Khinchin’s inequality,Dunford-Pettis and compact operators on the space <Emphasis Type="Italic">C</Emphasis>([0, 1], <Emphasis Type="Italic">X</Emphasis>)

We prove that if X, Y are Banach spaces, Ω a compact Hausdorff space and U:C(Ω, X) → Y is a bounded linear operator, and if U is a Dunford-Pettis operator the range of the representing measure G(Σ) ? DP(X, Y) is an uniformly Dunford-Pettis family of operators and ∥G∥ is continuous at Ø. As applications of this result we give necessary and/or sufficient conditions that some bounded linear operators on the space C([0, 1], X) with values in c ₀ or l _p, (1 ≤ p < ∞) be Dunford-Pettis and/or compact operators, in which, Khinchin’s inequality plays an important role.     

Supercyclicity in Spaces of Operators

A sufficient criterion for the map \({C_{A, B}(S) = ASB}\) to be supercyclic on certain algebras of operators on Banach spaces is given. If T is an operator satisfying the Supercyclicity Criterion on a Hilbert space H, then the linear map \({C_{T}(V) = TVT^*}\) is shown to be norm-supercyclic on the algebra \({\mathcal{K}(H)}\) of all compact operators, COT-supercyclic on the real subspace \({\mathcal{S}(H)}\) of all self-adjoint operators and weak*-supercyclic on \({\mathcal{L}(H)}\) of all bounded operators on H. Examples including operators of the form \({C_{B_w, F_\mu}}\) are provided, where B_w and \({F_\mu}\) are respectively backward and forward shifts on Banach sequence spaces.     

Disjoint hypercyclic powers of weighted translations on groups

Let G be a locally compact group and let 1 ≤ p < 1. Recently, Chen et al. characterized hypercyclic, supercyclic and chaotic weighted translations on locally compact groups and their homogeneous spaces. There has been an increasing interest in studying the disjoint hypercyclicity acting on various spaces of holomorphic functions. In this note, we will study disjoint hypercyclic and disjoint supercyclic powers of weighted translation operators on the Lebesgue space L ^p(G) in terms of the weights. Sufficient and necessary conditions for disjoint hypercyclic and disjoint supercyclic powers of weighted translations generated by aperiodic elements on groups will be given.     

Unbounded disjointness preserving linear functionals and operators

Let E and F be Banach lattices. We show first that the disjointness preserving linear functionals separate the points of any infinite dimensional Banach lattice E, which shows that in this case the unbounded disjointness preserving operators from \({E\to F}\) separate the points of E. Then we show that every disjointness preserving operator \({T:E\to F}\) is norm bounded on an order dense ideal. In case E has order continuous norm, this implies that every unbounded disjointness preserving map \({T:E\to F}\) has a unique decomposition T = R + S, where R is a bounded disjointness preserving operator and S is an unbounded disjointness preserving operator, which is zero on a norm dense ideal. For the case that E = C(X), with X a compact Hausdorff space, we show that every disjointness preserving operator \({T:C(X)\to F}\) is norm bounded on a norm dense sublattice algebra of C(X), which leads then to a decomposition of T into a bounded disjointness preserving operator and a finite sum of unbounded disjointness preserving operators, which are zero on order dense ideals.     

〈2, 1〉-Compact operators

We consider the class of the continuous L _2,1 linear operators in L ₂ that are sums of the operators of multiplication by bounded measurable functions and the operators sending the unit ball of L ₂ into a compact subset of L ₁. We prove that a functional equation with an operator from L _2,1 is equivalent to an integral equation with kernel satisfying the Carleman condition. We also prove that if T ∈ L _2,1 and VTV ^?1 ∈ L _2,1 for all unitary operators V in L ₂ then T = α1 + C, where α is a scalar, 1 is the identity operator in L ₂, and C is a compact operator in L ₂.     

On local spectral properties of Hamilton type operators

In this paper, we introduce the class of Hamilton type operators and study various properties of this class. We show that every Hamilton type operator with property (β) or (δ) is decomposable. In addition, we prove that a Hamilton type operator T satisfies property (β), Dunford’s property (C) and Weyl’s theorem if and only if its adjoint does.     

