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Let ( Ω , Σ , μ ) be a measure space, and 1 ≤ p ≤ ∞ . A subspace E ⊆ L p ( μ ) is said to do stable phase retrieval (SPR) if there exists a constant C ≥ 1 such that for any f , g ∈ E we have 0.1 inf | λ | = 1 ‖ f - λ g ‖ ≤ C ‖ | f | - | g | ‖ . In this case, if | f | is known, then f is uniquely determined up to an unavoidable global phase factor λ ; moreover, the phase recovery map is C -Lipschitz. Phase retrieval appears in several applied circumstances, ranging from crystallography to quantum mechanics. In this article, we construct various subspaces doing stable phase retrieval, and make connections with Λ ( p ) -set theory. Moreover, we set the foundations for an analysis of stable phase retrieval in general function spaces. This, in particular, allows us to show that Hölder stable phase retrieval implies stable phase retrieval, improving the stability bounds in a recent article of M. Christ and the third and fourth authors. We also characterize those compact Hausdorff spaces K such that C ( K ) contains an infinite dimensional SPR subspace.

In this article, we provide a comprehensive study on the continuity and essential norm of an operator defined by an infinite tridiagonal matrix, specifically when it operates from a weighted Orlicz sequence space or a weighted space into another space of similar nature. Our findings include significant characterizations regarding the compactness of this operator across various contexts of weighted Orlicz and sequence spaces.

Building on a recent construction of Plebanek and Salguero-Alarcón, which solved the Complemented Subspace Problem for $C(K)$ -spaces, and the subsequent work of De Hevia, Martínez-Cervantes, Salguero-Alarcón, and Tradacete solving the Complemented Subspace Problem for Banach lattices, we show that the class of Banach lattices is not primary. Specifically, we exhibit a compact Hausdorff space L such that $C(L) \\simeq X \\oplus \\tilde {X}$ and neither X nor $\\tilde {X}$ is isomorphic to a Banach lattice. In particular, it also follows that the class of $C(K)$ -spaces is not primary.

We find conditions on a function space$${\\varvec{L}}$$L that ensure that it behaves as an$$L_p$$L p -space in the sense that any unconditional basis of a complemented subspace of$${\\varvec{L}}$$L either is equivalent to the unit vector system of$$\\ell _2$$ℓ 2 or has a subbasis equivalent to a disjointly supported basic sequence. This dichotomy allows us to classify the symmetric basic sequences of$${\\varvec{L}}$$L . Several applications to Orlicz function spaces are provided.

In this paper, type n lattice-ordered algebras are introduced and a characterization is given for those of type 0 and type 1. Moreover we investigate the question: Let A be a lattice-ordered algebra with unit element e > 0 in which every positive element has an inverse. Under what conditions A is lattice and algebra isomorphic to R ? We have shown that for certain algebras the question has a positive answer, generalizing thus a result of Scheffold. We also obtained a result similar to Edwards’ Theorem for normed lattice-ordered algebras.

In this paper, we define the class of demi ab -continuous operators on Banach lattices where a and b are given as n , w , o , ru , uo , un , and uaw . The relations between ab -continuous and demi ab -continuous operators are given. Moreover, the relations between the class of demi ab -continuous operators for different a , b convergence are investigated.

We study a special type of infinite direct sums$$E({\\mathcal {X}})$$E ( X ) which can be seen as the amalgam spaces characterized by a local component given by a countable family$${\\mathcal {X}}=\\left( X_{\\alpha }\\right) _{\\alpha ın I}$$X = X α α ∈ I of quasi-normed function spaces and by a global component E , which is a quasi-normed sequence space. We characterize some fundamental properties of$$E({\\mathcal {X}})$$E ( X ) such as completeness, Köthe-duality, order continuity and the Fatou property. We also provide its Banach function space characterization. Then, we apply our general results to the appropriate amalgamations of Lorentz (Orlicz) function spaces and Lebesgue sequence spaces. Moreover, for the Lorentz-type amalgams, we derive interpolation results and prove the boundedness of a class of sublinear integral operators whose kernels satisfy a size condition.

We study when Aron–Berner extensions of (separately) almost Dunford–Pettis multilinear operators between Banach lattices are (separately) almost Dunford–Pettis. For instance, for a σ -Dedekind complete Banach lattice F containing a copy of ℓ ∞ , we characterize the Banach lattices E 1 , … , E m for which every continuous m -linear operator from E 1 × ⋯ × E m to F admits an almost Dunford–Pettis Aron–Berner extension. Illustrative examples are provided.

In this paper, we introduce and study a weak version of Grothendieck operators that we will call weak Grothendieck operators, these are operators between Banach spaces which exactly carry Dunford–Pettis sets into limited ones. We establish some characterizations of this class of operators. After that, we look for some conditions on the starting space under which this class of operators and that of Grothendieck operators coincide. Furthermore, we study the weak compactness of almost Grothendieck operators. Besides, we present some results concerning the domination property of positive Grothendieck operators. Finally, some connections between almost Grothendieck operators and those whose adjoint carries positive weak* null sequences into weakly null ones are obtained.

We define the weighted Orlicz-Lorentz-Morrey and weak weighted Orlicz-Lorentz-Morrey spaces to generalize the Orlicz spaces, the weighted Lorentz spaces, the Orlicz-Lorentz spaces, and the Orlicz-Morrey spaces. Furthermore, necessary and sufficient conditions for the boundedness of the Hardy-Littlewood maximal operator, generalized fractional integral, and maximal operators on the weighted Orlicz-Lorentz-Morrey and weak Orlicz-Lorentz-Morrey spaces are given, based on the exploration of properties of Young functions, weights, and weights. Specifying the weights and the Young functions, we recover the existing results and we obtain new results in the new and old settings.