The main result of this paper is the existence of a hyperinvariant subspace of weighted composition operator $Tf=vf\circτ$ on $L^p([0,1]^d)$, ($1 \leq p \leq \infty$) when the weight $v$ is in the class of ``generalized polynomials'' and the composition map is a bijective ergodic transform satisfying a given discrepancy. The work is based on the construction of a functional calculus initiated by Wermer and generalized by Davie.
We prove the existence of a non-trivial hyperinvariant subspace for several sets of polynomially compact operators. The main results of the paper are: (i) a non-trivial norm closed algebra $\mathcal A\subseteq \mathcal B(\mathscr X)$ which consists of polynomially compact quasinilpotent operators has a non-trivial hyperinvariant subspace; (ii) if there exists a non-zero compact operator in the norm closure of the algebra generated by an operator band $\mathcal S$, then $\mathcal S$ has a non-trivial hyperinvariant subspace.
Let $T$ be an absolutely continuous polynomially bounded operator, and let $θ$ be a singular inner function. It is shown that if $θ(T)$ is invertible and some additional conditions are fulfilled, then $T$ has nontrivial hyperinvariant subspaces.
In this paper, a sufficient condition for the existence of hyperinvariant subspace of compact perturbations of multiplication operators on some Banach spaces is presented. An interpretation of this result for compact perturbations of normal and diagonal operators on Hilbert space is also discussed. An improvement of a result of [FX12] for compact perturbations of diagonal operators is also obtained.
There has been a long-standing conjecture in Banach algebra that every amenable operator is similar to a normal operator. In this paper, we study the structure of amenable operators on Hilbert spaces. At first, we show that the conjecture is equivalent to every non-scalar amenable operator has a non-trivial hyperinvariant subspace and equivalent to every amenable operator is similar to a reducible operator and has a non-trivial invariant subspace; and then, we give two decompositions for amenable operators, which supporting the conjecture.
For a power bounded or polynomially bounded operator $T$ sufficient conditions for the existence of a nontrivial hyperinvariant subspace are given. The obtained hyperinvariant subspaces of $T$ have the form of the closure of the range of $φ(T)$. Here $φ$ is a singular inner function, if $T$ is polynomially bounded, or $φ$ is an analytic in the unit disc function with absolutely summable Taylor coefficients and singular inner part, if $T$ is supposed to be power bounded only. Also, an example of a quasianalytic contraction $T$ is given. The quasianalytic spectral set of $T$ is not the whole uni…
Let $f$ be an endomorphism of a finite dimensional vector space $V$ over a field $K$. An $f$-invariant subspace of $V$ is called hyperinvariant (respectively characteristic) if it is invariant under all endomorphisms (respectively automorphisms) that commute with $f$. We assume $|K| = 2$, since all characteristic subspaces are hyperinvariant if $|K| > 2$. The hyperinvariant hull $W^h$ of a subspace $ W$ of $ V$ is defined to be the smallest hyperinvariant subspace of $V$ that contains $ W$, the hyperinvariant kernel $W_H$ of $ W$ is the largest hyperinvariant subspace of $V$ that is contained …
Hyperinvariant subspaces of hyponormal operators: A constructive decomposition approach
Departamento de Análisis Matemático y Matemática Aplicada, Facultad de CC. Matematicas, Universidad Complutense de Madrid（马德里康普顿斯大学数学与计算科学学院）；Instituto de Ciencias Matemáticas ICMAT (CSIC-UAM-UC3M-UCM)（西班牙国家科学研究委员会-马德里自治大学-马德里理工大学-马德里康普顿斯大学联合数学科学研究所）
A subspace $X$ of a vector space over a field $K$ is hyperinvariant with respect to an endomorphism $f$ of $V$ if it is invariant for all endomorphisms of $V$ that commute with $f$. We assume that $f$ is locally nilpotent, that is, every $ x \in V $ is annihilated by some power of $f$, and that $V$ is an infinite direct sum of $f$-cyclic subspaces. In this note we describe the lattice of hyperinvariant subspaces of $V$. We extend results of Fillmore, Herrero and Longstaff (Linear Algebra Appl. 17 (1977), 125--132) to infinite dimensional spaces.
