Annals of Functional Analysis最新文献

In this article, the authors first introduce a class of Orlicz-amalgam spaces, which defined on a probabilistic setting. Based on these Orlicz-amalgam spaces, the authors introduce a new kind of Hardy type spaces, namely martingale Hardy–Orlicz-amalgam spaces, which generalize the martingale Hardy-amalgam spaces very recently studied by Bansah and Sehba. Their characterizations via the atomic decompositions are also obtained. As applications of these characterizations, the authors construct the dual theorems in the new framework. Furthermore, the authors also present the boundedness of fractional integral operators (I_alpha ) on martingale Hardy–Orlicz-amalgam spaces.

Abstract

Let ({mathcal {A}}) be a unital (C^*) -algebra with unit (1_{{mathcal {A}}}) and let (ain {mathcal {A}}) be a positive and invertible element. Suppose that ({mathcal {S}}({mathcal {A}})) is the set of all states on (mathcal {{mathcal {A}}}) and let $$begin{aligned} {mathcal {S}}_a ({mathcal {A}})=left{ dfrac{f}{f(a)} , : , f in {mathcal {S}}({mathcal {A}}), , f(a)ne 0right} . end{aligned}$$ The norm ( Vert xVert _a ) for every ( x in {mathcal {A}} ) is defined by $$begin{aligned} Vert xVert _a = sup _{varphi in {mathcal {S}}_a ({mathcal {A}}) } sqrt{varphi (x^* ax)}. end{aligned}$$ In this paper, we aim to investigate the notion of Birkhoff–James orthogonality with respect to the norm (Vert cdot Vert _a) in ({mathcal {A}},) namely a-Birkhoff–James orthogonality. The characterization of a-Birkhoff–James orthogonality in ({mathcal {A}}) by means of the elements of generalized state space ({mathcal {S}}_a({mathcal {A}})) is provided. As an application, a characterization for the best approximation to elements of ({mathcal {A}}) in a subspace ({mathcal {B}}) with respect to (Vert cdot Vert _a) is obtained. Moreover, a formula for the distance of an element of ({mathcal {A}}) to the subspace ({mathcal {B}}={mathbb {C}}1_{{mathcal {A}}}) is given. We also study the strong version of a-Birkhoff–James orthogonality in ( {mathcal {A}} .) The classes of (C^*) -algebras in which these two types orthogonality relationships coincide are described. In particular, we prove that the condition of the equivalence between the strong a-Birkhoff–James orthogonality and ({mathcal {A}}) -valued inner product orthogonality in ({mathcal {A}}) implies that the center of ({mathcal {A}}) is trivial. Finally, we show that if the (strong) a-Birkhoff–James orthogonality is right-additive (left-additive) in ({mathcal {A}},) then the center of ({mathcal {A}}) is trivial.

In this paper, we consider several questions about the eigenvalues, the numerical ranges, and the invariant subspaces of the Toeplitz operator on the Bergman space over the bidisk and we obtain the corresponding results.

In this paper, the main aim is to consider the mapping properties of the maximal or nonlinear commutator for the fractional maximal operator with the symbols belong to the Lipschitz spaces on variable Lebesgue spaces in the context of stratified Lie groups, with the help of which some new characterizations to the Lipschitz spaces and nonnegative Lipschitz functions are obtained in the stratified groups context. Meanwhile, some equivalent relations between the Lipschitz norm and the variable Lebesgue norm are also given.

We study the Bloch and the little Bloch spaces of harmonic functions on the real hyperbolic ball. We show that the Bergman projections from (L^infty ({mathbb {B}})) to ({mathcal {B}}), and from (C_0({mathbb {B}})) to ({mathcal {B}}_0) are onto. We verify that the dual space of the hyperbolic harmonic Bergman space ({mathcal {B}}^1_alpha ) is ({mathcal {B}}) and its predual is ({mathcal {B}}_0). Finally, we obtain atomic decompositions of Bloch functions as series of Bergman reproducing kernels.

We consider self-adjoint Dirac-type systems with rectangular matrix potentials on the interval [0, b), where (0<ble infty .) We present a new proof of the local Borg–Marchenko uniqueness theorem. The high-energy asymptotics of the Weyl–Titchmarsh functions and the local Borg–Marchenko uniqueness theorem are derived for locally smooth potentials at the right endpoint.

Applying an invariant measure on phase space, we study the Koopman representation of a group of symplectomorphisms in an infinite-dimensional Hilbert space equipped with a translation-invariant symplectic form. The phase space is equipped with a finitely additive measure, invariant under the group of symplectomorphisms generated by Liouville-integrable Hamiltonian systems. We construct an invariant measure of Lebesgue type by applying a special countable product of Lebesgue measures on real lines. An invariant measure of Banach type is constructed by applying a countable product of Banach measures (defined by the Banach limit) on real lines. One of the advantages of an invariant measure of Banach type compared to an invariant measure of Lebesgue type is finiteness of the values of this measure in the entire space. The introduced invariant measures help us to describe both the strong continuity subspaces of the Koopman unitary representation of an infinite-dimensional Hamiltonian flow and the spectral properties of the constraint generator of the unitary representation on the invariant strong continuity subspace.

Let E, F be Banach lattices, where E has the disjoint Riesz decomposition property. For a lattice homomorphism (T:Erightarrow F) and a bounded subset A of E, we establish a necessary and sufficient condition under which TA is b-order bounded. Based on this, we study the b-order boundedness of subsets of E and obtain several characterizations of AM-spaces. Furthermore, we introduce and investigate a novel type of operators referred to as M-serially summing operator. The connections of this category of operators with classical notions of operators, such as majorizing operators, preregular operators and serially summing operators, are considered.

In this paper, we initially introduce the concept of disjoint subspace-hypercyclic operators and illustrate that disjoint subspace-hypercyclic operators differ from disjoint hypercyclic operators. Furthermore, we obtain two different criteria for disjoint subspace-hypercyclic operators. Finally, we discover an equivalent condition regarding the bilateral forward weighted shift operators’ disjoint subspace-transitivity on (c_{0}(mathbb {Z})) or (l^{p}(mathbb {Z})) in a certain special case.