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Assume that is a faithful normal semifinite trace on a von Neumann algebra , is the unit of , is the -algebra of -measurable operators, and is the Banach space of -integrable operators. We present a new proof of the following generalization of Putnam's theorem (1951): No positive self-commutator with is invertible in . If is infinite then no positive self-commutator with can be of the form , where is a nonzero complex number and is a -compact operator. Given with we seek for the conditions that . If and with then . If and then . If a partial isometry lies in and for some integer then is a commutator and implies that . [ABSTRACT FROM AUTHOR]

Solutions of some partial differential equations are obtained as critical points of a real funtional. Then the Banach space where this functional is defined has to be real, otherwise, it is not differentiable. It follows that the equation is solved with respect to the real dual space of this Banach space. But if the solution is complex-valued there is the following problem: what does the multiplication of this equation by a complex number mean? In this note, we explain how to rigorously define this operation. [ABSTRACT FROM AUTHOR]

Let A and B be two closed linear relations acting between two Banach spaces X and Y , and let λ be a complex number. We study the stability of the nullity and deficiency of A when it is perturbed by λ B. In particular, we show the existence of a constant ρ > 0 for which both the nullity and deficiency of A remain stable under perturbation by λ B for all λ inside the disk λ < ρ. [ABSTRACT FROM AUTHOR]

In this paper, we solve the following quadratic ρ-functional inequalities ‖f (x + y + z/2) + f(x - y - z/2) + f(y - x - z/2) + f(z - x - y/2) -f(x) - f(y) - f(z)‖ ≤ ‖ρ(f(x + y + z) + f(x - y - z) + f(y - x - z) + f(z - x - y) (0.1) -4f(x) - 4f(y) - 4f(z))‖; where ρ is a fixed complex number with |ρ| < 1/8, and ‖f(x + y + z) + f(x - y - z) + f(y - x - z) + f(z - x - y) -4f(x) - 4f(y) - 4f(z)‖ (0.2) ≤ ‖ ρ(f(x + y + z/2) + f(x - y - z/2) + f(y - x - z/2) + f(z - x - y/2) -f(x) - f(y) - f(z))‖ ; where ρ is a fixed complex number with |ρ| < 4. Using the direct method, we prove the Hyers-Ulam stability of the quadratic ρ-functional inequalities (0.1) and (0.2) in complex Banach spaces and prove the Hyers-Ulam stability of quadratic ρ-functional equations associated with the quadratic ρ-functional inequalities (0.1) and (0.2) in complex Banach spaces. [ABSTRACT FROM AUTHOR]

In this paper, we solve the following quadratic ρ-functional inequalities ..., where ρ is a fixed complex number with |ρ| < 1/8, and ..., where ρ is a fixed complex number with |ρ| < 4. Using the direct method, we prove the Hyers-Ulam stability of the quadratic ρ-functional inequalities (0.1) and (0.2) in complex Banach spaces and prove the Hyers-Ulam stability of quadratic ρ-functional equations associated with the quadratic ρ-functional inequalities (0.1) and (0.2) in complex Banach spaces. [ABSTRACT FROM AUTHOR]

Let α be a fixed complex number, and let Ω be a simply connected region in complex plane C that is starlike with respect to α ∈ Ω. We define some Banach space of analytic functions on Ω and prove that it is a Banach algebra with respect to the α-Duhamel product defined by ... . We prove that its maximal ideal space consists of the homomorphism hα defined by hα(f)=f(α). Further, we characterize the lattice of invariant subspaces of the integration operator Jαf(z)=... f(t)dt. Moreover, we describe in terms of α-Duhamel operators the extended eigenvectors of Jα. [ABSTRACT FROM AUTHOR]

Let ? be a unimodular complex number, and let k be an integer. Then ?Ak is an isometry for any isometry A of a complex Banach space. It is shown that if f is an analytic function on the unit circle sending an isometry to an isometry for any norm, then f has the form z ? ?zk for some unimodular ? and integer k. The same conclusion on f can be deduced if f is merely continuous and preserves the isometries of some special classes of norms on a fixed finite-dimensional complex Banach space. The result is extended to real Banach spaces X with dimX = 4, and it is shown that one cannot get the same conclusion on f if dimX <4. Further extensions of these results are also considered. [ABSTRACT FROM AUTHOR]

