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We investigate a semilinear stochastic time-space fractional subdiffusion equation driven by fractionally integrated multiplicative noise. The equation involves the ψ -Caputo derivative of order α ∈ (0 , 1) and the spectral fractional Laplacian of order β ∈ (1 2 , 1 ] . The existence and uniqueness of the mild solution are proved in a suitable Banach space by using the Banach contraction mapping principle. The spatial and temporal regularities of the mild solution are established in terms of the smoothing properties of the solution operators. [ABSTRACT FROM AUTHOR]

This paper extends the well-known Ringrose theory for compact operators to polynomially Riesz operators on Banach spaces. The particular case of an ideal-triangularizable Riesz operator on an order continuous Banach lattice yields that the spectrum of such operator lies on its diagonal, which motivates the systematic study of an abstract diagonal of a regular operator on an order complete vector lattice E. We prove that the class of regular operators for which the diagonal coincides with the atomic diagonal is always a band in , which contains the band of abstract integral operators. If E is also a Banach lattice, then contains positive Riesz and positive AM-compact operators. [ABSTRACT FROM AUTHOR]

In this paper, we continue investigation of the lateral order on vector lattices started in [25]. We consider the complexification EC$E_{\mathbb {C}}$ of a real vector lattice E and introduce the lateral order on EC$E_{\mathbb {C}}$. Our first main result asserts that the set of all fragments Fv$\mathfrak {F}_v$ of an element v∈EC$v\in E_{\mathbb {C}}$ of the complexification of an uniformly complete vector lattice E is a Boolean algebra. Then, we study narrow operators defined on the complexification EC$E_{\mathbb {C}}$ of a vector lattice E, extending the results of articles [22, 27, 28] to the setting of operators defined on complex vector lattices. We prove that every order‐to‐norm continuous linear operator T:EC→X$\mathcal {T}: E_{\mathbb {C}} \rightarrow X$ from the complexification EC$E_{\mathbb {C}}$ of an atomless Dedekind complete vector lattice E to a finite‐dimensional Banach space X is strictly narrow. Then, we prove that every C‐compact order‐to‐norm continuous linear operator T$\mathcal {T}$ from EC$E_{\mathbb {C}}$ to a Banach space X is narrow. We also show that every regular order‐no‐norm continuous linear operator from EC$E_{\mathbb {C}}$ to a complex Banach lattice (ℓp(D)C$(\ell _p(\mathcal {D})_{\mathbb {C}}$ is narrow. Finally, in the last part of the paper we investigate narrow operators taking values in symmetric ideals of compact operators. [ABSTRACT FROM AUTHOR]

We characterize finite elements, an order-theoretic concept in Archimedean vector lattices, in the setting of ordered Banach spaces with positive unconditional basis as vectors having finite support with respect to their basis representations. Using algebraic vector space bases, we further describe a class of infinite dimensional vector lattices in which each element is finite and even self-majorizing. [ABSTRACT FROM AUTHOR]

Let p be a fixed number with 1 \leq p < \infty. It is shown that every surjective and biseparating linear map between L^p-spaces is continuous when the underlying measure space is non-atomic. We also prove that a separating isomorphism on l^p is both continuous and biseparating. Furthermore, these (bi-)separating maps take the form of a weighted composition operator. Our proofs are direct, elementary and do not invoke deep results about Riesz spaces or Banach lattices. [ABSTRACT FROM AUTHOR]

In this paper, we study a new class of locally dominated orthogonally additive operators on lattice-normed spaces (LNS). In the first part of the paper, sufficient conditions for the existence of a local exact majorant of a locally dominated operator and formulas for its calculation are given. The second part shows that the -compactness of a dominated orthogonally additive operator acting from a decomposable lattice-normed space to a Banach space with mixed norm implies the -compactness of its exact majorant. [ABSTRACT FROM AUTHOR]

