Dan Timotin
Using the tools of Sz.-Nagy--Foias theory of contractions, we describe in detail the invariant subspaces of the operator $ S\oplus S^* $, where $ S $ is the unilateral shift on a Hilbert space. This answers a question of Câmara and Ross.
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In this paper, we show that if the direct sum of two operators is subspace-hypercyclic (satisfies subspace hypercyclic criterion), then both operators are subspace-hypercyclic (satisfy subspace hypercyclic criterion). Moreover, if an operator $T$ satisfies subspace-hypercyclic criterion, then so $T\oplus T$ does. Also, we obtain that under certain conditions, if $T\oplus T$ is hypercyclic then $T$ satisfies subspace-hypercyclic criterion and, the subspace-hypercyclic operators satisfy subspace-hypercyclic criterion which gives the "subspace-hypercyclic" analogue of Theorem 2.3. (in Hereditaril…
Invariant Subspaces and the $C_{00}$-Property of $3$-Brownian Shifts
本文研究3布朗移位在特定不变子空间上的单位ary等价性及$C_{00}$-性质，基于布朗扩展理论，探讨其渐进行为。
Invariant subspaces of the direct sum of forward and backward shifts on vector-valued Hardy spaces
研究向量值Hardy空间上前向与后向移位直和算子的不变子空间，建立其与双边移位一类不变子空间的一一对应，并将其表示为混合Toeplitz与Hankel算子的核或值域，推广了一维情形下的已有结果。
We prove that if $X$ is a subspace of $L_p$ $(2<p<\infty)$, then either $X$ embeds isomorphically into $\ell_p \oplus \ell_2$ or $X$ contains a subspace $Y,$ which is isomorphic to $\ell_p(\ell_2)$. We also give an intrinsic characterization of when $X$ embeds into $\ell_p \oplus \ell_2$ in terms of weakly null trees in $X$ or, equivalently, in terms of the "infinite asymptotic game" played in $X$. This solves problems concerning small subspaces of $L_p$ originating in the 1970's. The techniques used were developed over several decades, the most recent being that of weakly null trees developed…
We present several operator and norm inequalities for Hilbert space operators. In particular, we prove that if $A_{1},A_{2},...,A_{n}\in {\mathbb B}({\mathscr H})$, then \[|||A_{1}A_{2}^{*}+A_{2}A_{3}^{*}+...+A_{n}A_{1}^{*}|||\leq|||\sum_{i=1}^{n}A_{i}A_{i}^{*}|||,\] for all unitarily invariant norms. We also show that if $A_{1},A_{2},A_{3},A_{4}$ are projections in ${\mathbb B}({\mathscr H})$, then &&|||(\sum_{i=1}^{4}(-1)^{i+1}A_{i})\oplus0\oplus0\oplus0|||&\leq&|||(A_{1}+|A_{3}A_{1}|)\oplus (A_{2}+|A_{4}A_{2}|)\oplus(A_{3}+|A_{1}A_{3}|)\oplus(A_{4}+|A_{2}A_{4}|)||| for any unitarily invaria…
We consider the problem of classification of invariant subspaces for the class of uniform Jordan operators. We show that given two invariant subspaces $M_1$ and $M_2$ of a uniform Jordan operator $T=S(θ)\oplus S(θ)\oplus \ldots$, the subspace $M_2$ belongs to the quasiaffine orbit of $M_1$ if and only if the restrictions $T|M_1$ and $T|M_2$ are quasisimilar and the compression $T_{M_2^\perp}$ can be injected in the compression $T_{M_1^\perp}$. Our result refines previous work on the subject by Bercovici and Smotzer.
We present a general method for constructing operators without non-trivial invariant closed subsets on a large class of non-reflexive Banach spaces. In particular, our approach unifies and generalizes several constructions due to Read of operators without non-trivial invariant subspaces on the spaces $\ell_{1}$, $c_{0}$ or $\oplus_{\ell_{2}}J$, and without non-trivial invariant subsets on $\ell_{1}$. We also investigate how far our methods can be extended to the Hilbertian setting, and construct an operator on a quasireflexive dual Banach space which has no non-trivial $w^{*}$-closed invariant…
We prove the existence of the invariant subspaces of some operators in a real Banach space. For example, linear isometries have invariant subspaces
Spectra and invariant subspaces of compressed shifts on nearly invariant subspaces
本文研究压缩位移在近不变子空间上的谱性质和不变子空间结构，通过单位等价性与变换理论揭示S*不变性放松对谱结构的影响，拓展了经典模型空间理论的应用范围。
We prove that a quotient of subspace of $C_p\oplus_pR_p$ ($1\le p<2$) embeds completely isomorphically into a noncommutative $L_p$-space, where $C_p$ and $R_p$ are respectively the $p$-column and $p$-row Hilbertian operator spaces. We also represent $C_q$ and $R_q$ ($p<q\le2$) as quotients of subspaces of $C_p\oplus_pR_p$. Consequently, $C_q$ and $R_q$ embed completely isomorphically into a noncommutative $L_p(M)$. We further show that the underlying von Neumann algebra $M$ cannot be semifinite.
