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Let X be a complex Banach space, and denote by \mathcal {B}(X) the algebra of all bounded linear operators on X. Let C,D\in \mathcal {B} \left (X\right) be fixed operators. In this paper, we characterize linear, continuous and bijective maps \varphi and \psi on \mathcal {B}\left (X\right) for which there exist invertible operators T_0, W_0 \in \mathcal { B}(X) such that \varphi (T_0), \psi (W_0) \in \mathcal {B}(X) are both invertible, having the property that \varphi \left (A\right) \psi \left (B\right) =D in \mathcal {B}(X) whenever AB=C in \mathcal {B}(X). As a corollary, we deduce the form of linear, bijective and continuous maps \varphi on \mathcal {B}(X) having the property that \varphi \left (A\right) \varphi \left (B\right) =D in \mathcal {B}(X) whenever AB=C. [ABSTRACT FROM AUTHOR]

The present paper continues investigation of narrow operators on complex vector lattices started in a recent paper by Dzhusoeva, Huang, Pliev and Sukochev. We answer their questions and strengthen one of their main results by proving that the extension T_{\Bbb {C}}\colon E_{\Bbb {C}}\to F_{\Bbb {C}} of a linear operator T \colon E \to F acting from a Dedekind complete real Riesz space E to a real Banach lattice F to their complexifications is narrow if and only if T is. [ABSTRACT FROM AUTHOR]

A study of the class \Delta consisting of topological \Delta-spaces was originated by Jerzy Ka̧kol and Arkady Leiderman [Proc. Amer. Math. Soc. Ser. B 8 (2021), pp. 86–99; Proc. Amer. Math. Soc. Ser. B 8 (2021), pp. 267–280]. The main purpose of this paper is to introduce and investigate new classes \Delta _2 \subset \Delta _1 properly containing \Delta. We observe that for every first-countable X the following equivalences hold: X\in \Delta _1 iff X\in \Delta _2 iff each countable subset of X is G_{\delta }. Thus, new proposed concepts provide a natural extension of the family of all \lambda-sets beyond the separable metrizable spaces. We prove that (1) A pseudocompact space X belongs to the class \Delta _1 iff countable subsets of X are scattered. (2) Every regular scattered space belongs to the class \Delta _2. We investigate whether the classes \Delta _1 and \Delta _2 are invariant under the basic topological operations. Similarly to \Delta, both classes \Delta _1 and \Delta _2 are invariant under the operation of taking countable unions of closed subspaces. In contrast to \Delta, they are not preserved by closed continuous images. Let Y be l-dominated by X, i.e. C_p(X) admits a continuous linear map onto C_p(Y). We show that Y \in \Delta _1 whenever X \in \Delta _1. Moreover, we establish that if Y is l-dominated by a compact scattered space X, then Y is a pseudocompact space such that its Stone–Čech compactification \beta Y is scattered. [ABSTRACT FROM AUTHOR]

In this paper, we study the dynamics of an operator \mathcal T naturally associated to the so-called Collatz map , which maps an integer n \geq 0 to n / 2 if n is even and 3n + 1 if n is odd. This operator \mathcal T is defined on certain weighted Bergman spaces \mathcal B ^2 _\omega of analytic functions on the unit disk. Building on previous work of Neklyudov, we show that \mathcal T is hypercyclic on \mathcal B ^2 _\omega, independently of whether the Collatz Conjecture holds true or not. Under some assumptions on the weight \omega, we show that \mathcal T is actually ergodic with respect to a Gaussian measure with full support, and thus frequently hypercyclic and chaotic. [ABSTRACT FROM AUTHOR]

This paper deals with the chaotic behavior of linear operators on Banach spaces in both discrete and continuous cases. The inheritances of chaos for linear operators and C_0-semigroups are obtained. More precisely, for any positive integer n\geq 2, both Li-Yorke n-chaos and distributional n-chaos are proved to be inherited under iterations for a linear operator. One further shows that a C_0-semigroup \{T_t\}_{t\geq 0} and every single operator T_t share the same Li-Yorke n-scrambled set and distributionally n-scrambled set for any positive integer n\geq 2. In particular, the Li-Yorke n-chaos and distributional n-chaos become the Li-Yorke chaos and distributional chaos when n=2, respectively. Some equivalent criteria for dense n-chaos and generic n-chaos of linear operators and C_0-semigroups are also established for any n\geq 2. [ABSTRACT FROM AUTHOR]

Let \mathcal {H} be a separable Hilbert space and let B(\mathcal {H}) be the *-algebra of all bounded linear operators on \mathcal {H}. In the present paper, we prove that a positive/regular operator from L_1(0,1) into an arbitrary separable operator ideal in B(\mathcal {H}) is necessarily Dunford–Pettis, extending and strengthening results due to Gretsky and Ostroy [Glasgow Math. J. 28 (1986), pp. 113–114], and Holub [Proc. Amer. Math. Soc. 104 (1988), pp. 89–95]. Consequently, for an arbitrary atomless von Neumann algebra \mathcal {M} and an arbitrary KB -ideal C_E in B(\mathcal {H}), the predual \mathcal {M}_* of \mathcal {M} is not isomorphic to any subspace of C_E. This observation complements several earlier results. [ABSTRACT FROM AUTHOR]