We establish that the summability of the series ∑ ε n is the necessary and sufficient criterion ensuring that every (1 + ε n) -bounded Markushevich basis in a separable Hilbert space is a Riesz basis. Further we show that if n ε n → ∞ , then in ℓ 2 there exists a (1 + ε n) -bounded Markushevich basis that under any permutation is non-equivalent to a Schauder basis. We extend this result to any separable Banach space. Finally we provide examples of Auerbach bases in 1-symmetric separable Banach spaces whose no permutations are equivalent to any Schauder basis or (depending on the space) any unconditional Schauder basis. [ABSTRACT FROM AUTHOR]

We study two-layer neural networks whose domain and range are Banach spaces with separable preduals. In addition, we assume that the image space is equipped with a partial order, i.e., it is a Riesz space. As the nonlinearity we choose the lattice operation of taking the positive part; in case of Rd-valued neural networks this corresponds to the ReLU activation function. We prove inverse and direct approximation theorems with Monte-Carlo rates for a certain class of functions, extending existing results for the finite-dimensional case. In the second part of the paper we study, from the regularization theory viewpoint, the problem of finding optimal representations of such functions via signed measures on a latent space from a finite number of noisy observations. We discuss regularity conditions known as source conditions and obtain convergence rates in a Bregman distance for the representing measure in the regime when both the noise level goes to zero and the number of samples goes to infinity at appropriate rates. [ABSTRACT FROM AUTHOR]

In this paper we characterize perturbation ideals of sets that generate the familiar spectra in Fredholm theory. [ABSTRACT FROM AUTHOR]

In the context of partially ordered vector spaces one encounters different sorts of order convergence and order topologies. This article investigates these notions and their relations. In particular, we study and relate the order topology presented by Floyd, Vulikh and Dobbertin, the order bound topology studied by Namioka and the concept of order convergence given in the works of Abramovich, Sirotkin, Wolk and Vulikh. We prove that the considered topologies disagree for all infinite dimensional Archimedean vector lattices that contain order units. For reflexive Banach spaces equipped with ice cream cones we show that the order topology, the order bound topology and the norm topology agree and that order convergence is equivalent to norm convergence. [ABSTRACT FROM AUTHOR]

We provide a general framework for no-arbitrage concepts in topological vector lattices, which covers many of the well-known no-arbitrage concepts as particular cases. The main structural condition we impose is that the outcomes of trading strategies with initial wealth zero and those with positive initial wealth have the structure of a convex cone. As one consequence of our approach, the concepts NUPBR, NAA 1 and NA 1 may fail to be equivalent in our general setting. Furthermore, we derive abstract versions of the fundamental theorem of asset pricing (FTAP), including an abstract FTAP on Banach function spaces, and investigate when the FTAP is warranted in its classical form with a separating measure. We also consider a financial market with semimartingales which does not need to have a numéraire, and derive results which show the links between the no-arbitrage concepts by only using the theory of topological vector lattices and well-known results from stochastic analysis in a sequence of short proofs. [ABSTRACT FROM AUTHOR]

We consider special class of C0-groups from [12], whose generators are unbounded, have pure point imaginary spectrum and corresponding dense and minimal family of eigenvectors, which however does not form a Schauder basis. We obtain two-sided estimates for norms of C0-groups from this class and thus prove that these C0-groups have linear growth. Moreover we show that C0-groups from the considered class do not have any maximal asymptotics. This means that the fastest growing orbits do not exist. [ABSTRACT FROM AUTHOR]

The theory of algebraic frames for a Hilbert space H is a generalization of the theory of frames and generalized frames. The paper applies the theory of unbounded operators to define the dual of algebraic frames with densely defined unbounded analysis operators. It is shown that every algebraic frame has an algebraic dual frame, and if an algebraic frame has a nonzero redundancy, then it is not Riesz-type. An example of an algebraic frame with finite redundancy is constructed which is not a Riesz-type algebraic frame. Finally, for a lower bounded analytic frame, the discreteness of its indexing measure space and the uniqueness of its algebraic dual are studied and shown to be interrelated. [ABSTRACT FROM AUTHOR]