In this paper, we solve the additive ρ-functional inequalities ‖f(2x - y) + f(y - x) - f(x)‖ ⩽ ‖ρ (f(x + y) - f(x) - f(y))‖ ; (0.1) where ρ is a fixed complex number with ¦ρ¦ < 1, and ‖f(x + y) - f(x) - f(y)‖ ⩽ ‖ρ(f(2x - y) + f(y - x) - f(x))‖k; (0.2) where ρ is a fixed complex number with ¦ρ¦ < 1/2. Furthermore, we prove the Hyers-Ulam stability of the additive ρ-functional in-equalities (0.1) and (0.2) in complex Banach spaces. [ABSTRACT FROM AUTHOR]

Let X and Y be locally compact Hausdorff spaces, where X is first-countable. Fix a positive integer n ⩾ 3 and a non-zero complex number λ. If a surjective map T : C0(X) → C0(Y ) satisfies the condition supy|Πnk+1T(fk)(y) + λ|= supϰ∊X|(Πnk=1 fk)(x) + λ| for all fk ∊ C0(X) (k = 1, . . . , n), then there exist a homeomorphism ϕ: Y → X, a continuous function w: Y → 핋, and a clopen set K ⊂ Y such that T(f) = w ⨰ (f ◦ ϕ) on K and T(f) = w ⨰ (f ◦ ϕ) on Y \ K for all f ∊ C0(X). Let A, B be function algebras on X, Y . Fix a positive integer n ⩾ 2. Suppose that X = Ch(A) and Y = Ch(B). If a surjective map T : A → B satisfies the condition σ л Πnk=1 T(fk)) ∩ σл(Πnk=1 fk)≠ø for all fk ∊ A (k = 1, . . . , n), then T is a weighted composition operator. [ABSTRACT FROM AUTHOR]

Let γ be a unimodular complex number, and let k be an integer. Then γAk is an isometry for any isometry A of a complex Banach space. It is shown that if f is an analytic function on the unit circle sending an isometry to an isometry for any norm, then f has the form z 7→ γzk for some unimodular γ and integer k. The same conclusion on f can be deduced if f is merely continuous and preserves the isometries of some special classes of norms on a fixed finite-dimensional complex Banach space. The result is extended to real Banach spaces X with dim X ≥ 4, and it is shown that one cannot get the same conclusion on f if dim X < 4. Further extensions of these results are also considered. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the latticetic operation preserving ρ-functional equation ... where ρ is a fixed non-Archimedean number or a fixed real or complex number with ρ ≠1. Then, using different methods, we prove the nonlinear stability of ρ-functional equation (*) in non-Archimedean Banach lattice spaces and non-Archimedean latticetic random Banach lattice spaces. [ABSTRACT FROM AUTHOR]

This paper deals with the semigroup generation of anti‐triangular operator matrices M with unbounded entries in Hilbert space. Based on the space decomposition, some necessary and sufficient conditions are given for M to generate contraction semigroups. In addition, the anti‐triangular differential system, converted from the damping wave equation, is used to explain our work, and it is proved that the corresponding anti‐triangular operator matrix satisfies the conditions and generates a contraction semigroup. [ABSTRACT FROM AUTHOR]

M. Daher [Canad. Math. Bull. 38 (1995), pp. 286–294] showed that if (X_0, X_1) is a regular couple of uniformly convex spaces then the unit spheres of the complex interpolation spaces X_{\theta } and X_{\eta } are uniformly homeomorphic for every 0 < \theta, \eta < 1. We show that this is a rather general phenomenon of the interpolation methods described by the discrete framework of interpolation of Lindemulder and Lorist [ A discrete framework for the interpolation of Banach spaces , https://arxiv.org/abs/2105.08373, 2021]. [ABSTRACT FROM AUTHOR]

