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

In this article we obtain, for 1 ? p ? ∞, a characterization of the invariant subspaces of spaces of vector-valued L^p functions defined on the unit circle—i.e., of those subspaces invariant under multiplication by e^ix. This result is then applied to extend, to the corresponding Hardy classes of vector-valued functions, the known characterizations of the extreme points of the unit ball in the scalar Hardy classes H¹ and H^∞. Finally, it is shown that the characterization of the closure of the set of extreme points of the unit ball in H¹ changes significantly when we pass from the scalar to the vector case.     

Operator-valued martingale transforms in rearrangement invariant spaces and applications

In this paper the operator-valued martingale transform inequalities in rearrangement invariant function spaces are proved. Some well-known results are generalized and unified. Applications are given to classical operators such as the maximal operator and the p-variation operator of vector-valued martingales, then we can very easily obtain some new vector-valued martingale inequalities in rearrangement invariant function spaces. These inequalities are closely related to both the geometrical properties of the underlying Banach spaces and the Boyd indices of the rearrangement invariant function spaces. Finally we give an equivalent characterization of UMD Banach lattices, and also prove the Fefferman-Stein theorem in the rearrangement invariant function spaces setting.     

Absolute continuity of vector-valued self-affine measures

This paper is devoted to the absolute continuity of (scalar-valued or vector-valued) self-affine measures and their properties on the boundary of an invariant set. We first extend the definition of WSC to self-affine IFS, and then obtain a necessary and sufficient condition for the vector-valued self-affine measures to be absolutely continuous with respect to the Lebesgue measure. In addition, we prove that, for any IFS and any invariant open set V, the corresponding (scalar-valued or vector-valued) invariant measure is supported either in V or in ∂V.     

Invariance of a Shift-Invariant Space in Several Variables

In this article we study invariance properties of shift-invariant spaces in higher dimensions. We state and prove several necessary and sufficient conditions for a shift-invariant space to be invariant under a given closed subgroup of \mathbb R^d{\mathbb {R}^d}, and prove the existence of shift-invariant spaces that are exactly invariant for each given subgroup. As an application we relate the extra invariance to the size of support of the Fourier transform of the generators of the shift-invariant space. This work extends recent results obtained for the case of one variable to several variables.     

Duality and operator algebras: automatic weak* continuity and applications

We investigate some subtle and interesting phenomena in the duality theory of operator spaces and operator algebras, and give several applications of the surprising fact that certain maps are always weak^*-continuous on dual spaces. In particular, if X is a subspace of a C^*-algebra A, and if a∈A satisfies aX⊂X, then we show that the function x?ax on X is automatically weak^* continuous if either (a) X is a dual operator space, or (b) a^*X⊂X and X is a dual Banach space. These results hinge on a generalization to Banach modules of Tomiyama's famous theorem on contractive projections onto a C^*-subalgebra. Applications include a new characterization of the σ-weakly closed (possibly nonunital and nonselfadjoint) operator algebras, and a generalization of the theory of W^*-modules to the framework of modules over such algebras. We also give a Banach module characterization of σ-weakly closed spaces of operators which are invariant under the action of a von Neumann algebra.     

Stable Manifolds and Nonuniform Hyperbolicity: Optimal Estimates

We establish the existence of invariant stable manifolds for C ¹ perturbations of a nonuniform exponential dichotomy with an arbitrary nonuniform part. We consider the general case of sequences of maps, which corresponds to a nonautonomous dynamics with discrete time. We also obtain optimal estimates for the decay of trajectories along the stable manifolds. The optimal C ¹ smoothness of the invariant manifolds is obtained using an invariant family of cones.     

Correlation Inequalities and Applications to Vector-Valued Gaussian Random Variables and Fractional Brownian Motion

In this paper we extend certain correlation inequalities for vector-valued Gaussian random variables due to Kolmogorov and Rozanov. The inequalities are applied to sequences of Gaussian random variables and Gaussian processes. For sequences of Gaussian random variables satisfying a correlation assumption, we prove a Borel-Cantelli lemma, maximal inequalities and several laws of large numbers. This extends results of Be?ka and Ciesielski and of Hytönen and the author. In the second part of the paper we consider a certain class of vector-valued Gaussian processes which are α-Hölder continuous in p-th moment. For these processes we obtain Besov regularity of the paths of order α. We also obtain estimates for the moments in the Besov norm. In particular, the results are applied to vector-valued fractional Brownian motions. These results extend earlier work of Ciesielski, Kerkyacharian and Roynette and of Hytönen and the author.     