Let $f$ be an endomorphism of a vector space $V$ over a field $K$. An $f$-invariant subspace $X \subseteq V$ is called hyperinvariant (respectively characteristic) if $X$ is invariant under all endomorphisms (respectively automorphisms) that commute with $f$. If $|K| > 2$ then all characteristic subspaces are hyperinvariant. If $|K| = 2$ then there are endomorphisms $f$ with invariant subspaces that are characteristic but not hyperinvariant. In this paper we give a new proof of a theorem of Shoda, which provides a necessary and sufficient condition for the existence of characteristic non-hyper…
If $f$ is an endomorphism of a finite dimensional vector space over a field $K$ then an invariant subspace $X \subseteq V$ is called hyperinvariant (respectively, characteristic) if $X$ is invariant under all endomorphisms (respectively, automorphisms) that commute with $f$. According to Shoda (Math. Zeit. 31, 611--624, 1930) only if $|K| = 2$ then there exist endomorphisms $f$ with invariant subspaces that are characteristic but not hyperinvariant. In this paper we obtain a description of the set of all characteristic non-hyperinvariant subspaces for nilpotent maps $f$ with exactly two unrepe…
It is well known that for a single bounded operator $A_0$ on a Hilbert $\mathfrak{H}$, if $\mathfrak{M}\subset \mathfrak{H}$ is hyperinvariant for $A_0$, then the spectrum of $A_0|_{\mathfrak{M}}$ is contained in the spectrum of $A_0$. In this note, we modify an example of Taylor to prove the following. There exist a quadruple $A=(A_1,A_2,A_3,A_4)$ of commuting bounded Hilbert space operators and a hyperinvariant subspace $\mathfrak{X}_1$ for $A$ such that the Taylor joint spectrum of $A$ restricted to $\mathfrak{X}_1$ is a not a subset of the Taylor joint spectrum of $A$.
The methods of "minimal vectors" were introduced by Ansari and Enflo and strengthened by Pearcy, in order to prove the existence of hyperinvariant subspaces for certain operators on Hilbert space. In this note we present the method of minimal vectors for operators on super-reflexive Banach spaces and we give a new sufficient condition for the existence of hyperinvariant subspaces of certain operators on these spaces.
For an operator $A$ on a complex Banach space $X$ and a closed subspace $M\subseteq X$, the local commutant of $A$ at $M$ is the set $C(A;M)$ of all operators $T$ on $X$ such that $TAx=ATx$ for every $x\in M$. It is clear that $ C(A;M)$ is a closed linear space of operators, however it is not an algebra, in general. For a given $A$, we show that $C(A;M)$ is an algebra if and only if the largest subspace $M_A$ such that $C(A;M)=C(A;M_A)$ is invariant for every operator in $C(A;M)$. We say that these are ultrainvariant subspaces of $A$. For several types of operators we prove that there exist no…
It is shown that if the Deddens algebra ${\mathcal D}_T$ associated with a quasinilpotent operator $T$ on a complex Banach space is closed and localizing then $T$ has a nontrivial closed hyperinvariant subspace.
It is shown that to every operator T in a general von Neumann factor M of type II_1 and to every Borel set B in the complex plane, one can associate a largest, closed, T-invariant subspace, K = K_T(B), affiliated with M, such that the Brown measure of T|_K is concentrated on B. Moreover, K is T-hyperinvariant, and the Brown measure of (1-P_K)T|_(1-P_K)(H) is concentrated on C\B. In particular, if T has a Brown measure which is not concentrated on a singleton, then there exists a non-trivial, closed, T-hyperinvariant subspace. Furthermore, it is shown that for every T in M, the limit A=\lim_{n\…
Let $T:D(T)\rightarrow H_2$ be a densely defined closed operator with domain $D(T)\subset H_1$. We say $T$ to be absolutely minimum attaining if for every closed subspace $M$ of $H_1$, the restriction operator $T|_M:D(T)\cap M\rightarrow H_2$ attains its minimum modulus $m(T|_{M})$. That is, there exists $x \in D(T)\cap M$ with $\|x\|= 1$ and $\|T(x)\| = \inf \{\|T(m)\|: m \in D(T) \cap M: \|m\|=1\}$. In this article, we prove several characterizations of this class of operators and show that every operator in this class has a nontrivial hyperinvariant subspace. We also prove a spectral theore…
We describe the spectrum of weighted $d$-isomorphisms of Banach lattices restricted on closed subspaces that are "rich" enough to preserve some "memory" of the order structure of the original lattice. The examples include (but are not limited to) weighted isometries of Hardy spaces on the polydisk and unit ball in $\mathds{C}^n$.