In this paper, we solve the quadratic ρ-functional equations f (x+ y)+ f (x− y)− 2f (x)− 2f (y)(0.1)= ρ(2f(x+y/2)+ 2f(x−y/2)− f (x)− f (y)),where ρ is a fixed non-Archimedean number or a fixed real or complex number with ρ≠− 1, 2,and2f(x+y/2)+ 2f(x−y/2)− f (x)− f (y)(0.2)= ρ (f (x+ y)+ f (x− y)− 2f (x)− 2f (y)),whereρ is a fixed non-Archimedean number or a fixed real or complex number with ρ≠− 1, 1/2.Using the direct method, we prove the Hyers-Ulam stability of the quadratic ρ-functionalequations (0.1) and (0.2) in non-Archimedean Banach spaces and in Banach spaces. [ABSTRACT FROM AUTHOR]

Let M1f(x; y) : = 3/4f(x+y) - 1/4 f(-x - y) + 1/4 f(x - y) + 1/4 f(y - x) - f(x) - f(y); M2f(x; y) := 2f (x + y/2 )+ f (x - y/ 2 )+ f (y - x/ 2 )- f(x) - f(y). We solve the additive-quadratic ρ-functional inequalities ‖M1f(x; y)‖ ≤ kρM2f(x; y)‖, (0.1) where ρ is a fixed complex number with ∣ρ∣ < 1/2 and ‖M2f(x, y)‖ ≤ kρM1f(x, y)‖, (0.2) where ρ is a fixed complex number with ∣ρ∣ < 1. Using the direct method, we prove the Hyers-Ulam stability of the additive-quadratic ρ-functional inequalities (0.1) and (0.2) in complex Banach spaces. [ABSTRACT FROM AUTHOR]

In this paper, we solve the following additive-quadratic ρ-functional inequalities ‖f(x + y) + f(x - y) - 2f(x) - f(y) - f(-y)‖ (0.1) ⩽ ǁρ 2f ( x + y 2 ) + 2f ( x - y 2 ) - 3/2 f(x) + 1 2 f(-x) - 1/2 f(y) - 1/2 f(-y)ǁ ; where ρ is a fixed complex number with ǀρǀ < 1, and ǁ2f ( x + y 2 ) + 2f (-x - y 2) - 3/2 f(x) + 1 2 f(-x) - 12 f(y) - 1/2 f(-y)ǁ ⩽‖ρ(f(x + y) + f(x - y) - 2f(x) - f(y) - f(-y)ǁk; (0.2) where ρ is a fixed complex number with ǀρǀ < 1 2 , and prove the Hyers-Ulam stability of the additivequadratic ρ-functional inequalities (0.1) and (0.2) in β-homogeneous complex Banach spaces and prove the Hyers-Ulam stability of additive-quadratic ρ-functional equations associated with the additive-quadratic β-functional inequalities (0.1) and (0.2) in β-homogeneous complex Banach spaces. [ABSTRACT FROM AUTHOR]

Let A and B be two closed linear relations acting between two Banach spaces X and Y and let λ be a complex number. We study the behavior of the nullity and deficiency of A when perturbed by λB. In particular, we show the existence of a constant ρ > 0 for which both the nullity and deficiency of A do not remain constant when A is perturbed by λB for all λ inside the disk |λ| < ρ. It turns out, however, that these quantities do not depend on λ in the specified disk, that is, both the nullity and deficiency of A - λB are uniform on the specified disk. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in the (recently introduced by Alavrez et al.) concept of the so-called asymptotically (ω,c) periodic functions. We investigate further properties of this class of functions which complement the currently known ones. These properties allowed us to prove the existence and uniqueness of asymptotically (ω,c) - periodic mild solution of an integro-differential equation in a Banach space, a result which generalizes some recent works in the class of S-asymptotically - periodic functions by Brindle and N'Guérékata. [ABSTRACT FROM AUTHOR]