For two Banach spaces X and Y, we denote by K(X,Y) (resp. B(X,Y)) the space of all compact (resp. bounded) linear operators from X to Y. In this paper, we show that for 1\leq p,q<\infty, K(\ell _{p},\ell _{q}) is an 8-Lipschitz retract of B(\ell _{p},\ell _{q}). [ABSTRACT FROM AUTHOR]

In his recent Proceedings of the AMS paper, ''Gossez's skew linear map and its pathological maximally monotone multifunctions'', Stephen Simons proved that the closure of the range of the sum of the Gossez operator and a multiple of the duality map is not convex whenever the scalar is between 0 and 4. The problem of the convexity of that range when the scalar is equal to 4 was explicitly stated. In this paper, we answer this question in the negative for any scalar greater than or equal to 4. We derive this result from an abstract framework that allows us to also obtain a corresponding result for the Fitzpatrick-Phelps integral operator. [ABSTRACT FROM AUTHOR]

We prove a Wiener-Wintner ergodic type theorem for a Markov representation S = {Sg : g ∊ G} of a right amenable semitopological semigroup G. We assume that S is mean ergodic as a semigroup of linear Markov operators acting on (C(K),sup), where K is a fixed Hausdorff, compact space. The main result of the paper is necessary and sufficient conditions for mean ergodicity of a distorted semigroup {X(g)Sg : g ∊ G}, where X is a semigroup character. Such conditions were obtained before under the additional assumption that S is uniquely ergodic. [ABSTRACT FROM AUTHOR]

In this note, we give a generalization of Gossez's example of a maximally monotone multifunction such that the closure of its range is not convex, using more elementary techniques than in Gossez's original papers. We also discuss some new properties of Gossez's skew linear operator and its adjoint. [ABSTRACT FROM AUTHOR]

Given a semifinite von Neumann algebra \mathcal M equipped with a faithful normal semifinite trace \tau, we prove that the spaces L^0(\mathcal M,\tau) and \mathcal R_\tau are complete with respect to pointwise—almost uniform and bilaterally almost uniform—convergences in L^0(\mathcal M,\tau). Then we show that the pointwise Cauchy property for a special class of nets of linear operators in the space L^1(\mathcal M,\tau) can be extended to pointwise convergence of such nets in any fully symmetric space E\subset \mathcal R_\tau, in particular, in any space L^p(\mathcal M,\tau), 1\leq p<\infty. Some applications of these results in the noncommutative ergodic theory are discussed. [ABSTRACT FROM AUTHOR]

Let T ϵ B(H) be a bounded linear operator on a Hilbert space Ⅿ, and let T ≡ V ∣T∣ be the polar decomposition of T. The mean transform of T is defined by T := 1/2 (V ∣T∣ + ∣T∣V ). In this paper we study the iterates of the mean transform and we define the mean limit of an operator as the limit (in the operator norm) of those iterates. We obtain new estimates for the numerical range and numerical radius of the mean transform in terms of the original operator. For the special class of unilateral weighted shifts we describe the precise relationship between the spectral radius and the mean limit, and obtain some sharp estimates. [ABSTRACT FROM AUTHOR]

The purpose of the present work is to answer an open problem which was raised by G. Costakis and A. Manoussos in their paper "J-class operators and hypercyclicity", J. Operator Theory 67 (2012), 101-119. More precisely, we give the spectral description of the closure of the set of J-class operators acting on a separable Hilbert space. [ABSTRACT FROM AUTHOR]

If X is a compact subset of a Banach space with X -X homogeneous (equivalently 'doubling' or with finite Assouad dimension), then X can be embedded into some R...n (with n sufficiently large) using a linear map L whose inverse is Lipschitz to within logarithmic corrections. More precisely, there exist c, α > 0 such that c ‖x - y‖/| log ‖x - y‖ |α ≤ |Lx - Ly| ≤ c‖x - y‖ for all x, y ∊ X, ‖x - y‖ < δ, for some δ sufficiently small. It is known that one must have α > 1 in the case of a general Banach space and α > 1/2 in the case of a Hilbert space. It is shown in this paper that these exponents can be achieved. While the argument in a general Banach space is relatively straightforward, the Hilbert space case relies on the fact that the maximum volume of a hyperplane slice of a k-fold product of unit volume N-balls is bounded independent of k (this provides a 'qualitative' generalisation of a result on slices of the unit cube due to Hensley (Proc.AMS 73 (1979), 95-100) and Ball (Proc.AMS 97 (1986), 465-473)). [ABSTRACT FROM AUTHOR]

