n = - \infty}^{\infty} | ψ ̂ (\cdot + n) | ²$ being > 0 a.e.

The Daugavet property and translation-invariant subspaces

Simon Lücking (2014)

Studia Mathematica

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Let G be an infinite, compact abelian group and let Λ be a subset of its dual group Γ. We study the question which spaces of the form $C_{Λ} (G)$ or $L ¹_{Λ} (G)$ and which quotients of the form $C (G) / C_{Λ} (G)$ or $L ¹ (G) / L ¹_{Λ} (G)$ have the Daugavet property. We show that $C_{Λ} (G)$ is a rich subspace of C(G) if and only if $Γ ∖ Λ^{- 1}$ is a semi-Riesz set. If $L ¹_{Λ} (G)$ is a rich subspace of L¹(G), then $C_{Λ} (G)$ is a rich subspace of C(G) as well. Concerning quotients, we prove that $C (G) / C_{Λ} (G)$ has the Daugavet property if Λ is a Rosenthal set, and that $L ¹_{Λ} (G)$ is a poor subspace of L¹(G)...

An obstruction to $ℓ^{p}$ -dimension

Nicolas Monod, Henrik Densing Petersen (2014)

Annales de l’institut Fourier

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Let $G$ be any group containing an infinite elementary amenable subgroup and let $2 < p < \infty$ . We construct an exhaustion of $ℓ^{p} G$ by closed invariant subspaces which all intersect trivially a fixed non-trivial closed invariant subspace. This is an obstacle to $ℓ^{p}$ -dimension and gives an answer to a question of Gaboriau.

Simultaneous stabilization in $A_{ℝ} ()$

Raymond Mortini, Brett D. Wick (2009)

Studia Mathematica

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We study the problem of simultaneous stabilization for the algebra $A_{ℝ} ()$ . Invertible pairs $(f_{j}, g_{j})$ , j = 1,..., n, in a commutative unital algebra are called simultaneously stabilizable if there exists a pair (α,β) of elements such that $α f_{j} + β g_{j}$ is invertible in this algebra for j = 1,..., n. For n = 2, the simultaneous stabilization problem admits a positive solution for any data if and only if the Bass stable rank of the algebra is one. Since $A_{ℝ} ()$ has stable rank two, we are faced here with a different...

On the range-kernel orthogonality of elementary operators

Said Bouali, Youssef Bouhafsi (2015)

Mathematica Bohemica

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Let $L (H)$ denote the algebra of operators on a complex infinite dimensional Hilbert space $H$ . For $A, B \in L (H)$ , the generalized derivation $δ_{A, B}$ and the elementary operator $Δ_{A, B}$ are defined by $δ_{A, B} (X) = A X - X B$ and $Δ_{A, B} (X) = A X B - X$ for all $X \in L (H)$ . In this paper, we exhibit pairs $(A, B)$ of operators such that the range-kernel orthogonality of $δ_{A, B}$ holds for the usual operator norm. We generalize some recent results. We also establish some theorems on the orthogonality of the range and the kernel of $Δ_{A, B}$ with respect to the wider class of unitarily invariant...

Convergence to stable laws and a local limit theorem for stochastic recursions

Mariusz Mirek (2010)

Colloquium Mathematicae

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We consider the random recursion $X ₙ^{x} = M ₙ X_{n - 1}^{x} + Q ₙ + N ₙ (X_{n - 1}^{x})$ , where x ∈ ℝ and (Mₙ,Qₙ,Nₙ) are i.i.d., Qₙ has a heavy tail with exponent α > 0, the tail of Mₙ is lighter and $N ₙ (X_{n - 1}^{x})$ is smaller at infinity, than $M ₙ X_{n - 1}^{x}$ . Using the asymptotics of the stationary solutions we show that properly normalized Birkhoff sums $S ₙ^{x} = \sum_{k = 0}^{n} X_{k}^{x}$ converge weakly to an α-stable law for α ∈ (0,2]. The related local limit theorem is also proved.

Stabilization of monomial maps in higher codimension

Jan-Li Lin, Elizabeth Wulcan (2014)

Annales de l’institut Fourier

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A monomial self-map $f$ on a complex toric variety is said to be $k$ -stable if the action induced on the $2 k$ -cohomology is compatible with iteration. We show that under suitable conditions on the eigenvalues of the matrix of exponents of $f$ , we can find a toric model with at worst quotient singularities where $f$ is $k$ -stable. If $f$ is replaced by an iterate one can find a $k$ -stable model as soon as the dynamical degrees $λ_{k}$ of $f$ satisfy $λ_{k}^{2} > λ_{k - 1} λ_{k + 1}$ . On the other hand, we give examples of monomial maps $f$ , where...