Lushness, numerical index one and duality

We construct an example of a Banach space which is not lush, but whose dual space is lush. This example shows that lushness is not equivalent to numerical index one. A characterization of lushness for some quotient spaces of L₁(μ) spaces and new results on C-rich subspaces of (scalar- or vector-valued) C(K) spaces are also presented.     

A generalization of the cellular indecomposable property via fiber dimension

The cellular indecomposable property, introduced by Olin and Thomson in 1984 [11], is well known for the Dirichlet space, but it fails trivially for the vector-valued case. The purpose of this paper is to use the fiber dimension to reformulate the property such that it naturally extends the scalar-valued case, yet fix the vector-valued case in a meaningful way. Using the new formulation, we are able to generalize several previous results to the vector-valued setting. In particular, we extend a theorem of Bourdon relating the cellular indecomposable property and the codimension-one property to codimension-N. Several of our results appear to be new even for the Hardy space over the unit disc.     

Algebras of bounded functions invariant under the restricted backward shift

Let T be the unit circle in the complex plane and let A be a vector space of bounded Lebesgue measurable functions on T. A is said to be invariant under the restricted backward shift if, whenever ? is in A and the 0-th Fourier coefficient of ? vanishes, then $\text{e}{}^{\mathrm{?i\theta }}\text{?(e}{}^{\mathrm{i\theta }}\text{)}$ is also in A. The theorems of this paper provide a characterization of the uniformly closed subalgebras of C(T) which contain the constants and which are invariant under the restricted backward shift and, a similar characterization of the weak-¹ closed subalgebras of L^∞(T, dθ) which contain the constants and which are invariant under the restricted backward shift.     

Malliavin calculus and decoupling inequalities in Banach spaces

We develop a theory of Malliavin calculus for Banach space-valued random variables. Using radonifying operators instead of symmetric tensor products we extend the Wiener-Itô isometry to Banach spaces. In the white noise case we obtain two sided Lp-estimates for multiple stochastic integrals in arbitrary Banach spaces. It is shown that the Malliavin derivative is bounded on vector-valued Wiener-Itô chaoses. Our main tools are decoupling inequalities for vector-valued random variables. In the opposite direction we use Meyer's inequalities to give a new proof of a decoupling result for Gaussian chaoses in UMD Banach spaces.     

Tight frames of compactly supported multivariate multi-wavelets

This paper is devoted to the study and construction of compactly supported tight frames of multivariate multi-wavelets. In particular, a necessary condition for their existence is derived to provide some useful guide for constructing such MRA tight frames, by reducing the factorization task of the associated polyphase matrix-valued Laurent polynomial to that of certain scalar-valued non-negative ones. We illustrate our construction method with examples of both multivariate scalar- and vector-valued subdivision schemes. Since our constructions for C¹ and C² piecewise cubic schemes are quite involved, we also include the corresponding Matlab code in the Appendix.     

Ordered K-groups associated to substitutional dynamics

The Matsumoto K₀-group is an interesting invariant of flow equivalence for symbolic dynamical systems. Because of its origin as the K-theory of a certain C^∗-algebra, which is also a flow invariant, this group comes equipped with a flow invariant order structure. We emphasize this order structure and demonstrate how methods from operator algebra and symbolic dynamics combine to allow a computation of it in certain cases, including Sturmian and primitive substitutional shifts. In the latter case we show by example that the ordered group is a strictly finer invariant than the group itself.     