For each sequence $\{c_n\}_n$ in $l_{1}(\N)$ we define an operator $A$ in the hyperfinite $\mathrm{II}_1$-factor $\mathcal{R}$. We prove that these operators are quasinilpotent and they generate the whole hyperfinite $\mathrm{II}_1$-factor. We show that they have non-trivial, closed, invariant subspaces affiliated to the von Neumann algebra and we provide enough evidence to suggest that these operators are interesting for the hyperinvariant subspace problem. We also present some of their properties. In particular, we show that the real and imaginary part of $A$ are equally distributed, and we …
A Hilbert space operator $U$ is called universal (in the sense of Rota) if every Hilbert space operator is similar to a multiple of $U$ restricted to one of its invariant subspaces. It follows that the Invariant Subspace Problem for Hilbert spaces is equivalent to the statement that all minimal invariant subspaces for $U$ are one dimensional. In this article we characterize all linear fractional composition operators $C_ϕ f=f\circϕ$ that have universal translates on both the classical Hardy spaces $H^2(\mathbb{C}_+)$ and $H^2(\mathbb{D})$ of the half-plane and the unit disk respectively. The s…
We discuss the continuity of the composition on several spaces of holomorphic mappings on open subsets of a complex Banach space. On the Fréchet space of the entire mappings that are bounded on bounded sets the composition turns to be even holomorphic. In such space we consider linear subspaces closed under left and right composition. We discuss the relationship of such subspaces with ideals of operators and give several examples of them. We also provide natural examples of spaces of holomorphic mappings where the composition is not continuous.
We consider the invariant subspace of composition operators on Hardy space $H^p$ where the composition operators corresponding to a function $φ$ that is a holomorphic self-map of $\mathbb D$. Firstly, we discuss composition operators $C_φ$ on subspace $H_{α,β}^p$ of Hardy space $H^p$. We will explore the invariant subspaces for $C_φ$ in various special cases. Secondly, we consider Beurling type invariant subspace for $C_φ$. When $θ$ is a inner function, we prove that $θH^p$ is invariant for $C_φ$ if and only if $\displaystyle{\frac{θ\circφ}θ}$ belongs to $\mathcal S(\mathbb D)$. Thirdly, we ob…
In general there exists no relationship between the fixed point sets of the composition and of the average of a family of nonexpansive operators in Hilbert spaces. In this paper, we establish an asymptotic principle connecting the cycles generated by under-relaxed compositions of nonexpansive operators to the fixed points of the average of these operators. In the special case when the operators are projectors onto closed convex sets, we prove a conjecture by De Pierro which has so far been established only for projections onto affine subspaces.
The Invariant Subspace Problem ("ISP") for Hilbert space operators is known to be equivalent to a question that, on its surface, seems surprisingly concrete: For composition operators induced on the Hardy space H^2 by hyperbolic automorphisms of the unit disc, is every nontrivial minimal invariant subspace one dimensional (i.e., spanned by an eigenvector)? In the hope of reviving interest in the contribution this remarkable result might offer to the studies of both composition operators and the ISP, I revisit some known results, weaken their hypotheses and simplify their proofs. Sample results…
Controlled frames which presented to improve the numerical output of iterative algorithms for inverting the frame operator, have been introduced by Balazs and et al. Also, these frames are used by Bogdanova and et al. for spherical wavelets. In this note, we aim to study some new results about controlled fusion frames and behavior of them under erasures. Finally, we will introduce the approximation operator for these frames and a result of the composition of the operators of two controlled fusion frames will be proved.
We study topologizability and power boundedness of weigh\-ted composition operators on (certain subspaces of) $\mathscr{D}'(X)$ for an open subset $X$ of $\mathbb{R}^d$. For the former property we derive a characterization in terms of the symbol and the weight of the weighted composition operator, while for the latter property necessary and sufficient conditions on the weight and the symbol are presented. Moreover, for an unweighted composition operator a characterization of power boundedness in terms of the symbol is derived for the special case of a bijective symbol.
We give a simple characterization of chaos for weighted composition $C_0$-semigroups on $L^p_ρ(Ω)$ for an open interval $Ω\subseteq\mathbb{R}$. Moreover, we characterize chaos for these classes of $C_0$-semigroups on the closed subspace $W^{1,p}_*(Ω)$ of the Sobolev space $W^{1,p}(Ω)$ for a bounded interval $Ω\subset\mathbb{R}$. These characterizations simplify previously obtained characterization of chaos for these classes of $C_0$-semigroups.