In this paper, we derive novel families of inclusion sets for the spectra and pseudo- spectra of large classes of bounded linear operators, and establish convergence of particular sequences of these inclusion sets to the spectrum or pseudospectrum, as appropriate. Our results apply, in particular, to bounded linear operators on a separable Hilbert space that, with respect to some orthonormal basis, have a representation as a bi-infinite matrix that is banded or band-dominated. More generally, our results apply in cases where the matrix entries themselves are bounded linear operators on some Banach space. In the scalar matrix entry case, we show that our methods, given the input information we assume, lead to a sequence of approximations to the spectrum, each element of which can be computed in finitely many arithmetic operations, so that, with our assumed inputs, the problem of determining the spectrum of a band-dominated operator has solvability complexity index one in the sense of Ben-Artzi et al. (2020). As a concrete and substantial application, we apply our methods to the determination of the spectra of non-self-adjoint bi-infinite tridiagonal matrices that are pseudoergodic in the sense of Davies [Commun. Math. Phys. 216 (2001), 687-704]. [ABSTRACT FROM AUTHOR]

We present new results for the generalized Drazin inverse of 2 × 2 anti-triangular matrices with commuting entries over a Banach algebra. As an application, the g-Drazin invertibility of block-operator matrices is obtained under new wider conditions. [ABSTRACT FROM AUTHOR]

In this paper, with the help of some new atomic decomposition theorems, several Φ-moment Banach space valued martingale inequalities associated with concave functions in the context of Lorentz spaces are deduced. Our results are closely related with the geometrical properties of the underlying Banach spaces. [ABSTRACT FROM AUTHOR]

Let X, Y be Banach spaces and S : X →Y be an invertible Lipschitz map. Let T :X →Y be a map and there exist 1, 2 = [0,1) such that Then we prove that T is an invertible Lipschitz map. This is non-linear version of 26 years old Casazza-Kalton-Christensen-van Eijndhoven perturbation. It also a non-linear version of 29 years old Soderlind-Campanato perturbation and 3 years old Barbagallo-Ernst-Thera perturbation. We give applications to the theory of metric frames. The notion of Lipschitz atomic decomposition for Banach spaces is also introduced. [ABSTRACT FROM AUTHOR]

In this paper, we solve the following cubic ρ-functional inequality ‖f(2x + y) + f(2x - y) - 2f(x + y) - 2f(x - y) - 12f(x)‖ ⩽ ǁ ρ 4f (x + y/2 )+ 4f (x - y/2) - f(x + y) - f(x - y) - 6f(x))ǁ ; (0.1) where ρ is a ρ xed complex number with ǀρǀ < 2, and the quartic ρ-functional inequality ‖f(2x + y) + f(2x - y) - 4f(x + y) - 4f(x - y) - 24f(x) + 6f(y)‖ (0.2) ⩽ ǁ ρ(8f x + y/2( + 8f (x - y/2) - 2f(x + y) - 2f(x - y) - 12f(x) + 3f(y) ǁ ; where ρ is a fixed complex number with ǀρǀ <2. Using the direct method, we prove the Hyers-Ulam stability of the cubic ρ-functional in- equality (0.1) and the quartic ρ-functional inequality (0.2) in complex Banach spaces. [ABSTRACT FROM AUTHOR]

This work computes the spectrum of the backward fractional difference operator $ \Delta ^{(\alpha)} $ Δ (α) $ (0 (0 < α < 1) over various Banach spaces $ \ell _{1},c $ ℓ 1 , c and $ c_0 $ c 0 . Note that recently, the spectrum of the fractional difference operator $ \Delta _\nu ^{(\alpha)} $ Δ ν (α) , ( $ \nu = (\nu _k) $ ν = (ν k) being either a constant or strictly decreasing sequence of positive real numbers satisfying some conditions) on the sequence space $ \ell _{1} $ ℓ 1 has been computed by Baliarsingh P. [On the spectrum of fractional difference operator. Linear Multilinear Algebra. 2021;1–13] and it has been noticed that the results obtained are unified and more general, but lack of accuracy for the case $ 0 0 < α < 1. In this work, we improve the results by providing the sharper estimations and some illustrative examples. Also, individual sets describing the point, the residual and the continuous spectrum of the operator $ \Delta ^{(\alpha)} $ Δ (α) on the Banach space c and $ c_0 $ c 0 are computed. [ABSTRACT FROM AUTHOR]