A study on compactly supported orthogonal vector-valued wavelets and wavelet packets

In this paper, vector-valued multiresolution analysis and orthogonal vector-valued wavelets are introduced. The definition for orthogonal vector-valued wavelet packets is proposed. A necessary and sufficient condition on the existence of orthogonal vector-valued wavelets is derived by means of paraunitary vector filter bank theory. An algorithm for constructing a class of compactly supported orthogonal vector-valued wavelets is presented. The properties of the vector-valued wavelet packets are investigated by using operator theory and algebra theory. In particular, it is shown how to construct various orthonormal bases of L²(R, C^s) from the orthogonal vector-valued wavelet packets.     

Applications of the Connes Spectrum to C1-dynamical systems,III

For a C¹-dynamical system (A, G, α) we show that the crossed product C¹-algebra is induced from a simple C¹-algebra equipped with an action of the Connes Spectrum, provided that A is G-simple and all isotropy subgroups of G under the action on the primitive ideal space of A are discrete. We then study the Borchers Spectrum of α and characterize its annihilator in G as the group of locally derivable automorphisms, under the assumption that the Arveson Spectrum of α is compact modulo the Borchers Spectrum. Finally a properly outer automorphism α is characterized by a series of equivalent conditions, one of which says that α is not close to the inner automorphisms on any ideal, another that α is not universally weakly inner on any ideal, and a third that the Borchers Spectrum of α on any invariant hereditary C¹-subalgebra is non-zero. This characterization leads to the conclusion that α is aperiodic (i.e., every non-zero power is properly outer) precisely when the Connes Spectrum of α is the full circle group.     

A characteristic operator function for the class of n-hypercontractions

We consider a class of bounded linear operators on Hilbert space called n-hypercontractions which relates naturally to adjoint shift operators on certain vector-valued standard weighted Bergman spaces on the unit disc. In the context of n-hypercontractions in the class C0⋅ we introduce a counterpart to the so-called characteristic operator function for a contraction operator. This generalized characteristic operator function Wn,T is an operator-valued analytic function in the unit disc whose values are operators between two Hilbert spaces of defect type. Using an operator-valued function of the form Wn,T, we parametrize the wandering subspace for a general shift invariant subspace of the corresponding vector-valued standard weighted Bergman space. The operator-valued analytic function Wn,T is shown to act as a contractive multiplier from the Hardy space into the associated standard weighted Bergman space.     

Quadratic Lyapunov functions and nonuniform exponential dichotomies

For nonautonomous linear equations x^′=A(t)x, we show how to characterize completely nonuniform exponential dichotomies using quadratic Lyapunov functions. The characterization can be expressed in terms of inequalities between matrices. In particular, we obtain converse theorems, by constructing explicitly quadratic Lyapunov functions for each nonuniform exponential dichotomy. We note that the nonuniform exponential dichotomies include as a very special case (uniform) exponential dichotomies. In particular, we recover in a very simple manner a complete characterization of uniform exponential dichotomies in terms of quadratic Lyapunov functions. We emphasize that our approach is new even in the uniform case.Furthermore, we show that the instability of a nonuniform exponential dichotomy persists under sufficiently small perturbations. The proof uses quadratic Lyapunov functions, and in particular avoids the use of invariant unstable manifolds which, to the best of our knowledge, are not known to exist in this general setting.     

Quantum invariant families of matrices in free probability

We consider (self-adjoint) families of infinite matrices of noncommutative random variables such that the joint distribution of their entries is invariant under conjugation by a free quantum group. For the free orthogonal and hyperoctahedral groups, we obtain complete characterizations of the invariant families in terms of an operator-valued R-cyclicity condition. This is a surprising contrast with the Aldous-Hoover characterization of jointly exchangeable arrays.     

A characterization of the minimal invariant sets of Alspach’s mapping

In 1981, Dale Alspach modified the baker’s transform to produce the first example of a nonexpansive mapping T on a weakly compact convex subset C of a Banach space that is fixed point free. By Zorn’s lemma, there exist minimal weakly compact, convex subsets of C which are invariant under T and are fixed point free.In this paper we produce an explicit formula for the nth power of T, Tⁿ, and prove that the sequence (Tⁿf)_n∈N converges weakly to , for all f∈C. From this we derive a characterization of the minimal invariant sets of T.