Vector invariant ideals of abelian group algebras under the actions of the unitary groups and orthogonal groups

Lingli Zeng, Jizhu Nan (2016)

Czechoslovak Mathematical Journal

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Let $F$ be a finite field of characteristic $p$ and $K$ a field which contains a primitive $p$ th root of unity and $char K \neq p$ . Suppose that a classical group $G$ acts on the $F$ -vector space $V$ . Then it can induce the actions on the vector space $V \oplus V$ and on the group algebra $K [V \oplus V]$ , respectively. In this paper we determine the structure of $G$ -invariant ideals of the group algebra $K [V \oplus V]$ , and establish the relationship between the invariant ideals of $K [V]$ and the vector invariant ideals of $K [V \oplus V]$ , if $G$ is a unitary group or orthogonal...

Invariant theory and the $𝒲_{1 + \infty}$ algebra with negative integral central charge

Andrew Linshaw (2011)

Journal of the European Mathematical Society

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The vertex algebra $𝒲_{1 + \infty, c}$ with central charge $c$ may be defined as a module over the universal central extension of the Lie algebra of differential operators on the circle. For an integer $n \geq 1$ , it was conjectured in the physics literature that $𝒲_{1 + \infty, - n}$ should have a minimal strong generating set consisting of $n^{2} + 2 n$ elements. Using a free field realization of $𝒲_{1 + \infty, - n}$ due to Kac–Radul, together with a deformed version of Weyl’s first and second fundamental theorems of invariant theory for the standard representation...

Invariant subspaces for operators in a general II1-factor

Uffe Haagerup, Hanne Schultz (2009)

Publications Mathématiques de l'IHÉS

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Let ℳ be a von Neumann factor of type II1 with a normalized trace τ. In 1983 L. G. Brown showed that to every operator T∈ℳ one can in a natural way associate a spectral distribution measure μ T (now called the Brown measure of T), which is a probability measure in ℂ with support in the spectrum σ(T) of T. In this paper it is shown that for every T∈ℳ and every Borel set B in ℂ, there is a unique closed T-invariant subspace $𝒦 = 𝒦_{T} (B)$ affiliated with ℳ, such that the Brown measure of ${T |}_{𝒦}$ is concentrated...

The Embeddability of c₀ in Spaces of Operators

Ioana Ghenciu, Paul Lewis (2008)

Bulletin of the Polish Academy of Sciences. Mathematics

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Results of Emmanuele and Drewnowski are used to study the containment of c₀ in the space $K_{w *} (X *, Y)$ , as well as the complementation of the space $K_{w *} (X *, Y)$ of w*-w compact operators in the space $L_{w *} (X *, Y)$ of w*-w operators from X* to Y.

n-supercyclic and strongly n-supercyclic operators in finite dimensions

Romuald Ernst (2014)

Studia Mathematica

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We prove that on $ℝ^{N}$ , there is no n-supercyclic operator with 1 ≤ n < ⌊(N + 1)/2⌋, i.e. if $ℝ^{N}$ has an n-dimensional subspace whose orbit under $T \in (ℝ^{N})$ is dense in $ℝ^{N}$ , then n is greater than ⌊(N + 1)/2⌋. Moreover, this value is optimal. We then consider the case of strongly n-supercyclic operators. An operator $T \in (ℝ^{N})$ is strongly n-supercyclic if $ℝ^{N}$ has an n-dimensional subspace whose orbit under T is dense in $ℙ ₙ (ℝ^{N})$ , the nth Grassmannian. We prove that strong n-supercyclicity does not occur non-trivially...

Structure properties of D-R spaces

Hartmut von Trotha

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CONTENTSIntroduction................................................................................................................................... 5 Notations.......................................................................................................................... 5§ 1. Preliminaries........................................................................................................................ 6 1. Right invertible operators.....................................................................................................

On Lindenstrauss-Pełczyński spaces

Jesús M. F. Castillo, Yolanda Moreno, Jesús Suárez (2006)

Studia Mathematica

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We consider some stability aspects of the classical problem of extension of C(K)-valued operators. We introduce the class ℒ of Banach spaces of Lindenstrauss-Pełczyński type as those such that every operator from a subspace of c₀ into them can be extended to c₀. We show that all ℒ-spaces are of type $ℒ_{\infty}$ but not conversely. Moreover, $ℒ_{\infty}$ -spaces will be characterized as those spaces E such that E-valued operators from w*(l₁,c₀)-closed subspaces of l₁ extend to l₁. Regarding examples we will...