In this paper, we prove a local surjection theorem with continuous right-inverse for maps between Banach spaces, and we apply it to a class of inversion problems with loss of derivatives. [ABSTRACT FROM AUTHOR]

In this paper we study the asymptotic behavior of solutions of fractional differential equations of the form D^{\alpha }_Cu(t)=Au(t)+f(t), u(0)=x, 0<\alpha \le 1, (*) where D^{\alpha }_Cu(t) is the derivative of the function u in the Caputo's sense, A is a linear operator in a Banach space \mathbb {X} that may be unbounded and f satisfies the property that \lim _{t\to \infty } (f(t+1)-f(t))=0 which we will call asymptotic 1-periodicity. By using the spectral theory of functions on the half line we derive analogs of Katznelson-Tzafriri and Massera Theorems. Namely, we give sufficient conditions in terms of spectral properties of the operator A for all asymptotic mild solutions of Eq. (*) to be asymptotic 1-periodic, or there exists an asymptotic mild solution that is asymptotic 1-periodic. [ABSTRACT FROM AUTHOR]

The theory of modular flow has proved extremely useful for applying thermodynamic reasoning to out-of-equilibrium states in quantum field theory. However, the standard proofs of the fundamental theorems of modular flow use machinery from Fourier analysis on Banach spaces, and as such are not especially transparent to an audience of physicists. In this article, I present a construction of modular flow that differs from existing treatments. The main pedagogical contribution is that I start with thermal physics via the KMS condition, and derive the modular operator as the only operator that could generate a thermal time-evolution map, rather than starting with the modular operator as the fundamental object of the theory. The main technical contribution is a new proof of the fundamental theorem stating that modular flow is a symmetry. The new proof circumvents the delicate issues of Fourier analysis that appear in previous treatments, but is still mathematically rigorous. [ABSTRACT FROM AUTHOR]

As an extension of some classes of generalized almost periodic functions, in this paper we develop the notion of c-almost periodicity in the sense of Stepanov and Weyl approaches. In fact, we extend some basic results of this theory which were already demonstrated for the standard cases. In particular, we prove that every c-almost periodic function in the sense of Stepanov approach (in the sense of equi-Weyl or Weyl approaches, respectively) is also c m -almost periodic in the sense of Stepanov approach (in the sense of equi-Weyl or Weyl approaches, respectively) for each non-zero integer number m. This study is performed for both representative cases of functions defined on the real axis and with values in a Banach space and the complex functions defined on vertical strips in the complex plane. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the following new additive s-functional inequalities ||f(x - y) + f(y) + f(-x)|| ≤ ||s (f(x + y) - f(x) - f(y)) k; (0.1) ||kf(x + y) - f(x) - f(y)|| ≤ ||s (f(x - y) + f(y) + f(-x)) k; (0.2) where s is a fixed complex number with |s| < 1, and prove the Hyers-Ulam stability of linear derivations on Banach algebras associated to the additive s-functional inequalities (0.1) and (0.2). [ABSTRACT FROM AUTHOR]

Let A be the generator of a strongly continuous periodic one-parameter group of bounded linear operators in a Banach space H, and let Rλ be the resolvent of A at the point λ and Φ(A) is a polynomial of A with complex coefficients. We have obtained conditions for compactness be the operator Rλ. Using this result, conditions of the existence of a solution to the nonlinear equation Φ(A)x = f(x) are obtained, where f is a continuous mapping of H into itself. [ABSTRACT FROM AUTHOR]