The <Emphasis Type="Italic">p</Emphasis>-Gelfand–Phillips property in spaces of operators and Dunford–Pettis like sets

The p-Gelfand–Phillips property (1 \({\leq}\) p < ∞) is studied in spaces of operators. Dunford–Pettis type like sets are studied in Banach spaces. We discuss Banach spaces X with the property that every p-convergent operator T:X \({\rightarrow}\) Y is weakly compact, for every Banach space Y.     

Traceability of Positive Integral Operators

A necessary and sufficient condition is given for a positive bounded linear operator with an integral kernel to be trace class on L²(μ) for a σ-finite measure μ. The condition refines earlier criteria for positive Hilbert–Schmidt operators and positive integral operators with continuous kernels on a locally compact space.     

Operator algebras associated with multiplicative convolutions of arithmetic functions

The action of N on l~2(N) is studied in association with the multiplicative structure of N. Then the maximal ideal space of the Banach algebra generated by N is homeomorphic to the product of closed unit disks indexed by primes, which reflects the fundamental theorem of arithmetic. The C*-algebra generated by N does not contain any non-zero projection of finite rank. This assertion is equivalent to the existence of infinitely many primes. The von Neumann algebra generated by N is B(l~2(N)), the set of all bounded operators on l~2(N).Moreover, the differential operator on l~2(N,1/n(n+1)) defined by ▽f = μ * f is considered, where μ is the Mbius function. It is shown that the spectrum σ(▽) contains the closure of {ζ(s)-1: Re(s) 1}. Interesting problems concerning are discussed.     

<Emphasis Type="Italic">C</Emphasis>*-Algebras Generated by Mappings. Criterion of Irreducibility

We study the operator algebra associated with a self-mapping ? on a countable set X which can be represented as a directed graph. The algebra is generated by the family of partial isometries acting on the corresponding l²(X). We study the structure of involutive semigroup multiplicatively generated by the family of partial isometries. We formulate the criterion when the algebra is irreducible on the Hilbert space. We consider the concrete examples of operator algebras. In particular, we give the examples of nonisomorphic C*-algebras, which are the extensions by compact operators of the algebra of continuous functions on the unit circle.     

Li–Yorke chaos translation set for linear operators

In order to study Li–Yorke chaos by the scalar perturbation for a given bounded linear operator T on a Banach space X, we introduce the Li–Yorke chaos translation set of T, which is defined by \(S_{LY}(T)=\{\lambda \in {\mathbb {C}};\lambda +T \text { is Li--Yorke chaotic}\}\). In this paper, some operator classes are considered, such as normal operators, compact operators, shift operators, and so on. In particular, we show that the Li–Yorke chaos translation set of the Kalisch operator on the Hilbert space \(\mathcal {L}^2[0,2\pi ]\) is a simple point set \(\{0\}\).     

Essential spectrum of difference operators on periodic metric spaces

The paper deals with the study of Fredholm property and essential spectrum of general difference (or band) operators acting on the spaces l ^p (X) on a discrete metric space X periodic with respect to the action of a finitely generated discrete group. The Schrödinger operator on a combinatorial periodic graph is a prominent example of a band operator of this kind.     

Weyl-Type Theorems on Banach Spaces Under Compact Perturbations

In this paper, we study Browder-type and Weyl-type theorems for operators \(T+K\) defined on a Banach space X, where K is (a non necessarily commuting) compact operator on X. In the last part, the theory is exemplified in the case of isometries, analytic Toeplitz operators, semi-shift operators, and weighted right shifts.     

Weighted Composition Operators on Spaces of Analytic Functions on the Complex Half-Plane

In this paper we will show how the boundedness condition for the weighted composition operators on a class of spaces of analytic functions on the open right complex half-plane called Zen spaces (which include the Hardy spaces and weighted Bergman spaces) can be stated in terms of Carleson measures and Bergman kernels. In Hilbertian setting we will also show how the norms of causal weighted composition operators on these spaces are related to each other and use it to show that an (unweighted) composition operator \(C_\varphi \) is bounded on a Zen space if and only if \(\varphi \) has a finite angular derivative at infinity. Finally, we will show that there is no compact composition operator on Zen spaces.     