Refuting a Recent Proof of the Invariant Subspace Problem
本文指出Khalil等人关于不变子空间问题的证明存在错误，并予以反驳。
For graded Hilbert spaces $H$ and shift-like commuting tuples $T \in B(H)^n$, we show that each homogeneous joint invariant subspace $M$ of $T$ has finite index and is generated by its wandering subspace. Under suitable conditions on the grading $(H_k)_{k\geq 0}$ of $H$ the algebraic direct sum $\tilde{M} = \oplus_{k\geq 0} M \cap H_k$ becomes a finitely generated module over the polynomial ring $\mathbb C[z]$. We show that the wandering subspace $W_T(M)$ of $M$ is contained in $\tilde{M}$ and that each linear basis of $W_T(M)$ forms a minimal set of generators for the $\mathbb C[z]$-module $\…
We prove that if T is an operator on an infinite-dimensional Hilbert space whose spectrum and essential spectrum are both connected and whose Fredholm index is only 0 or 1, then the only nontrivial norm-stable invariant subspaces of T are the finite-dimensional ones. We also characterize norm-stable invariant subspaces of any weighted unilateral shift operator. We show that quasianalytic shift operators are points of norm continuity of the lattice of the invariant subspaces. We also provide a necessary condition for strongly stable invariant subspaces for certain operators.
In the almost periodic context, any $H_0^2-$space cannot be generated by one of its elements. Together with cocycle argument, this derives that there exist all kinds of invariant subspaces without single generator, from which we can answer some questions on invariant subspace theory.
On invariant subspaces of a linear operator
本文针对无界线性算子的不变子空间问题，分析已有定义的不足并给出新的定义。
We prove that the nearly invariant subspaces of a de Branges space which have no common zeros are precisely of the form an exponential function times a de Branges space.
Nearly invariant subspaces in the real Hardy space
本文研究实 Hardy 空间中 backward shift 算子的近不变子空间，探讨有限缺陷近不变子空间，并为 backward shift 算子的近似不变子空间提供特征描述。
In this paper we study subspaces which are invariant under squares and cubes (separately as well as jointly) of unicellular backward weighted shift operators on a separable Hilbert space. The finite-dimensional subspaces are characterized for all weights and the infinite-dimensional subspaces are characterized for two classes of weights.
Almost invariant subspaces of shift operators and products of Toeplitz and Hankel operators
本文通过Toeplitz与Hankel算子乘积的值域或核，给出了后向移位算子几乎不变子空间的更显式刻画，并证明其与前向移位算子的几乎不变子空间相同。
We relax assumptions for a dissipative operator in Krein space to possess a maximal non-negative invariant subspace. Our main result is a generalization of a well-known Pontrjagin-Krein-Langer-Azizov theorem. Then we investigate the semigroup properties of the restriction of the operator onto the invariant subspace.
In this note we extend D. Singh and A. A. W. Mehanna's invariant subspace theorem for $RH^1$ (the real Banach space of analytic functions in $H^1$ with real Taylor coefficients) to the simply invariant subspaces of $RL^1$ (the real Banach space of functions in $L^1$ with real Fourier coefficients).
By applying methods of Duhamel algebra and reproducing kernels, we prove that every linear bounded operator on the Hardy-Hilbert space H^{2}(D) has a nontrivial invariant subspace. This solves affirmatively the Invariant Subspace Problem in the Hilbert space.