We study geometric and isometric properties of two classes of sequence spaces. First, we consider the Banach sequence space X_n, the completion of c_{00} relatively to the norm \[ \|x\|_n = \sup \left \{\sum _{i \in F}|x_i|, \text { with }, F \text { of cardinality at most } n\right \}. \] For this space, we show that the isometry group of X_n consists of standard isometries. Second, we consider a Tsirelson-type space, T[\theta, A_n] (0<\theta <1) associated with the regular family A_n, comprised of all subsets of \mathbb {N} with at most n elements. For this space, we characterize the extreme points and show that for \theta \leq \frac {1}{n}, T[\theta, A_n] supports standard isometries. We also derive the form for the surjective isometries of T[\theta, A_3]. [ABSTRACT FROM AUTHOR]

Operator-valued Schatten spaces were introduced by G. Pisier as a noncommutative counterpart of vector-valued ℓ p -spaces. This family of operator spaces forms an interpolation scale which makes it a powerful and convenient tool in a variety of applications. In particular, as the norms coming from this family naturally appear in the definition of certain entropic quantities in quantum information theory (QIT), one may apply Pisier's theory to establish some features of those quantities. Nevertheless, it could be quite challenging to follow the proofs of the main results of this theory from the existing literature. In this article, we attempt to fill this gap by presenting the underlying concepts and ideas of Pisier's theory in a self-contained way which we hope to be more accessible, especially for the QIT community at large. Furthermore, we describe some applications of this theory in QIT. In particular, we prove a new uniform continuity bound for the quantum conditional Rényi entropy. [ABSTRACT FROM AUTHOR]

We construct a Cesaro-type operator acting on Dirichlet space of the upper half plane using the approach of strongly continuous semigroups of composition operators on Banach spaces. We then determine the spectral and norm properties of the obtained Cesaro-type operator. [ABSTRACT FROM AUTHOR]

Let X be an infinite-dimensional complex Banach space, B (X) the algebra of all bounded linear operators on X . Denote the spectral domain by σ γ (T) = { λ ∈ σ a (T) : T that is semi-Fredholm and a s c (T − λ I) < ∞ } . In this paper, we characterize the structure of additive surjective maps φ : B (X) → B (X) with σ γ (φ (T)) = σ γ (T) for all T ∈ B (X) . [ABSTRACT FROM AUTHOR]

In the paper, we study various countably generated algebras of entire analytic functions on complex Banach spaces and their homomorphisms. Countably generated algebras often appear as algebras of symmetric analytic functions on Banach spaces with respect to a group of symmetries, and are interesting for their possible applications. Some conditions of the existence of topological isomorphisms between such algebras are obtained. We construct a class of countably generated algebras, where all normalized algebraic bases are equivalent. On the other hand, we find non-isomorphic classes of such algebras. In addition, we establish the conditions of the hypercyclicity of derivations in countably generated algebras of entire analytic functions of the bounded type. We use methods from the theory of analytic functions of several variables, the theory of commutative Fréchet algebras, and the theory of linear dynamical systems. [ABSTRACT FROM AUTHOR]

In this paper, we develop some generalized Fredholm perturbation results for bounded linear operators acting on non-reflexive Banach spaces satisfying certain conditions. Furthermore, we investigate the stability of the generalized Schechter S-essential spectrum by means of the so-called generalized Fredholm index. This concept is introduced as a semigroup homomorphism satisfying certain properties. Some generalized Fredholm results for block operator matrices acting on a non-reflexive product of two Banach spaces are also developed. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the additive (α β)-functional equation f(x) + αf(αy) = β-1f(β(x + y)) for all complex numbers α with ¦α¦ = 1 and for a fixed nonzero complex number β. Using the fixed point method and the direct method, we prove the Hyers-Ulam stability of the additive (α, β)-functional equation (0.1) in complex Banach spaces. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the additive (α, β)-functional equation ... for all complex numbers α with [α] D 1 and for a fixed nonzero complex number β. Using the fixed point method and the direct method, we prove the Hyers-Ulam stability of the additive (α, β)-functional equation (0.1) in complex Banach spaces. [ABSTRACT FROM AUTHOR]