We prove that the algebraic dimension of an infinite dimensional C-σ-complete Riesz space (in particular, of a Dedekind σ-complete and a laterally σ-complete Riesz space) with the principal projection property which either has a weak order unit or is not purely atomic, is at least continuum. A similar (incomparable to ours) result for complete metric linear spaces is well known. [ABSTRACT FROM AUTHOR]

We say that a Riesz space E has the horizontal Egorov property if for every net (ƒα) in E, order convergent to ƒ ∈ E with |ƒα| + |ƒ| ≤ e ∈ E+ for all α, there exists a net (eβ) of fragments of e laterally convergent to e such that for every nβ, the net (|ƒ − ƒα | ∧ eβ)α e-uniformly tends to zero. Our main result asserts that every Dedekind complete Riesz space which satisfies the weak distributive law possesses the horizontal Egorov property. A Riesz space E is said to satisfy the weak distributive law if for every e ∈ E+ \ 0 the Boolean algebra Fe of fragments of e satisfies the weak distributive law; that is, whenever (Πn)n ∈ N is a sequence of partitions of Fe, there is a partition Π of Fe such that every element of Π is finitely covered by each of Πn (e.g., every measurable Boolean algebra is so). Using a new technical tool, we show that for every net (ƒα) order convergent to ƒ in a Riesz space with the horizontal Egorov property there are a horizontally vanishing net (vβ) and a net (uα,β)(α,β) ∈ A × B, which uniformly tends to zero for every fixed β such that |ƒ − ƒα| ≤ uα, β + vβ for all α, β. [ABSTRACT FROM AUTHOR]

We introduce the (ϕ, 2, α)-bounded variation spaces, which are a common generalization between Riesz's spaces, p-variation and (ϕ, 2)-bounded variation spaces. We also study its structure as Banach spaces, as well as some embedding results. [ABSTRACT FROM AUTHOR]

We prove, in particular, that if E is a Dedekind complete atomless Riesz space and X is a Banach space then the sum of a narrow and a C-compact laterally continuous orthogonally additive operators from E to X is narrow. This generalizes in several directions known results on narrowness of the sum of a narrow and a compact operators for the settings of linear and orthogonally additive operators defined on Köthe function spaces and Riesz spaces. [ABSTRACT FROM AUTHOR]

We show that the free Banach lattice FBL (A) may be constructed as the completion of FVL (A) with respect to the maximal lattice seminorm ν on FVL (A) with ν (a) ⩽ 1 for all a ∈ A . We present a similar construction for the free Banach lattice FBL [ E ] generated by a Banach space E. [ABSTRACT FROM AUTHOR]

We prove that if E is a Dedekind complete atomless Riesz space and X is a Banach space, then the sum of two laterally continuous orthogonally additive operators from E to X, one of which is strictly narrow and the other one is hereditarily strictly narrow with finite variation (in particular, has finite rank), is strictly narrow. Similar results were previously obtained for narrow operators by different authors; however, no theorem of the kind was known for strictly narrow operators. [ABSTRACT FROM AUTHOR]

Let L=-Δ+V be a Schrödinger operator, where Δ is the Laplacian on Rd and the nonnegative potential V belongs to the reverse Hölder class RHq for q≥d. The Riesz transform associated with the operator L=-Δ+V is denoted by R=∇(-Δ+V)-1/2 and the dual Riesz transform is denoted by R⁎=(-Δ+V)-1/2∇. In this paper, we first introduce some kinds of weighted Morrey spaces related to certain nonnegative potentials belonging to the reverse Hölder class RHq for q≥d. Then we will establish the mapping properties of the operator R and its adjoint R⁎ on these new spaces. Furthermore, the weighted strong-type estimate and weighted endpoint estimate for the corresponding commutators [b,R] and [b,R⁎] are also obtained. The classes of weights, classes of symbol functions, and weighted Morrey spaces discussed in this paper are larger than Ap, BMO(Rd), and Lp,κ(w) corresponding to the classical Riesz transforms (V≡0). [ABSTRACT FROM AUTHOR]

