Shift-type invariant subspaces of contractions

Invariant subspaces and extremum problems in spaces of vector-valued functions

Invariant subspace problem for classical spaces of functions

Let S be the shift operator on the Hardy space H2 and let S* be its adjoint. A closed subspace \({\mathcal F}\) of H2 is said to be nearly S*-invariant if every element \({f\in\mathcal F}\) with f(0) = 0 satisfies \({S^*f\in\mathcal F}\). In particular, the kernels of Toeplitz operators are nearly S*-invariant subspaces. Hitt gave the description of these subspaces. They are of the form \({\mathcal F=g (H^2\ominus u H^2)}\) with \({g\in H^2}\) and u inner, u(0) = 0. A very particular fact is that the operator of multiplication by g acts as an isometry on \({H^2\ominus uH^2}\). Sarason obtained a characterization of the functions g which act isometrically on \({H^2\ominus uH^2}\). Hayashi obtained the link between the symbol \({\varphi}\) of a Toeplitz operator and the functions g and u to ensure that a given subspace \({\mathcal F=gK_u}\) is the kernel of \({T_\varphi}\). Chalendar, Chevrot and Partington studied the nearly S*-invariant subspaces for vector-valued functions. In this paper, we investigate the generalization of Sarason’s and Hayashi’s results in the vector-valued context.

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: If φ is a hyperbolic disc automorphism with fixed points at a and β (both necessarily on the unit circle), and C φ the composition operator it induces on H 2, then for every \( f \in \sqrt {(z - \alpha )(z - \beta )} \) H 2, the doubly C φ-cyclic subspace generated by f contains many independent eigenvectors; more precisely, the point spectrum of C φ’s restriction to that subspace intersects the unit circle in a set of positive measure. Moreover, this restriction of C φ is hypercyclic (some forward orbit is dense). Under the stronger restriction \( f \in \sqrt {(z - \alpha )(z - \beta )} \) H p for some p > 2, the point spectrum of the restricted operator contains an open annulus centered at the origin.KeywordsComposition operatorhyperbolic automorphismInvariant Subspace ProblemMathematics Subject Classification (2000)Primary 47B33Secondary 47A15

Operators that commute with a unilateral shift on an invariant subspace

Publication in the conference proceedings of EUSIPCO, Marrakech, Morocco, 2013

A commutative Banach algebra is called symmetric if, regarded as a function algebra on its maximal ideal space, it is closed under complex conjugation. Varopoulos, [5], proved the asymmetry of the tensor algebra , where T is the unit circle. It is the object of this paper to prove the asymmetry of the Schur multipliers of the space , where m is the Lebesgue measure. In the second part of the paper we study the Hankel multipliers of the space and give an application to it.1. The asymmetry of. Let C(T) denote the space of continuous functions on T and A(T) be the space of those functions in C(T) that have absolutely convergent Fourier series. Consider the mapping F: C(T) → C(T × T) defined by F(f)(x, y) = f (x + y).

Frames by Orbits of Two Operators that Commute

We prove that the existence of a solution operator for a convolution operator from the space of ultradifferntiable functions to the corresponding space of ultradistributions is equivalent to the existence of a continuous solution operator in the space of functions. Our results are in the spirit of a classical characterization of the surjectivity of convolution operators due to Hormander. The behaviour of a fixed convolution operator in different classes of ultradifferentiable functions of Beurling type concerning the existence of a continuous linear right inverse is also considered.

We characterize boundedness and compactness of pullback operators under holomorphic maps between Bargmann spaces of entire holomorphic functions with quadratic strictly plurisubharmonic exponential weights, extending the result of Carswell, MacCluer, and Schuster [Acta Sci. Math. (Szeged) 69 (2003), pp. 871–887], obtained in the case of the radial quadratic weight. We also show that the pullback operator between Bargmann spaces is compact precisely when it is of trace class, with sub-exponentially decaying singular values.

Invariant subspaces for polynomially bounded operators

On considere le produit Ψn de n matrices aleatoires M i.i.d. ayant les proprietes suivantes: la suite (Ψn)n est bornee et M possede un vecteur invariant deterministe (constant). On suppose que la probabilite pour que, sur le cercle unite, M possede une unique valeur propre simple en 1 soit non nulle. On montre que Ψn est la somme d’un processus fluctuant et d’un processus decroissant. Ce dernier tend vers zero presque surement et exponentiellement rapidement lorsque n tend vers l’infini. Le terme fluctuant converge en moyenne de Cesaro vers une limite caracterisee explicitement par le vecteur invariant deterministe et par les donnees spectrales associees a la valeur propre 1 de la matrice $\mathbb{E}[M]$. Aucune hypothese supplementaire n’est faite sur les matrices M; elles peuvent etre a valeurs complexes et pas necessairement inversibles. On applique les resultats generaux a deux classes de systemes dynamiques: les chaines de Markov inhomogenes avec matrices de transition aleatoires (matrices stochastiques), ainsi que les systemes quantiques avec interactions repetees et aleatoires. Dans les deux cas, on prouve des resultats d’ergodicite pour la dynamique et on determine les etats limites.

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. 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 use the method of minimal vectors to prove that certain classes of positive quasinilpotent operators on Banach lattices have invariant subspaces. We say that a collection of operators F on a Banach lattice X satisfies condition ( ) if there exists a closed ball B(x0,r) in X such that x0 > 0 and kx0k > r, and for every sequence (xn) in B(x0,r) (0,x0) there exists a subsequence (xni) and a sequence Ki 2 F such that Kixni converges to a non-zero vector. Let Q be a positive quasinilpotent operator on X, one-to-one, with dense range. Denote hQ) = {T > 0 : TQ 6 QT}. If either the set of all operators dominated by Q or the set of all contractions in hQ) satisfies ( ), then hQ) has a common invariant subspace. We also show that if Q is a one-to-one quasinilpotent interval preserving operator on C0(), then hQ) has a common invariant subspace. Lomonosov proved in (Lom73) that if T is not a multiple of the identity and com- mutes with a non-zero compact operator K, then T has a hyperinvariant subspace, that is, a proper closed nontrivial subspace invariant under every operator S in the commutant {T} 0 = {S 2 L(X) : ST = TS}. There has been numerous extensions and generalizations of the result of Lomonosov. In particular, Abramovich, Aliprantis, and Burkinshaw produced several generalizations of Lomonosov's theorem for Banach lattice setting (AAB93, AAB94, AAB98), see also (AA02). In these generalizations commutation relations are substituted by a super-commutation relation ST 6 TS or ST > TS and domination 0 6 K 6 T. They proved a series of results of the fol- lowing type: if S is related to a compact operator via a certain rather loose chain of super-commutations and dominations, then S has an invariant subspace. Ansari and Enflo (AE98) have recently introduced the so-called technique of minimal vectors in order to prove the existence of invariant subspaces for certain classes of operators on a Hilbert space. The method was later modified so that it could be used in arbitrary Banach spaces in (JKP03, And03, CPS04, Tr04). In particular, the method of minimal vectors allows to prove Lomonosov-type results where a compact operator is replaced with a family of operators that mimic a compact operator.

We develop an approach to describe invariant subspaces of the integration operator on various scales of weighted spaces of holomorphic functions on the unit disk and the complex plane. It allows us to solve the problem for wide classes of Bergman, Bloch, Dirichlet, and Fock spaces, while all previous known results concern spaces defined by some weights of a special form. In addition, we also show that an analogous method works as well for the differentiation operator on weighted spaces of entire functions.