Let M1f(x; y) : 3/4f(x+y)-1/4f(-x-y) + 1/4f(x-y)+1/4f(y-x)+1/4f(y-x)-f(x)-f(y), M2f(x,y): = 2f(x+y/2)+f(x-y/2)+f(y-x/2)-f(x)-f(x). We solve the additive-quadratic ρ-functional inequalities ‖M1f(x,y)‖⩽‖ρM2f(x,y)ρ, where ρ is a fixed complex number with |ρ| < 1 Using the direct method, we prove the Hyers-Ulam stability of the additive-quadratic ρ-functional inequalities (0.1) and (0.2) in complex Banach spaces. [ABSTRACT FROM AUTHOR]

This paper considers L^1-spaces of covariant functions of characters of normal subgroups. Assume that G is a locally compact group with the group algebra L^1(G) and N is a closed and normal subgroup of G. Suppose that \mathbb {T} is the circle group and \xi :N\to \mathbb {T} is a character. We introduce the Banach space L_\xi ^1(G,N) of covariant functions of \xi on G and present an operator theoretic approach to study structure of L^1_\xi (G,N). It is proven that L^1_\xi (G,N) and a quotient space of L^1(G) are isometrically isomorphic. The paper is concluded by constructive characterizations for covariant functions of \xi. [ABSTRACT FROM AUTHOR]

This paper is devoted to an invariance principle for Kemperman's model of oscillating random walk on Z. This result appears as an extension of the invariance principal theorem for classical random walks on Z or reflected random walks on N0. Relying on some natural Markov sub-process which takes into account the oscillation of the random walks between Z and Z+, we first construct an aperiodic sequence of renewal operators acting on a suitable Banach space and then apply a powerful theorem proved by S. Gouëzel. [ABSTRACT FROM AUTHOR]

This work investigates a weighted Banach space second order pantograph fractional differential equation. The considered equation is of second order, expressed in terms of the Caputo–Hadamard fractional operator, and constructed in a general manner to accommodate many specific situations. The asymptotic stability of the main equation's trivial solution has been given. The primary theorem was demonstrated in a unique manner by employing the Krasnoselskii's fixed point theorem. We provide a concrete example that supports the theoretical findings. [ABSTRACT FROM AUTHOR]

An operator T acting on a Banach space X obeys property (gt) if the isolated points of the spectrum σ(T) of T which are eigenvalues are exactly those points λ of the spectrum for which T −λ is an upper semi-B-Fredholm with index less than or equal to 0. In this paper we study the stability of property (gt) under perturbations by finite rank operators, by nilpotent operators and, more generally, by algebraic operators commuting with T . Moreover, we study the transfer of property (gt) from a bounded linear operator T acting on a Banach space X and a bounded linear operator S acting on a Banach space Y to their tensor product T ⊗S. [ABSTRACT FROM AUTHOR]

In this article, we establish some new properties of the two-parameter Mittag-Leffler function and use them to prove that, mild solutions of the evolution equation CDαtu(t) =Au(t)+ f (t) (t ∈ R) are (ω,c) -asymptotically periodic, where A is the generator of a strongly continuous semigroup {T(θ)}θ≥0 (which is exponentially stable) on a Banach space X and CDαt denotes the Caputo fractional derivative of order 0 < α ≤ 1. We further establish an existence and uniqueness result for optimal (ω,c) -asymptotically periodic mild solution if X is a uniformly convex Banach space. [ABSTRACT FROM AUTHOR]

M1f(x,y): = 3/4f(x+y)-1/4f(-x-y) +1/4f(x-y)+1/4f(y-x)-f(x)-f(y), M2f(x,y):=2f(x+y/2)+f(x-y/2)+f(y-x/2)-f(x)-f(y). We solve the additive-quadratic ρ-functional inequalities ‖M1f(x,y)‖≤‖ρM2f(x,y)‖≤‖ρM1f(x,y)‖, where ρ is a fixed complex number with ∣ρ∣<1. Using the fixed point method, we prove the Hyers-Ulam stability of the additive-quadratic ρ-functional inequalities (0.1) and (0.2) in complex Banach spaces. [ABSTRACT FROM AUTHOR]