In this paper, we focus on direct limits and inverse limits in the category with generalized pseudo-effect algebras (GPEAs for short) as objects and GPEA-morphisms as morphisms. We show that direct limits exist in the category of GPEAs and direct limits of GPEAs satisfy the Riesz decomposition properties whenever the directed systems of GPEAs satisfy the Riesz decomposition properties. Then, we give a condition under which the quotient of a direct limit of GPEAs is a direct limit of quotients of GPEAs. Moreover, we prove that if inverse systems of GPEAs satisfy the Riesz decomposition properties, then inverse limits also satisfy the Riesz decomposition properties. [ABSTRACT FROM AUTHOR]

Abstract Every infinite dimensional Banach space admits Riesz operators that are not finite rank. In this note we discuss conditions under which a Riesz operator, or some power thereof, is a finite rank operator. [ABSTRACT FROM AUTHOR]

Martingale-like sequences in vector lattice and Banach lattice frameworks are defined in the same way as martingales are defined in [Positivity 9 (2005), 437-456]. In these frameworks, a collection of bounded X-martingales is shown to be a Banach space under the supremum norm, and under some conditions it is also a Banach lattice with coordinatewise order. Moreover, a necessary and sufficient condition is presented for the collection of E-martingales to be a vector lattice with coordinate-wise order. It is also shown that the collection of bounded E-martingales is a normed lattice but not necessarily a Banach space under the supremum norm. [ABSTRACT FROM AUTHOR]

We introduce and study new class of sets (almost L-limited sets). Also, we introduce new concept of property in Banach lattice (almost Gelfand-Phillips property) and we characterize this property using almost L-limited sets. On the other hand, we introduce the class of disjoint limited completely continuous operators which is a largest class than that of limited completely continuous operators, we characterize this class of operators and we study some of its properties. [ABSTRACT FROM AUTHOR]

This paper proves several fixed point results for mappings satisfying the Hardy-Rogers inequality on order-metric spaces. Using ordered vector space valued norm-type mappings, the concept of completeness is redefined resulting order-Banach spaces. One of the main outcomes proves that each Dedekind σ-complete Riesz space is complete with respect to its natural absolute value. This statement leads to consistent examples of order-Banach structures. Ultimately, the Thakur iterative process is analyzed in the newly defined order-Banach framework; it results that the Thakur iteration orderconverges faster than the Picard iterative process, for the class of Hardy-Rogers contractions. [ABSTRACT FROM AUTHOR]

The commutators of bilinear Calderón-Zygmund operators and pointwise multiplication with a symbol in CMO are bilinear compact operators on products of Lebesgue spaces. We show that, for certain non-degenerate Calderón-Zygmund operators, the symbol being in CMO is not only sufficient but actually necessary for the compactness of the commutators. [ABSTRACT FROM AUTHOR]

In this paper we find sufficient and necessary conditions under which vector lattice C ( X ) and its sublattices C b ( X ) , C 0 ( X ) and C c ( X ) have the countable sup property. It turns out that the countable sup property is tightly connected to the countable chain condition of the underlying topological space X . We also consider the countable sup property of C ( X × Y ) . Even when both C ( X ) and C ( Y ) have the countable sup property it is possible that C ( X × Y ) fails to have it. For this construction one needs to assume the continuum hypothesis. In general, we present a positive result in this direction and also address the question when C ( ∏ λ ∈ Λ X λ ) has the countable sup property. Our results can be understood as vector lattice theoretical versions of results regarding products of spaces satisfying the countable chain condition. We also present new results for general vector lattices that are of an independent interest. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we present a new approach to the problem of finding a common zero for a system of m‐accretive mappings in a uniformly convex Banach space with a uniformly Gâteaux differentiable norm. We propose an implicit iteration method and two explicit ones, based on compositions of resolvents with the steepest‐descent method. We show that our results contain some iterative methods in literature as special cases. An extension of the Xu's regularization method for the proximal point algorithm from Hilbert spaces onto Banach ones under simple conditions of convergence and a new variant for the method of alternating resolvents are obtained. Numerical experiments are given to affirm efficiency of the methods. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept of isotone cones in Banach spaces. Then, we apply the order monotonic property of the metric projection operator to prove the existence of best approximations for some operators without continuity conditions in partially ordered Banach spaces. [ABSTRACT FROM AUTHOR]

