Spaces Of Vector-valued Functions Research Articles

Well-posedness of second order degenerate integro-differential equations in vector-valued function spaces

We consider the well-posedness of the second order degenerate integrodifferential equations (P2): , (0 ≤ t ≤ 2π) with periodic boundary conditions u(0) = u(2π), (Mu′)(0) = (Mu′)(2π), in periodic Lebesgue-Bochner spaces Lp(𝕋, X), periodic Besov spaces (𝕋, X) and and periodic Triebel-Lizorkin spaces (𝕋, X), where A and M are closed linear operators on a Banach space X satisfying D(A) ⊂ D(M), a ∊ L1 (ℝ+) and α is a scalar number. Using known operator-valued Fourier multiplier theorems, we give necessary and sufficient conditions for the well-posedness of (P2) in the above three function spaces.

Weak forms of Banach–Stone theorem for C0(K,X) spaces via the αth derivatives of K

Let X be a Banach space and S be a locally compact Hausdorff space. By C0(S,X) we will stand the Banach space of all continuous X-valued functions on S endowed with the supremum norm.Suppose that C0(S,X) contains a copy of some C0(K) space with K infinite. Does it follow that the cardinality of the αth derivative of K is less than or equal to the αth derivative of S, for every ordinal number α? In general the answer is no, even when α=0.In the present paper we prove that the answer is yes whenever X contains no copy of c0 and α=0. Moreover, in the case where α>0 and the αth derivative of S is infinite, we show that the existence an isomorphism from C0(K) into C0(S,X) with distortion ‖T‖‖T−1‖ strictly less than 3 provides also a positive answer to this question.As a consequence, we improve a classical Cengiz theorem and a recent result on isomorphisms between spaces of vector-valued continuous functions by obtaining two weak forms of Banach–Stone theorem for C0(S,X) spaces via the αth derivatives of S.

Coherent states and Berezin quantization for non-scalar holomorphic representations

Let \(G\) be a quasi-Hermitian Lie group and let \(K\) be a maximal compactly embedded subgroup of \(G\). Let \(\pi \) be a unitary representation of \(G\) which is holomorphically induced from a unitary representation \(\rho \) of \(K\). Then \(\pi \) is usually realized in a Hilbert space of vector-valued holomorphic functions. Here we introduce a realization of \(\pi \) in a reproducing kernel Hilbert space of complex-valued functions and we compute the (scalar-valued) coherent states. We study the corresponding Berezin quantization map and, in particular, we show that this map is equivalent to that introduced by Cahen (Rend Istit Mat Univ Trieste 46:157–180, 2014).

Surjective isometries on the vector-valued differentiable function spaces

This paper investigates the surjective linear isometries between the differentiable function spaces C0n(Ω,E) and C0m(Σ,F) (where Ω,Σ are open subsets of Euclidean spaces and E,F are reflexive, strictly convex Banach spaces), and show that such isometries can be written as weighted composition operators.

On universal enveloping algebras in a topological setting

We study some embeddings of suitably topologized spaces of vector-valued smooth functions on topological groups, where smoothness is defined via differentiability along continuous one-parameter subgroups. As an application, we investigate the canonical co

Algebraic and topological properties of the group of isometries on classes of vector valued function spaces

We study algebraic and topological properties of the group of all surjective isometries on several spaces of vector valued analytic functions and vector valued $L^p$ spaces ($1 \leq p \leq \infty$). We also derive the form for the surjective linear isometries on the vector valued little Zygmund space.

Projections on some vector valued function spaces

We determine when the average of two isometries on a Banach space of analytic vector valued functions is a projection. We also characterize the generalized bi-circular projections on a Banach space of absolutely continuous vector valued functions.

Pointwise Multiplication on Vector-Valued Function Spaces with Power Weights

We investigate pointwise multipliers on vector-valued function spaces over \({\mathbb {R}}^d\), equipped with Muckenhoupt weights. The main result is that in the natural parameter range, the characteristic function of the half-space is a pointwise multiplier on Bessel-potential spaces with values in a UMD Banach space. This is proved for a class of power weights, including the unweighted case, and extends the classical result of Shamir and Strichartz. The multiplication estimate is based on the paraproduct technique and a randomized Littlewood–Paley decomposition. An analogous result is obtained for Besov and Triebel–Lizorkin spaces.

Multitask Classification Hypothesis Space With Improved Generalization Bounds.

This paper presents a pair of hypothesis spaces (HSs) of vector-valued functions intended to be used in the context of multitask classification. While both are parameterized on the elements of reproducing kernel Hilbert spaces and impose a feature mapping that is common to all tasks, one of them assumes this mapping as fixed, while the more general one learns the mapping via multiple kernel learning. For these new HSs, empirical Rademacher complexity-based generalization bounds are derived, and are shown to be tighter than the bound of a particular HS, which has appeared recently in the literature, leading to improved performance. As a matter of fact, the latter HS is shown to be a special case of ours. Based on an equivalence to Group-Lasso type HSs, the proposed HSs are utilized toward corresponding support vector machine-based formulations. Finally, experimental results on multitask learning problems underline the quality of the derived bounds and validate this paper's analysis.

Finite codimensional isometries on spaces of vector-valued continuous functions

Based on the vector-valued generalization of Holsztyński's theorem by M. Cambern, we provide a complete description of the linear isometries of C(X,E) into C(Y,F) whose range has finite codimension.

