Mathematics

Let X be a complex topological vector space and L(X) the set of all continuous linear operators on X. In this paper, we extend the notion of the codiskcyclicity of a single operator T?�L(X) to a set of operators ??�L(X). We prove some results for codiskcyclic sets of operators and we establish a codiskcyclicity criterion. As an application, we study the codiskcyclicity of C 0 -semigroups of operators.

The Hilbert space effect algebra is a fundamental mathematical structure which is used to describe unsharp quantum measurements in Ludwig's formulation of quantum mechanics. Each effect represents a quantum (fuzzy) event. The relation of coexistence plays an important role in this theory, as it expresses when two quantum events can be measured together by applying a suitable apparatus. This paper's first goal is to answer a very natural question about this relation, namely, when two effects are coexistent with exactly the same effects? The other main aim is to describe all automorphisms of the effect algebra with respect to the relation of coexistence. In particular, we will see that they can differ quite a lot from usual standard automorphisms, which appear for instance in Ludwig's theorem. As a byproduct of our methods we also strengthen a theorem of Molnar.

Using the group G(1) of invertible elements and the maximal ideals m x of the commutative algebra C(X) of real-valued functions on a compact regular space X , we define a Borel action of the algebra on the measure space (X,μ) with μ a Radon measure. The zero sets Z(X) of the algebra C(X) is used to study the ergodicity of the G(1) -action via its action on the maximal ideals m x which defines an action groupoid G= m x ?�G(1) trivialized on X . The resulting measure groupoid (G,C) is used to define a proper action on the generalized space M(X) . The existence of slice at each point of M(X) present it as a cohomogeneity-one G -space. The dynamical system of the algebra C(X) is defined by the action of the measure groupoid (G,C)?M(X)?�M(X) .

We provide with criteria for a family of sequences of operators to share a frequently universal vector. These criteria are variants of the classical Frequent Hypercyclicity Criterion and of a recent criterion due to Grivaux, Matheron and Menet where periodic points play the central role. As an application, we obtain for any operator T in a specific class of operators acting on a separable Banach space, a necessary and sufficient condition on a subset Λ of the complex plane for the family { λ T : λ ∈ Λ } to have a common frequently hypercyclic vector. In passing, this permits us to easily exhibit frequent hypercyclic weighted shifts which do not possess common frequent hypercyclic vectors. We also provide with criteria for families of the recently introduced operators of C-type to share a common frequently hypercyclic vector. Further, we prove that the same problem of common α -frequent hypercyclicity may be vacuous, where the notion of α -frequent hypercyclicity extends that of frequent hypercyclicity replacing the natural density by more general weighted densities. Finally, it is already known that any operator satisfying the classical Frequent Universality Criterion is α -frequently universal for any sequence α satisfying a suitable condition. We complement this result by showing that for any such operator, there exists a vector x which is α -frequently universal for T , with respect to all such α .

A characteristic function is a special operator-valued analytic function defined on the open unit ball of C n associated with an n -tuple of commuting row contraction on some Hilbert space. In this paper, we continue our study of the representations of n -tuples of commuting row contractions on Hilbert spaces, which have polynomial characteristic functions. Gleason's problem plays an important role in the representations of row contractions. We further complement the representations of our row contractions by proving theorems concerning factorizations of characteristic functions. We also emphasize the importance and the role of the noncommutative operator theory and noncommutative varieties to the classification problem of polynomial characteristic functions.

Comparing with univariate framelets, the main challenge involved in studying multivariate framelets is that we have to deal with the highly non-trivial problem of factorizing multivariate polynomial matrices. As a consequence, multivariate framelets are much less studied than univariate framelets in the literature. Among existing works on multivariate framelets, multivariate multiframelets are much less considered comparing with the exitensively studied scalar framelets. Hence multiframelets are far from being well understood. In this paper, we focus on multivariate dual multiframelets (or dual vector framelets) obtained through the popular oblique extension principle (OEP), which are called OEP-based dual multiframelets. We will show that from any given pair of compactly supported refinable vector functions, one can always construct an OEP-based dual mltiframelet, such that its generators have the highest possible order of vanishing moments. Moreover, the associated discrete framelet transform is compact and sparse.

Let X be a ball Banach function space on R n . Let Ω be a Lipschitz function on the unit sphere of R n ,which is homogeneous of degree zero and has mean value zero, and let T Ω be the convolutional singular integral operator with kernel Ω(??/|??| n . In this article, under the assumption that the Hardy--Littlewood maximal operator M is bounded on both X and its associated space, the authors prove that the commutator [b, T Ω ] is compact on X if and only if b?�CMO( R n ) . To achieve this, the authors mainly employ three key tools: some elaborate estimates, given in this article, on the norm in X of the commutators and the characteristic functions of some measurable subset,which are implied by the assumed boundedness of M on X and its associated space as well as the geometry of R n ; the complete John--Nirenberg inequality in X obtained by Y. Sawano et al.; the generalized Fréchet--Kolmogorov theorem on X also established in this article. All these results have a wide range of applications. Particularly, even when X:= L p(?? ( R n ) (the variable Lebesgue space), X:= L p ??( R n ) (the mixed-norm Lebesgue space), X:= L Φ ( R n ) (the Orlicz space), and X:=( E q Φ ) t ( R n ) (the Orlicz-slice space or the generalized amalgam space), all these results are new.

We specify the structure of completely positive operators and quantum Markov semigroup generators that are symmetric with respect to a family of inner products, also providing new information on the order strucure an extreme points in some previously studied cases.

We show that well-established methods from the theory of Banach modules and time-frequency analysis allow to derive completeness results for the collection of shifted and dilated version of a given (test) function in a quite general setting. While the basic ideas show strong similarity to the arguments used in a recent paper by V.~Katsnelson we extend his results in several directions, both relaxing the assumptions and widening the range of applications. There is no need for the Banach spaces considered to be embedded into ( L 2 (R),||⋅| | 2 ) , nor is the Hilbert space structure relevant. We choose to present the results in the setting of the Euclidean spaces, because then the Schwartz space S ′ ( R d ) ( d≥1 ) of tempered distributions provides a well-established environment for mathematical analysis. We also establish connections to modulation spaces and Shubin classes ( Q s ( R d ),||⋅| | Q s ) , showing that they are special cases of Katsnelson's setting (only) for s≥0 .

We prove that any non-complete orthonormal system in a Hilbert space can be transformed into a basis by small perturbations.