The universal completion of the Archimedean Riesz space C ( X ) of continuous, real valued functions on a completely regular space X is characterised as the space NL ( X ) of nearly finite, normal lower semi-continuous functions on X . As an application, we obtain, under additional assumptions on X , a characterisation of unbounded order convergence in C ( X ) as pointwise convergence everywhere except possibly on a set of first Baire category. This result is analogous to the situation in spaces of (real) p -summable functions, the sets of first Baire category now playing the role of null sets. We pursue this analogy further. First it is shown that, for a Baire space X , NL ( X ) is Riesz and algebra isomorphic to the space of real Borel measurable functions on X , with identification of functions differing at most on a set of first category. Secondly, through the use of density topologies and category measures, the extent to which our results can be cast in a measure-theoretic setting, and vice versa, is explored. Finally, through an application of the Maeda–Ogasawara Representation Theorem, we obtain a characterisation of those completely regular spaces X and Z such that C ( X ) and C ( Z ) have Riesz isomorphic universal completions. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to introduce and to study fuzzy normed Riesz spaces. By the direct method, we prove the Hyers-Ulam stability of the following lattice preserving functional equation in fuzzy Banach Riesz space N2 (f(τx V ηy) - τ f(x) ⋁ ηf(y),t) ≤ N1(φ(τx ⋀ ηy, τx ⋀ ηy),t) Where (X,N1), (y,N2) are fuzzy normed Riesz space and fuzzy Banach Riesz space, respectively; and φ : X × X → X is a mapping such that φ(x,y)≥(Ƭη)α/2φ(x/τ,y/η) for all τη≤ 1 and α ∊[0,1). [ABSTRACT FROM AUTHOR]

Let C be an axiomatizable class of order continuous real or complex Banach lattices, that is, this class is closed under isometric vector lattice isomorphisms and ultraproducts, and the complementary class is closed under ultrapowers. We show that if linear isometric embeddings of members of C in their ultrapowers preserve disjointness, the class CB of Banach spaces obtained by forgetting the Banach lattice structure is still axiomatizable. Moreover if C coincides with its “script class” SC, so does CB with SCB. This allows us to give new examples of axiomatizable classes of Banach spaces, namely certain Musielak–Orlicz spaces, Nakano spaces, and mixed norm spaces. [ABSTRACT FROM AUTHOR]

Let Y be a sublattice of a vector lattice X. We consider the problem of identifying the smallest order closed sublattice of X containing Y. It is known that the analogy with topological closure fails. Let ... be the order closure of Y consisting of all order limits of nets of elements from Y. Then ... need not be order closed. We show that in many cases the smallest order closed sublattice containing Y is in fact the second order closure .... Moreover, if X is a σ-order complete Banach lattice, then the condition that ... is order closed for every sublattice Y characterizes order continuity of the norm of X. The present paper provides a general approach to a fundamental result in financial economics concerning the spanning power of options written on a financial asset. [ABSTRACT FROM AUTHOR]

Nonstandard hulls of a vector lattice were introduced and studied in many papers. Recently, these notions were extended to ordered vector spaces. In the present paper, following the construction of associated Banach-Kantorovich space due to Emelyanov, we describe and investigate the nonstandard hull of a lattice-normed space, which is the foregoing generalization of Luxemburg's nonstandard hull of a normed space. [ABSTRACT FROM AUTHOR]