Composition operators on vector-valued analytic function spaces: a survey

We survey recent results about composition operators induced by analytic self-maps of the unit disk in the complex plane on various Banach spaces of analytic functions taking values in infinite-dimensional Banach spaces. We mostly concentrate on the research line into qualitative properties such as weak compactness, initiated by Liu, Saksman and Tylli (1998), and continued in several other papers. We discuss composition operators on strong, respectively weak, spaces of vector-valued analytic functions, as well as between weak and strong spaces. As concrete examples, we review more carefully and present some new observations in the cases of vector-valued Hardy and BMOA spaces, though the study of composition operators has been extended to a wide range of spaces of vector-valued analytic functions, including spaces defined on other domains. Several open problems are stated.

How Lipschitz Functions Characterize the Underlying Metric Spaces

AbstractLetXandYbe metric spaces andE,Fbe Banach spaces. Suppose that bothXandYare realcompact, or bothE, Fare realcompact. The zero set of a vector-valued function ƒ is denoted byz(ƒ). A linear bijection T between local or generalized Lipschitz vector-valued function spaces is said to preserve zero-set containments or nonvanishing functions ifrespectively. Every zero-set containment preserver, and every nonvanishing function preserver when dimE= dimF<+∞, is a weighted composition operator. We show that the mapτ:Y→Xis a locally (little) Lipschitz homeomorphism.

THE VECTOR-VALUED TENT SPACES AND

AbstractTent spaces of vector-valued functions were recently studied by Hytönen, van Neerven and Portal with an eye on applications to $H^{\infty }$-functional calculi. This paper extends their results to the endpoint cases $p=1$ and $p=\infty $ along the lines of earlier work by Harboure, Torrea and Viviani in the scalar-valued case. The main result of the paper is an atomic decomposition in the case $p=1$, which relies on a new geometric argument for cones. A result on the duality of these spaces is also given.

Subdifferential calculus and doubly nonlinear evolution equations in [formula omitted]-spaces with variable exponents

This paper is concerned with the Cauchy–Dirichlet problem for a doubly nonlinear parabolic equation involving variable exponents and provides some theorems on existence and regularity of strong solutions. In the proof of these results, we also analyze the relations occurring between Lebesgue spaces of space–time variables and Lebesgue–Bochner spaces of vector-valued functions, with a special emphasis on measurability issues and particularly referring to the case of space-dependent variable exponents. Moreover, we establish a chain rule for (possibly nonsmooth) convex functionals defined on variable exponent spaces. Actually, in such a peculiar functional setting the proof of this integration formula is nontrivial and requires a proper reformulation of some basic concepts of convex analysis, like those of resolvent, of Yosida approximation, and of Moreau–Yosida regularization.

Inverse-Closed Algebras of Integral Operators on Locally Compact Groups

We construct some inverse-closed algebras of bounded integral operators with operator-valued kernels, acting in spaces of vector-valued functions on locally compact groups. To this end we make use of covariance algebras associated to C*-dynamical systems defined by the C*-algebras of right uniformly continuous functions with respect to the left regular representation.

Multikernel least mean square algorithm.

The multikernel least-mean-square algorithm is introduced for adaptive estimation of vector-valued nonlinear and nonstationary signals. This is achieved by mapping the multivariate input data to a Hilbert space of time-varying vector-valued functions, whose inner products (kernels) are combined in an online fashion. The proposed algorithm is equipped with novel adaptive sparsification criteria ensuring a finite dictionary, and is computationally efficient and suitable for nonstationary environments. We also show the ability of the proposed vector-valued reproducing kernel Hilbert space to serve as a feature space for the class of multikernel least-squares algorithms. The benefits of adaptive multikernel (MK) estimation algorithms are illuminated in the nonlinear multivariate adaptive prediction setting. Simulations on nonlinear inertial body sensor signals and nonstationary real-world wind signals of low, medium, and high dynamic regimes support the approach.

Weak versions of Banach spaces of vector-valued functions

Weak versions of Banach spaces of vector-valued functions

Weak versions of Banach spaces of vector-valued functions

Weak versions of Banach spaces of vector-valued functions

ON THE ADMISSIBILITY OF THE SPACE L0($\mathcal{A}$,X) OF VECTOR-VALUED MEASURABLE FUNCTIONS

We prove the admissibility of the space L0(A,X) of vector- valued measurable functions determined by real-valued finitely additive set functions defined on algebras of sets. The notion of admissibility introduced by Klee (7) guarantees that a com- pact mapping into an admissible Hausdorff topological vector space E can be approximated by compact finite dimensional mappings. This notion is very important in degree theory and fixed point theory. It is known that locally convex spaces are admissible (see (10)). There are some classes of nonlocally convex spaces which are admissible. Riedrich in (13) proved the admissibility of the space S(0,1) of measurable functions and in (12) the admissibility of the space Lp(0,1) for 0 < p < 1. The admissibility of other function spaces has been proved by Mach (6) and Ishii (8). In (14) it is proved the admissibility of spaces of Besov-Triebel-Lizorkin type. Definition 1 ((7)). Let E be a Haudorff topological vector space. A subset Z of E is said to be admissible if for every compact subset K of Z and for every neighborhood V of zero in E there exists a continuous mapping H : K → Z such that dim(span (H(K)))< +∞ and x−Hx ∈ V for every x ∈ K. If Z = E we say that the space E is admissible. In this paper we deal with spaces of vector-valued measurable functions and, as a major fact, instead of σ-additive measures we consider finitely additive set functions defined on algebras of sets. Let X be a Banach space, a nonempty set, A a subalgebra of the power set P() of and µ : A → R a finitely additive set function. We prove

Korovkin theory for vector-valued functions on a locally compact space

We consider ordered spaces of continuous vector-valued functions on a locally compact Hausdorff space, endowed with appropriate locally convex topologies. Using suitable sets of such functions as test systems a Korovkin type approximation theorem for equicontinuous nets of positive operators is established. As in the classical theory, the Korovkin closure is characterized both through envelopes of functions and through measure theoretical conditions.