Let T:=[T_1,..., T_n] be an n-tuple of operators on a Hilbert space such that T is a completely non-coisometric row contraction. We establish the existence of a "one-to-one" correspondence between the joint invariant subspaces under T_1,..., T_n, and the regular factorizations of the characteristic function associated with T. In particular, we prove that there is a non-trivial joint invariant subspace under the operators T_1,..., T_n, if and only if there is a non-trivial regular factorization of the characteristic function. We also provide a functional model for the joint invariant subspaces in terms of the regular factorizations of the characteristic function, and prove the existence of joint invariant subspaces for certain classes of n-tuples of operators.
We obtain criterions for joint similarity of n-tuples of operators to Cuntz row isometries. In particular, we prove that a completely non-coisometric row contraction T is jointly similar to a Cuntz row isometry if and only if the characteristic function of T is an invertible multi-analytic operator.
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.
We obtain a Shimorin-Wold-type decomposition for a doubly commuting covariant representation of a product system of $C^*$-correspondences. This extends a recent Wold-type decomposition by Jeu and Pinto for a $q$-doubly commuting isometries. Application to the wandering subspaces of doubly commuting induced representations is explored, and a version of Mandrekar's Beurling type theorem is obtained to study doubly commuting invariant subspaces using Fock space approach due to Popescu.
We define the (convex) joint numerical range for an infinite family of compact operators in a Hilbert space H. We use this set to determine whether a self-adjoint compact operator A with {||A||, -||A||} in its spectrum is minimal respect to the set of diagonals in a fixed basis E of H in the operator norm, that is ||A|| <= ||A+D||, for all diagonal D. We also describe the moment set m_S = conv{ |v|^2 : v in S and ||v|| = 1 } of a subspace S of H in terms of joint numerical ranges and obtain equivalences between the intersection of moments of two subspaces and of its two related joint numerical…
In this paper, motivated by the Berger, Coburn and Lebow and Bercovici, Douglas and Foias theory for tuples of commuting isometries, we study analytic representations and joint invariant subspaces of a class of commuting $n$-isometries and prove that the $C^*$-algebra generated by the $n$-shift restricted to an invariant subspace of finite codimension in $H^2(\mathbb{D}^n)$ is unitarily equivalent to the $C^*$-algebra generated by the $n$-shift on $H^2(\mathbb{D}^n)$.
We give a complete characterization of invariant subspaces for $(M_{z_1}, \ldots, M_{z_n})$ on the Hardy space $H^2(\mathbb{D}^n)$ over the unit polydisc $\mathbb{D}^n$ in $\mathbb{C}^n$, $n >1$. In particular, this yields a complete set of unitary invariants for invariant subspaces for $(M_{z_1}, \ldots, M_{z_n})$ on $H^2(\mathbb{D}^n)$, $n > 1$. As a consequence, we classify a large class of $n$-tuples, $n > 1$, of commuting isometries. All of our results hold for vector-valued Hardy spaces over $\mathbb{D}^n$, $n > 1$. Our invariant subspace theorem solves the well-known open problem on cha…
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.
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.
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.
Invariant subspaces for finite index shifts in Hardy spaces
University of Rome Tor Vergata（罗马托尔维加塔大学）；Universidad Complutense de Madrid（马德里康普顿斯大学）；Instituto de Ciencias Matemáticas ICMAT(CSIC-UAM-UC3M-UCM)（数学科学研究所 ICMAT）
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.
A classical result of Norbert Wiener characterises doubly shift-invariant subspaces for square integrable functions on the unit circle with respect to a finite positive Borel measure $μ$, as being the ranges of the multiplication maps corresponding to the characteristic functions of $μ$-measurable subsets of the unit circle. An analogue of this result is given for the Besicovitch Hilbert space of `square integrable almost periodic functions'.
This paper focuses on representations of contractively embedded invariant subspaces in several variables. We present a version of the de Branges theorem for $n$-tuples of multiplication operators by the coordinate functions on analytic reproducing kernel Hilbert spaces over the unit ball $\mathbb{B}^n$ and the Hardy space over the unit polydics $\mathbb{D}^n$ in $\mathbb{C}^n$.
This paper is a follow-up contribution to our work [20] where we discussed some invariant subspace results for contractions on Hilbert spaces. Here we extend the results of [20] to the context of n-tuples of bounded linear operators on Hilbert spaces. Let T = (T_1, \ldots, T_n) be a pure commuting co-spherically contractive n-tuple of operators on a Hilbert space \mathcal{H} and \mathcal{S} be a non-trivial closed subspace of \mathcal{H}. One of our main results states that: \mathcal{S} is a joint T-invariant subspace if and only if there exists a partially isometric operator Π\in \mathcal{B}(…
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}.
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.
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.
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.
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 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…