In this paper, we obtain new Riesz representations of continuous linear maps associated with certain boundary value problems in the set of all closed bounded convex non-empty subsets of any Banach space. As applications, the Riesz integral representation results are also given. [ABSTRACT FROM AUTHOR]

Let X be a Banach function space over the unit circle T and let H[X] be the abstract Hardy space built upon X. If the Riesz projection P is bounded on X and a∈L∞, then the Toeplitz operator Taf=P(af) is bounded on H[X]. We extend well-known results by Brown and Halmos for X=L2 and show that, under certain assumptions on the space X, the Toeplitz operator Ta is bounded (resp., compact) if and only if a∈L∞ (resp., a=0). Moreover, aL∞≤TaB(H[X])≤PB(X)aL∞. These results are specified to the cases of abstract Hardy spaces built upon Lebesgue spaces with Muckenhoupt weights and Nakano spaces with radial oscillating weights. [ABSTRACT FROM AUTHOR]

Let ${\mathcal C}[{\mathcal X}]$ be any class of operators on a Banach space ${\mathcal X}$, and let ${Holo}^{-1}({\mathcal C})$ denote the class of operators A for which there exists a holomorphic function f on a neighbourhood ${\mathcal N}$ of the spectrum σ(A) of A such that f is non-constant on connected components of ${\mathcal N}$ and f(A) lies in ${\mathcal C}$. Let ${{\mathcal R}[{\mathcal X}]}$ denote the class of Riesz operators in ${{\mathcal B}[{\mathcal X}]}$. This paper considers perturbation of operators $A\in\Phi_{+}({\mathcal X})\Cup\Phi_{-}({\mathcal X})$ (the class of all upper or lower [semi] Fredholm operators) by commuting operators in $B\in{Holo}^{-1}({\mathcal R}[{\mathcal X}])$. We prove (amongst other results) that if πB(B) = ∏mi = 1(B − μi) is Riesz, then there exist decompositions ${\mathcal X}=\oplus_{i=1}^m{{\mathcal X}_i}$ and $B=\oplus_{i=1}^m{B|_{{\mathcal X}_i}}=\oplus_{i=1}^m{B_i}$ such that: (i) If λ ≠ 0, then $\pi_B(A,\lambda)=\prod_{i=1}^m{(A-\lambda\mu_i)^{\alpha_i}} \in\Phi_{+}({\mathcal X})$ (resp., $\in\Phi_{-}({\mathcal X})$) if and only if $A-\lambda B_0-\lambda\mu_i\in\Phi_{+}({\mathcal X})$ (resp., $\in\Phi_{-}({\mathcal X})$), and (ii) (case λ = 0) $A\in\Phi_{+}({\mathcal X})$ (resp., $\in\Phi_{-}({\mathcal X})$) if and only if $A-B_0\in\Phi_{+}({\mathcal X})$ (resp., $\in\Phi_{-}({\mathcal X})$), where B0 = ⊕mi = 1(Bi − μi); (iii) if $\pi_B(A,\lambda)\in\Phi_{+}({\mathcal X})$ (resp., $\in\Phi_{-}({\mathcal X})$) for some λ ≠ 0, then $A-\lambda B\in\Phi_{+}({\mathcal X})$ (resp., $\in\Phi_{-}({\mathcal X})$). [ABSTRACT FROM PUBLISHER]

A net $$(x_\alpha )$$ in a vector lattice X is unbounded order convergent to $$x \in X$$ if $$|x_\alpha - x| \wedge u$$ converges to 0 in order for all $$u\in X_+$$ . This convergence has been investigated and applied in several recent papers by Gao et al. It may be viewed as a generalization of almost everywhere convergence to general vector lattices. In this paper, we study a variation of this convergence for Banach lattices. A net $$(x_\alpha )$$ in a Banach lattice X is unbounded norm convergent to x if for all $$u\in X_+$$ . We show that this convergence may be viewed as a generalization of convergence in measure. We also investigate its relationship with other convergences. [ABSTRACT FROM AUTHOR]