Recent perspectives on the Invariant Subspace Problem
本文综述算子不变子空间问题近15年的研究新进展，涉及几乎不变子空间、通用算子、特定算子类及Banach空间框架下的相关新结果。
Following Beurling's theorem and a study of the topology of invariant subspaces by R. Douglas and C. Pearcy description of path connected components of invariant subspace lattice for shift of multiplicity one has been given by R.Yang. This paper generalizes result to arbitrary finite multiplicity. We show that there exists one to one correspondence between the invariant subspace lattice of shift of arbitrary finite multiplicity and the space of inner functions.
In this article, we briefly describe nearly $T^{-1}$ invariant subspaces with finite defect for a shift operator $T$ having finite multiplicity acting on a separable Hilbert space $\mathcal{H}$ as a generalization of nearly $T^{-1}$ invariant subspaces introduced by Liang and Partington in \cite{YP}. In other words we characterize nearly $T^{-1}$ invariant subspaces with finite defect in terms of backward shift invariant subspaces in vector-valued Hardy spaces by using Theorem 3.5 in \cite{CDP}. Furthermore, we also provide a concrete representation of the nearly $T_B^{-1}$ invariant subspaces…
In this article, we characterize nearly invariant subspaces of finite defect for the backward shift operator acting on the vector-valued Hardy space which is a vectorial generalization of a result of Chalendar-Gallardo-Partington (C-G-P). Using this characterization of nearly invariant subspace under the backward shift we completely describe the almost invariant subspaces for the shift and its adjoint acting on the vector valued Hardy space.
We construct a complete locally convex topological vector space $X$ of countable algebraic dimension and a continuous linear operator $T:X\to X$ such that $T$ has no non-trivial closed invariant subspaces.
Operators of multiplication by independent variables on the space of square summable functions over the torus and its Hardy subspace are considered. Invariant subspaces where the operators are compatible are described.
In this paper, we discuss to the nonuniform sampling problem in principal shift-invariant subspaces of mixed Lebesgue spaces. We proposed a fast reconstruction algorithm which allows to exactly reconstruct the functions in the principal shift-invariant subspaces as long as the sampling set $X=\{(x_{j},y_{k}):k,j\in \mathbb{J}\}$ is sufficiently dense. Our results improve the result in \cite{LiLiu}.
In this paper we first study the structure of the scalar and vector-valued nearly invariant subspaces with a finite defect. We then subsequently produce some fruitful applications of our new results. We produce a decomposition theorem for the vector-valued nearly invariant subspaces with a finite defect. More specifically, we show every vector-valued nearly invariant subspace with a finite defect can be written as the isometric image of a backwards shift invariant subspace. We also show that there is a link between the vector-valued nearly invariant subspaces and the scalar-valued nearly invar…
We construct a continuous linear operator acting on the space of smooth functions on the real line without non-trivial invariant subspaces. This is a first example of such an operator acting on a Fréchet space without a continuous norm. The construction is based on the ideas due to C. Read who constructed a continuous operator without non-trivial invariant subspaces on the Banach space $\ell_1$.
It is proved that a commutative algebra $A$ of operators in a reflexive real Banach space has an invariant subspace if each operator $T\in A$ satisfies the condition $$\|1- \varepsilon T^2\|_e \le 1 + o(\varepsilon) \text{ when } \varepsilon\searrow 0,$$ where $\|\cdot\|_e$ is the essential norm. This implies the existence of an invariant subspace for every commutative family of essentially selfadjoint operators in a real Hilbert space.
It is proved that a commutative algebra $A$ of operators on a reflexive real Banach space has an invariant subspace if each operator $T\in A$ satisfies the condition $$\|1- \varepsilon T^2\|_e \le 1 + o(\varepsilon) \text{ when } \varepsilon\searrow 0,$$ where $\|\cdot\|_e$ is the essential norm. This implies the existence of an invariant subspace for every commutative family of essentially selfadjoint operators on a real Hilbert space.
In this note, we describe the backward shift invariant subspaces for a large class of reproducing kernel Hilbert spaces. This class includes in particular de Branges-Rovnyak spaces (the non-extreme case) and the range space of co-analytic Toeplitz operators.
In this article we obtain some positive results about the existence of a common nontrivial invariant subspace for $N$-tuples of not necessarily commuting operators on Banach spaces with a Schauder basis. The concept of joint quasinilpotence plays a basic role. Our results complement recent work by Kosiek \cite{6} and Ptak \cite{8}.