A new method of defining hereditarily indecomposable Banach spaces is presented. This method provides a unified approach for constructing reflexive HI spaces and also HI spaces with no reflexive subspace. All the spaces presented here satisfy the property that the composition of any two strictly singular operators is a compact one. This yields the first known example of a Banach space with no reflexive subspace such that every operator has a non-trivial closed invariant subspace.
A conditional weighted composition operator $T_u: L^p(Σ)\rightarrow L^p(\mathcal{A})$ ($1\leq p<\infty$), is defined by $T_u(f):= E^{\mathcal{A}}(u f\circ φ)$, where $φ: X\rightarrow X$ is a measurable transformation, $u$ is a weight function on $X$ and $E^{\mathcal{A}}$ is the conditional expectation operator with respect to $\mathcal{A}$. In this paper, we study the subspace-hypercyclicity of $T_u$ with respect to $L^p(\mathcal{A})$. First, we show that if $φ$ is a periodic nonsingular transformation, then $T_u$ is not $L^p(\mathcal{A})$-hypercyclic. The necessary conditions for the subspace…
Let $\mathbb{D}$ be the open unit disk in $\mathbb{C}$, let $H^2$ denote the Hardy space on $\mathbb{D}$ and let $φ: \mathbb{D} \rightarrow \mathbb{D}$ be a holomorphic self map of $\mathbb{D}$. The composition operator $C_φ$ on $H^2$ is defined by \[ (C_φ f)(z)=f(φ(z)) \quad \quad (f \in H^2,\, z \in \mathbb{D}). \] Denote by $\mathcal{S}(\mathbb{D})$ the set of all functions that are holomorphic and bounded by one in modulus on $\mathbb{D}$, that is \[ \mathcal{S}(\mathbb{D}) = \{ψ\in H^\infty(\mathbb{D}): \|ψ\|_{\infty} := \sup_{z \in \mathbb{D}} |ψ(z)| \leq 1\}. \] The elements of $\mathca…
In this paper we propose a different (and equivalent) norm on $S^{2} ({\mathbb{D}})$ which consists of functions whose derivatives are in the Hardy space of unit disk. The reproducing kernel of $S^{2}({\mathbb{D}})$ in this norm admits an explicit form, and it is a complete Nevanlinna-Pick kernel. Furthermore, there is a surprising connection of this norm with $3$ -isometries. We then study composition and multiplication operators on this space. Specifically, we obtain an upper bound for the norm of $C_φ$ for a class of composition operators. We completely characterize multiplication operators…
Let $ E $ be a space of holomorphic functions on the unit ball $ B_X $ of a Banach space $ X.$ In this work, we introduce a Banach structure associated to $ E $ on the linear space $ WE(Y) $ containing $ Y$-valued holomorphic functions on $ B_X $ such that $ w \circ f \in E $ for every $ w \in W, $ a separating subspace of the dual $ Y' $ of a Banach $ Y. $ We establish the relation between the boundedness, the (weak) compactness of the weighted composition operators $ W_{ψ,φ}: f\mapsto ψ\cdot(f \circ φ) $ on $ E $ and $ \widetilde{W}_{ψ,φ}: g\mapsto ψ\cdot(g \circ φ) $ on $ WE(Y)$ via some ch…
We show that a composition operator induced by an analytic self-map of the unit disc in the complex plane is weakly compact on the space BMOA precisely when the operator is compact on BMOA. As a crucial step we simplify the compactness criterion due to Smith for composition operators on BMOA and show that his condition on the Nevanlinna counting function alone characterizes compactness. In addition, other equivalent compactness criteria are established for composition operators on both BMOA and its subspace VMOA.
Given a holomorphic self-map $φ$ of $\D$ (the open unit disc in $\mathbb{C}$), the composition operator $C_φ f = f \circ φ$, $f \in H^2(\mathbb{\D})$, defines a bounded linear operator on the Hardy space $H^2(\mathbb{\D})$. The model spaces are the backward shift-invariant closed subspaces of $H^2(\mathbb{\D})$, which are canonically associated with inner functions. In this paper, we study model spaces that are invariant under composition operators. Emphasis is put on finite-dimensional model spaces, affine transformations, and linear fractional transformations.
We characterize the convex-cyclic weighted composition operators $W_{(u,ψ)}$ and their adjoints on the Fock space in terms of the derivative powers of $ ψ$ and the location of the eigenvalues of the operators on the complex plane. Such a description is also equivalent to identifying the operators or their adjoints for which their invariant closed convex sets are all invariant subspaces. We further show that the space supports no supercyclic weighted composition operators with respect to the pointwise convergence topology and hence with the weak and strong topologies and answers a question rais…