Let Ω and I denote a compact metrizable space with card(Ω)≥2 and the unit interval, respectively. We prove Milutin and Cantor-Bernstein type theorems for the spaces M(Ω) of Radon measures on compact Hausdorff spaces Ω. In particular, we obtain the following results: (1) for every infinite closed subset K of βN the spaces M(K), M(βN), and M(Ω2ℵ0) are order-isometric; (2) for every discrete space Γ with m≔card(Γ)>ℵ0 the spaces M(βΓ) and M(I2m) are order-isometric, whereas there is no linear homeomorphic injection from C(βT) into C(I2m). [ABSTRACT FROM AUTHOR]

We shall say that a bounded linear operatorTacting on a Banach spaceXadmits a generalized Kato–Riesz decomposition if there exists a pair ofT-invariant closed subspaces (M, N) such that, the reductionis Kato andis Riesz. In this paper, we define and investigate the generalized Kato–Riesz spectrum of an operator. ForTis said to be generalized Drazin-Riesz invertible if there exists a bounded linear operatorSacting onXsuch that,,is Riesz. We investigate generalized Drazin-Riesz invertible operators and also characterize bounded linear operators which can be expressed as a direct sum of a Riesz operator and a bounded below (resp. surjective, upper (lower) semi-Fredholm, Fredholm, upper (lower) semi-Weyl, Weyl) operator. In particular, we characterize the single-valued extension property at a pointin the case thatadmits a generalized Kato–Riesz decomposition. [ABSTRACT FROM PUBLISHER]

Using the existence of solutions for equilibrium equations with a Neumann type boundary condition as developed by Shi and Liao (J. Inequal. Appl. 2015:363, 2015), we obtain the Riesz integral representation for continuous linear maps associated with additive set-valued maps with values in the set of all closed bounded convex non-empty subsets of any Banach space, which are generalizations of integral representations for harmonic functions proved by Leng, Xu and Zhao (Comput. Math. Appl. 66:1-18, 2013). We also deduce the Riesz integral representation for set-valued maps, for the vector-valued maps of Diestel-Uhl and for the scalar-valued maps of Dunford-Schwartz. [ABSTRACT FROM AUTHOR]

For a Riesz spaceE, we introduceE-valued sequence spaces generated by an order-function. It is proved that these spaces equipped with a Luxemburg norm are ideal Banach lattices. We also determine necessary and sufficient conditions so that the Luxemburg norm on these spaces has the-order continuous property. Finally, we present some criteria for strictly monotonicity, uniformly monotonicity and locally monotonicity of the spaces. [ABSTRACT FROM AUTHOR]

It is known that the sum of every two narrow operators on L1 is narrow, however the same is false for Lp with 1 < p < to. The present paper continues numerous investigations of the kind. Firstly, we study narrowness of a linear and orthogonally additive operators on Kothe function spaces and Riesz spaces at a fixed point. Theorem 1 asserts that, for every Kothe Banach space E on a finite atomless measure space there exist continuous linear operators S, T : E ->A E which are narrow at some fixed point but the sum S + T is not narrow at the same point. Secondly, we introduce and study uniformly narrow pairs of operators S, T : E ^ X, that is, for every e Є E and every ε > 0 there exists a decomposition e = e' + e" to disjoint elements such that ||S(e') — S(e")|| < ε and ||T(e') — T(e'')|| < ε. The standard tool in the literature to prove the narrowness of the sum of two narrow operators S + T is to show that the pair S, T is uniformly narrow. We study the question of whether every pair of narrow operators with narrow sum is uniformly narrow. Having no counterexample, we prove several theorems showing that the answer is affirmative for some partial cases. [ABSTRACT FROM AUTHOR]