We prove that every lattice homomorphism acting on a Banach space $\mathcal{X}$ with the lattice structure given by an unconditional basis has a non-trivial closed invariant subspace. In fact, it has a non-trivial closed invariant ideal, which is no longer true for every positive operator on such a space. Motivated by these later examples, we characterize tridiagonal positive operators without non-trivial closed invariant ideals on $\mathcal{X}$ extending to this context a result of Grivaux on the existence of non-trivial closed invariant subspaces for tridiagonal operators.
On the invariant subspace problem in Hilbert spaces
该研究针对 Hilbert 空间中的不变子空间问题，证明任意有界线性算子均存在闭的非平凡不变子空间。
We show that for any bounded operator $T$ acting on infinite dimensional, complex Banach space, and for any $\varepsilon>0$, there exists an operator $F$ of rank at most one and norm smaller than $\varepsilon$ such that $T+F$ has an invariant subspace of infinite dimension and codimension. A version of this result was proved in \cite{T19} under additional spectral conditions for $T$ or $T^*$. This solves in full generality the quantitative version of the invariant subspace problem for rank-one perturbations.
In this paper, we prove the Mohebi-Radjabalipour Conjecture under a little additional condition, and obtain a new invariant subspace theorem for subdecomposable operators. Our main results contain known results in this topic as special cases. Moreover, the condition in our theorem is really much weaker, much simpler and much more nature than that in the known results.
A regular generalized sampling theory in some structured T-invariant subspaces of a Hilbert space H, where T denotes a bounded invertible operator in H, is established in this paper. This is done by walking through the most important cases which generalize the usual sampling settings.
We define an affine structure on $\ltwor\oplus...\oplus\ltwor$ and, following some ideas developed in \cite{Dut1}, we construct a local trace function for this situation. This trace function is a complete invariant for a shift invariant subspace and it has a variety of properties which make it easily computable. The local trace is then used to give a characterization of super-wavelets and to analyze their multiplicity function, dimension function and spectral function. The "$n\times$" oversampling result of Chui and Shi \cite{CS} is refined to produce super-wavelets.
We show that a bounded quasinilpotent operator $T$ acting on an infinite dimensional Banach space has an invariant subspace if and only if there exists a rank one operator $F$ and a scalar $α\in\mathbb{C}$, $α\neq 0$, $α\neq 1$, such that $T+F$ and $T+αF$ are also quasinilpotent. We also prove that for any fixed rank-one operator $F$, almost all perturbations $T+αF$ have invariant subspaces of infinite dimension and codimension.
The use of unitary invariant subspaces of a Hilbert space $\mathcal{H}$ is nowadays a recognized fact in the treatment of sampling problems. Indeed, shift-invariant subspaces of $L^2(\mathbb{R})$ and also periodic extensions of finite signals are remarkable examples where this occurs. As a consequence, the availability of an abstract unitary sampling theory becomes a useful tool to handle these problems. In this paper we derive a sampling theory for tensor products of unitary invariant subspaces. This allows to merge the cases of finitely/infinitely generated unitary invariant subspaces former…
We describe necessary and sufficient conditions for a $J$-dissipative operator in a Krein space to have maximal semidefinite invariant subspaces. The semigroup properties of the restrictions of an operator to these subspaces are studied. Applications are given to the case when an operator admits matrix representation with respect to the canonical decomposition of the space. The main conditions are given in the terms of the interpolation theory of Banach spaces.
The theorem on the existence of maximal nonnegative invariant subspaces for a special class of dissipative operators in Hilbert space with indefinite inner product is proved in the paper. It is shown in addition that the spectra of the restrictions of these operators on the corresponding invariant subspaces lie in the closed upper half-plane. The obtained theorem is a generalization of well-known results of L.S.Pontrjagin, H.K.Langer, M.G.Krein and T.Ja.Azizov devoted to this subject.
We obtaine the full characterization of proper closed invariant subspaces of a generalized backward shift operator (Pommiez operator) in the Frechet space of all holomorphic functions on a simply connected domain $Ω$ of the complex plane, containing the origin. In the case when the function, which defines this operator, is not has zeros in $Ω$ all such subspaces are finite-dimensional. If additionally $Ω$ coincides with the complex plane, then the considered operator is unicellular. If this function has zeros in $Ω$, then the family of mentioned invariant subspaces splits into two classes: the…
It is well known that the structure of nontrival invariant subspaces for the shift operator on the Bergman space is extremely complicated and very little is known about their specific structures, and that a complete description of the structure seems unlikely. In this paper, we find that any invariant subspace $M(\neq \{0\})$ for the shift operator $M_z$ on the Bergman space $A^{^p}_α(\D)~(1\leq p< \infty,~~-1<α<\infty)$ contains a nonempty subset that lies in $A^{^p}_α(\D)\backslash H^{^p}(\D)$. To a certain extent, this result describes the specific structure of every nonzero invariant subsp…
In the setting of operators on Hilbert spaces, we prove that every quasinilpotent operator has a non-trivial closed invariant subspace if and only if every pair of idempotents with a quasinilpotent commutator has a non-trivial common closed invariant subspace. We also present a geometric characterization of invariant subspaces of idempotents and classify operators that are essentially idempotent.