In this paper, we solve the additive ρ-functional equations ... where ρ is a fixed non-Archimedean number or a fixed real or complex number with ρ ≠ 1. Using the direct method, we prove the Hyers-Ulam stability of the additive ρ- functional equations (0.1) and (0.2) in non-Archimedean Banach spaces and in Ba- nach spaces. [ABSTRACT FROM AUTHOR]

We provide a method for constructing (possibly non-trivial) measures on non-locally compact Polish subspaces of infinite-dimensional separable Banach spaces which, under suitable assumptions, are minimizers of causal variational principles in the non-locally compact setting. Moreover, for non-trivial minimizers the corresponding Euler–Lagrange equations are derived. The method is to exhaust the underlying Banach space by finite-dimensional subspaces and to prove existence of minimizers of the causal variational principle restricted to these finite-dimensional subsets of the Polish space under suitable assumptions on the Lagrangian. This gives rise to a corresponding sequence of minimizers. Restricting the resulting sequence to countably many compact subsets of the Polish space, by considering the resulting diagonal sequence, we are able to construct a regular measure on the Borel algebra over the whole topological space. For continuous Lagrangians of bounded range, it can be shown that, under suitable assumptions, the obtained measure is a (possibly non-trivial) minimizer under variations of compact support. Under additional assumptions, we prove that the constructed measure is a minimizer under variations of finite volume and solves the corresponding Euler–Lagrange equations. Afterwards, we extend our results to continuous Lagrangians vanishing in entropy. Finally, assuming that the obtained measure is locally finite, topological properties of spacetime are worked out and a connection to dimension theory is established. [ABSTRACT FROM AUTHOR]

In this paper, we characterize the classes (L, Lk), (Lk, L) and (L∞, Lk), 1 ≤ k < ∞, of all four dimensional infinite matrices, where Lk and L∞ are the spaces of all absolutely k-summable and bounded double sequences, respectively. Using them, we establish some relations between N, pn, qn and N, p' n, q' n k summability methods which extend some results of Bosanquet (1950), Sarıgöl (1993), Sarıgöl & Bor (1995), and Sunouchi (1949) to double summability methods, and give a relation between single and double summability methods. [ABSTRACT FROM AUTHOR]

In this paper, we analyze Levitan and Bebutov metrical approximations of functions F : Λ x X → Y by trigonometric polynomials and ρ-periodic type functions, where ∅= Λ ⊆ Rn, X and Y are complex Banach spaces, and ρ is a general binary relation on Y . We also analyze various classes of multidimensional Levitan almost periodic functions in general metric and multidimensional Bebutov uniformly recurrent functions in general metric. We provide several applications of our theoretical results to the abstract Volterra integro-differential equations and the partial differential equations. [ABSTRACT FROM AUTHOR]

In this paper, we establish some new properties of (ω , c) -periodic and asymptotically (ω , c) -periodic functions, then we apply them to study the existence and uniqueness of mild solutions of these types to the following semilinear fractional differential equations: (1) { c D t α u (t) = A u (t) + c D t α − 1 f (t , u (t)) , 1 < α < 2 , t ∈ R , u (0) = 0 and (2) { c D t α u (t) = A u (t) + c D t α − 1 f (t , u (t − h)) , 1 < α < 2 , t , h ∈ R + , u (0) = 0 where c D t α (⋅) (1 < α < 2) stands for the Caputo derivative and A is a linear densely defined operator of sectorial type on a complex Banach space X and the function f (t , x) is (ω , c) -periodic or asymptotically (ω , c) -periodic with respect to the first variable. Our results are obtained using the Leray–Schauder alternative theorem, the Banach fixed point principle and the Schauder theorem. Then we illustrate our main results with an application to fractional diffusion-wave equations. [ABSTRACT FROM AUTHOR]