Let X be a reflexive separable Banach space, and let pB the set of p-Bessel sequences in X* for X. We show that pB is a non-commutative unital Banach algebra isometrically isomorphic to B(X). Also, we classify p-Bessel sequences for X in terms of different kind of operators in B(X) and B(X*), and we give important characterizations of p-frames and q-Riesz sequences. Using an isomorphism between the sets p B and B(X) we obtain interesting results for p-frames in Banach spaces. Using operator theory tools, we investigate the geometry of p-Bessel sequences. Also, we show that the set of all q-Riesz bases for X* is a topological group. [ABSTRACT FROM AUTHOR]

The theory of discrete and continuous frames was introduced for the purpose of analyzing and reconstructing signals mainly in Hilbert spaces. However, in many interesting applications the analyzed space is usually a Banach space, and consequently the stable analysis/reconstruction schemes need to be investigated for general Banach spaces. Parallel to discrete Hilbert space frames, the theory of atomic decompositions, p -frames and framings have been introduced in the literature to address this problem. In this paper we focus on continuous frames and continuous framings (alternatively, integral reconstructions) for Banach spaces by the means of g-Köthe function spaces, in which the involved measure space is σ -finite, positive and complete. Necessary and sufficient conditions for a measurable function to be an L ρ -frame are obtained, and we obtain a decomposition result for the analysis operators of continuous frames in terms of simple Köthe–Bochner operators. As a byproduct we show that a Riesz type continuous frame doesn't exist unless the measure space is purely atomic. One of our main results shows that there is an intrinsic connection between continuous framings and g-Köthe function spaces. [ABSTRACT FROM AUTHOR]

We investigate pointwise nonnegativity as an obstruction to various types of structured completeness in Lp(R). For example, we prove that if each element of the system {fn}∞ n=1⊂ Lp(R) is pointwise nonnegative, then {fn}∞n=1 cannot be an unconditional basis or unconditional quasibasis (unconditional Schauder frame) for Lp(R). In particular, in L2(R) this precludes the existence of nonnegative Riesz bases and frames. On the other hand, there exist pointwise nonnegative conditional quasibases in Lp(R), and there also exist pointwise nonnegative exact systems and Markushevich bases in Lp(R). [ABSTRACT FROM AUTHOR]

This work is dedicated to the generalization of frames and Riesz bases in Banach spaces with respect to the Banach space of vector-valued sequences. The concepts of X-frame and X-Riesz basis generated by a bilinear mapping are introduced. Criteria for X-frameness and X-Riesz basicity are found. [ABSTRACT FROM AUTHOR]

We prove an analog of the Dodds-Fremlin-Wickstead Theorem on compact domination for homogeneous orthogonally additive polynomials in Banach lattices. The proof is based on linearization of the polynomials which was established earlier by the author. [ABSTRACT FROM AUTHOR]

In this article, the orthogonal projection and the Riesz representation theorem are mainly formalized. In the first section, we defined the norm of elements on real Hilbert spaces, and defined Mizar functor RUSp2RNSp, real normed spaces as real Hilbert spaces. By this definition, we regarded sequences of real Hilbert spaces as sequences of real normed spaces, and proved some properties of real Hilbert spaces. Furthermore, we defined the continuity and the Lipschitz the continuity of functionals on real Hilbert spaces. Referring to the article [15], we also defined some definitions on real Hilbert spaces and proved some theorems for defining dual spaces of real Hilbert spaces. As to the properties of all definitions, we proved that they are equivalent properties of functionals on real normed spaces. In Sec. 2, by the definitions [11], we showed properties of the orthogonal complement. Then we proved theorems on the orthogonal decomposition of elements of real Hilbert spaces. They are the last two theorems of existence and uniqueness. In the third and final section, we defined the kernel of linear functionals on real Hilbert spaces. By the last three theorems, we showed the Riesz representation theorem, existence, uniqueness, and the property of the norm of bounded linear functionals on real Hilbert spaces. We referred to [36], [9], [24] and [3] in the formalization. [ABSTRACT FROM AUTHOR]