For a shift operator $T$ with finite multiplicity acting on a separable infinite dimensional Hilbert space we represent its nearly $T^{-1}$ invariant subspaces in terms of invariant subspaces under the backward shift. Going further, given any finite Blaschke product $B$, we give a description of the nearly $T_{B}^{-1}$ invariant subspaces for the operator $T_B$ of multiplication by $B$ in a scale of Dirichlet-type spaces.
A Fréchet space $X$ satisfies the Hereditary Invariant Subspace (resp. Subset) Property if for every closed infinite-dimensional subspace $M$ in $X$, each continuous operator on $M$ possesses a non-trivial invariant subspace (resp. subset). In this paper, we exhibit a family of non-normable separable infinite-dimensional Fréchet spaces satisfying the Hereditary Invariant Subspace Property and we show that many non-normable Fréchet spaces do not satisfy this property. We also state sufficient conditions for the existence of a continuous operator without non-trivial invariant subset and deduce a…
It is known that the structure of invariant subspaces of the Hardy space $H^2(\mathbb D^n)$ on the polydisc $\mathbb{D}^n$ is very complicated; hence, we need good examples help us to understand the structure of invariant subspaces of $H^2(\mathbb D^n)$. In this paper, we define two types of invariant subspaces of $H^2(\mathbb D^n)$. Then, we give a characterization of these types invariant subspaces in view of the Beurling-Lax-Halmos Theorem. Unitary equivalence is also studied in this paper.
A complete characterization of nearly-invariant subspaces of finite defect for the backward shift operator acting on the Hardy space is provided in the spirit of Hitt and Sarason's theorem. As a corollary we describe the almost-invariant subspaces for the shift and its adjoint.
A Proof of the Invariant Subspace Conjecture for Separable Hilbert Spaces
该研究完成了可分希尔伯特空间下不变子空间猜想的证明，在泛函分析核心猜想领域取得了关键理论突破。
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…
We study unitary representations of groups in Krein spaces, irreducibility criteria and integral decompositions. Our main tool is the theory of Krein subspaces and their (reproducing) kernels and a variant of Choquet's theorem.
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.
We study the structure of the backward shift invariant and nearly invariant subspaces in weighted Fock-type spaces $\mathcal{F}_W^p$, whose weight $W$ is not necessarily radial. We show that in the spaces $\mathcal{F}_W^p$ which contain the polynomials as a dense subspace (in particular, in the radial case) all nontrivial backward shift invariant subspaces are of the form $\mathcal{P}_n$, i.e., finite dimensional subspaces consisting of polynomials of degree at most $n$. In general, the structure of the nearly invariant subspaces is more complicated. In the case of spaces of slow growth (up to…
The lattice of closed invariant subspaces of the Volterra operator acting on $L^2(0,1)$ was completely described by Sarason. On the other hand, he explicitly found the lattice of closed invariant subspaces of the shift plus Volterra operator on $L^2(0,1)$. Inspired by Sarason's results, we find the lattice of closed invariant subspaces of the shift plus complex Volterra operator acting on the Hardy space
For an operator $T$ of class $C_0$ with multiplicity two, we show that the quasisimilarity class of an invariant subspace $M$ is determined by the quasisimilarity classes of the restriction $T|M$ and of the compression $T_{M^\perp}$. We also provide a canonical form for the subspace $M$.
A reflexive hereditarily indecomposable Banach space $\mathfrak{X}_{_{^\text{ISP}}}$ is presented, such that for every $Y$ infinite dimensional closed subspace of $\mathfrak{X}_{_{^\text{ISP}}}$ and every bounded linear operator $T:Y\rightarrow Y$, the operator $T$ admits a non-trivial closed invariant subspace.
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