On pathwise uniqueness for stochastic differential equations driven by stable Lévy processes

Nicolas Fournier (2013)

Annales de l'I.H.P. Probabilités et statistiques

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We study a one-dimensional stochastic differential equation driven by a stable Lévy process of order $α$ with drift and diffusion coefficients $b$ , $σ$ . When $α \in (1, 2)$ , we investigate pathwise uniqueness for this equation. When $α \in (0, 1)$ , we study another stochastic differential equation, which is equivalent in law, but for which pathwise uniqueness holds under much weaker conditions. We obtain various results, depending on whether $α \in (0, 1)$ or $α \in (1, 2)$ and on whether the driving stable process is symmetric or not. Our...

Finite-energy sign-changing solutions with dihedral symmetry for the stationary nonlinear Schrödinger equation

Monica Musso, Frank Pacard, Juncheng Wei (2012)

Journal of the European Mathematical Society

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We address the problem of the existence of finite energy solitary waves for nonlinear Klein-Gordon or Schrödinger type equations $Δ u - u + f (u) = 0$ in $ℝ^{N}$ , $u \in H^{1} (ℝ^{N})$ , where $N \geq 2$ . Under natural conditions on the nonlinearity $f$ , we prove the existence of $𝑖𝑛𝑓𝑖𝑛𝑖𝑡𝑒𝑙𝑦𝑚𝑎𝑛𝑦𝑛𝑜𝑛𝑟𝑎𝑑𝑖𝑎𝑙𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛𝑠$ in any dimension $N \geq 2$ . Our result complements earlier works of Bartsch and Willem $(N = 4 𝚘𝚛 N \geq 6)$ and Lorca-Ubilla $(N = 5)$ where solutions invariant under the action of $O (2) \times O (N - 2)$ are constructed. In contrast, the solutions we construct are invariant under the action of $D_{k} \times O (N - 2)$ where $D_{k} \subset O (2)$ denotes the dihedral...

A finite multiplicity Helson-Lowdenslager-de Branges theorem

Sneh Lata, Meghna Mittal, Dinesh Singh (2010)

Studia Mathematica

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We prove two theorems. The first theorem reduces to a scalar situation the well known vector-valued generalization of the Helson-Lowdenslager theorem that characterizes the invariant subspaces of the operator of multiplication by the coordinate function z on the vector-valued Lebesgue space L²(;ℂⁿ). Our approach allows us to prove an equivalent version of the vector-valued Helson-Lowdenslager theorem in a completely scalar setting, thereby eliminating the use of range functions and partial...

On invariant, dual invariant and absolute formulas

Andrzej Mostowski

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CONTENTS Introduction..............................................................................................................................................................3 1. Lemmas concerning first order formulas.....................................................................................................5 2. Representability of recursively enumerable sets........................................................................................9 3. Simple theory of types.......................................................................................................................................10...

Structure of Rademacher subspaces in Cesàro type spaces

Sergey V. Astashkin, Lech Maligranda (2015)

Studia Mathematica

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The structure of the closed linear span of the Rademacher functions in the Cesàro space $C e s_{\infty}$ is investigated. It is shown that every infinite-dimensional subspace of either is isomorphic to l₂ and uncomplemented in $C e s_{\infty}$ , or contains a subspace isomorphic to c₀ and complemented in . The situation is rather different in the p-convexification of $C e s_{\infty}$ if 1 < p < ∞.

Real deformations and invariants of map-germs

J. H. Rieger, M. A. S. Ruas, R. Wik Atique (2008)

Banach Center Publications

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A stable deformation $f^{t}$ of a real map-germ $f : ℝ ⁿ, 0 \to ℝ^{p}, 0$ is said to be an M-deformation if all isolated stable (local and multi-local) singularities of its complexification $f_{ℂ}^{t}$ are real. A related notion is that of a good real perturbation $f^{t}$ of f (studied e.g. by Mond and his coworkers) for which the homology of the image (for n < p) or discriminant (for n ≥ p) of $f^{t}$ coincides with that of $f_{C}^{t}$ . The class of map germs having an M-deformation is, in some sense, much larger than the one having a good...

Local density of diffeomorphisms with large centralizers

Christian Bonatti, Sylvain Crovisier, Gioia M. Vago, Amie Wilkinson (2008)

Annales scientifiques de l'École Normale Supérieure

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Given any compact manifold $M$ , we construct a non-empty open subset $𝒪$ of the space ${Diff}^{1} (M)$ of $C^{1}$ -diffeomorphisms and a dense subset $𝒟 \subset 𝒪$ such that the centralizer of every diffeomorphism in $𝒟$ is uncountable, hence non-trivial.