Two-weighted estimates for generalized fractional maximal operators on non-homogeneous spaces

Let μ be a nonnegative Borel measure on R ^d satisfying that μ(Q) ? l(Q)ⁿ for every cube Q ? R ⁿ , where l(Q) is the side length of the cube Q and 0 < n ? d.We study the class of pairs of weights related to the boundedness of radial maximal operators of fractional type associated to a Young function B in the context of non-homogeneous spaces related to the measure μ. Our results include two-weighted norm and weak type inequalities and pointwise estimates. Particularly, we give an improvement of a two-weighted result for certain fractional maximal operator proved in W.Wang, C. Tan, Z. Lou (2012).     

Miyamoto involutions in axial algebras of Jordan type half

Nonassociative commutative algebras A, generated by idempotents e whose adjoint operators ad _e : A → A, given by x ? xe, are diagonalizable and have few eigenvalues, are of recent interest. When certain fusion (multiplication) rules between the associated eigenspaces are imposed, the structure of these algebras remains rich yet rather rigid. For example, vertex operator algebras give rise to such algebras. The connection between the Monster algebra and Monster group extends to many axial algebras which then have interesting groups of automorphisms.Axial algebras of Jordan type η are commutative algebras generated by idempotents whose adjoint operators have a minimal polynomial dividing (x-1)x(x-η), where η ? {0, 1} is fixed, with well-defined and restrictive fusion rules. The case of η ≠1/2 was thoroughly analyzed by Hall, Rehren and Shpectorov in a recent paper, in which axial algebras were introduced. Here we focus on the case where η = 1/2, which is less understood and is of a different nature.     

On perturbation theory for points of discrete spectrum of dissipative operator functions

We consider the operator function L(α, θ) = A(α) ? θR of two complex arguments, where A(α) is an analytic operator function defined in some neighborhood of a real point α ₀ ∈ ? and ranging in the space of bounded operators in a Hilbert space ?. We assume that A(α) is a dissipative operator for real α in a small neighborhood of the point α ₀ and R is a bounded positive operator; moreover, the point α ₀ is a normal eigenvalue of the operator function L(α, θ ₀) for some θ ₀ ∈ ?, and the number θ ₀ is a normal eigenvalue of the operator function L(α ₀ θ). We obtain analogs and generalizations of well-known results for self-adjoint operator functions A(α) on the decomposition of α- and θ-eigenvalues in neighborhoods of the points α ₀ and θ ₀, respectively, and on the representation of the corresponding eigenfunctions by series.     

Fractal Perturbation of Shaped Functions: Convergence Independent of Scaling

In this paper, we introduce a new class of fractal approximants as a fixed points of the Read–Bajraktarevi? operator defined on a suitable function space. In the development of our fractal approximants, we used the suitable bounded linear operators defined on the space \({\mathcal {C}}(I)\) of continuous functions and \(\alpha \)-fractal functions. The convergence of the proposed fractal approximants towards the continuous function f does not need any condition on the scaling vector. Owing to this reason, the proposed fractal approximants approximate the function f without losing their fractality. We establish constrained approximation by a new class of fractal polynomials. In particular, our constrained fractal polynomials preserve positivity and fractality of the original function simultaneously whenever the original function is positive and irregular. Calculus of the proposed fractal approximants is studied using suitable bounded linear operators defined on the space \({\mathcal {C}}^r(I)\) of all real-valued functions on the compact interval I that are r-times differentiable with continuous r-th derivative. We identify the IFS parameters so that our \(\alpha \)-fractal functions preserve fundamental shape properties such as monotonicity and convexity in addition to the smoothness of f in the given compact interval.     

Optimal cubature formulas for calculation of multidimensional integrals in weighted Sobolev spaces

Optimal cubature formulas are constructed for calculations of multidimensional integrals in weighted Sobolev spaces. We consider some classes of functions defined in the cube Ω = [-1, 1]^l, l = 1, 2,..., and having bounded partial derivatives up to the order r in Ω and the derivatives of jth order (r < j ≤ s) whose modulus tends to infinity as power functions of the form (d(x, Г))^-(j-r), where x ∈ Ω Г, x = (x₁,..., x_l), Г = ?Ω, and d(x, Г) is the distance from x to Г.