We study Hilbert generated versions of nonseparable Banach spaces \mathcal {X} considered by Shelah, Steprāns and Wark where the behavior of the norm on nonseparable subsets is so irregular that it does not allow any linear bounded operator on \mathcal {X} other than a diagonal operator (or a scalar multiple of the identity) plus a separable range operator. We address the questions as to whether these spaces admit uncountable equilateral sets and whether their unit spheres admit uncountable (1+)-separated or (1+\varepsilon)-separated sets. We resolve some of the above questions for two types of these spaces by showing both absolute and undecidability results. The corollaries are that the continuum hypothesis (in fact: the existence of a nonmeager set of reals of the first uncountable cardinality) implies the existence of an equivalent renorming of the nonseparable Hilbert space \ell _2(\omega _1) which does not admit any uncountable equilateral set and it implies the existence of a nonseparable Hilbert generated Banach space containing an isomorphic copy of \ell _2 in each nonseparable subspace, whose unit sphere does not admit an uncountable equilateral set and does not admit an uncountable (1+\varepsilon)-separated set for any \varepsilon >0. This could be compared with a recent result by Hájek, Kania and Russo saying that all nonseparable reflexive Banach spaces admit uncountable (1+\varepsilon)-separated sets in their unit spheres. [ABSTRACT FROM AUTHOR]

In this paper, by using Gröbner-Shirshov bases for Rota-Baxter algebras, we prove that every dendriform algebra over a field of characteristic 0 can be embedded into its universal enveloping Rota-Baxter algebra. [ABSTRACT FROM AUTHOR]

Let Sn be the set of all n-by-n symmetric real matrices, and let Cn be the copositive cone, that is, the set of all matrices a ∈ Sn that fulfill the condition utop a u ≥ 0 for all n-vectors u with nonnegative entries. We prove that a linear mapping φ : Sn → Sn satisfies φ (Cn)= Cn if and only if φ (x)=mtop xm for a fixed monomial matrix m with nonnegative entries. [ABSTRACT FROM AUTHOR]

In this paper we estimate norms of integral operator induced by the Green function related to the Poisson equation in the unit ball with vanishing boundary data. [ABSTRACT FROM AUTHOR]

In the present paper we answer a question raised by J. Borcea and P. Brändén and give a description of the class of operators preserving roots in open circular domains, i.e., in images of the open upper half-plane under the Möbius transformations. Our second result is a description of the difference between A(G) (the class of operators preserving roots in an open set G) and A(...) (the class of operators preserving roots in ...). [ABSTRACT FROM AUTHOR]

In this paper we give a geometric characterization of mean ergodic convergence in the Calkin algebras for Banach spaces that have the bounded compact approximation property. [ABSTRACT FROM AUTHOR]

In this paper, we first show that homogeneous Keller maps are injective on lines through the origin. We subsequently formulate a generalization which states that under some conditions, a polynomial endomorphism with r homogeneous parts of positive degree does not have r times the same image point on a line through the origin, in case its Jacobian determinant does not vanish anywhere on that line. As a consequence, a Keller map of degree r does not take the same values on r > 1 collinear points, provided r is a unit in the base field. Next, we show that for invertible maps x+H of degree d such that ker JH has n - r independent vectors over the base field, in particular for invertible power linear maps x+(Ax)*d with rkA = r, the degree of the inverse of x+H is at most dr. [ABSTRACT FROM AUTHOR]

For a bounded linear operator T on a complex Hilbert space H, let ΔT,m = Σ m k=0(-1)k ( m k ) T"m-k for m ∊ N. T is called an misometry if ΔT,m = 0. In this paper, for every even number m, we give an example of invertible (m + 1)-isometry which is not an m-isometry. Next we show that if T is an m-isometry, then the operator Δm-k for m ∊ N. T is called an misometry if ΔT,m = 0. In this paper, for every even number m, we give an example of invertible (m + 1)-isometry which is not an m-isometry. Next we show that if T is an m-isometry, then the operator ΔT,m-1 is not invertible. [ABSTRACT FROM AUTHOR]

Let E be a Banach function space on a probability measure space (Ω,Σ,μ). Let X be a Banach space and E(X) be the associated Kothe-Bochner space. An operator on E(X) is called a multiplication operator if it is given by multiplication by a function in L∞(μ). In the main result of this paper, we show that an operator T on E(X) is a multiplication operator if and only if T commutes with L∞(μ) and leaves invariant the cyclic subspaces generated by the constant vector-valued functions in E(X). As a corollary we show that this is equivalent to T satisfying a functional equation considered by Calabuig, Rodr'ıguez, and S'anchez-P'erez. [ABSTRACT FROM AUTHOR]

This paper concerns the question whether the cone spectral radius rC(f) of a continuous compact order-preserving homogenous map f : C → C on a closed cone C in Banach space X depends continuously on the map. Using the fixed point index we show that if there exists 0 < a1 < a2 < a3 < ... not in the cone spectrum, σC(f), and limk→∞ ak = rC(f), then the cone spectral radius is continuous. An example is presented showing that if such a sequence (ak)k does not exist, continuity may fail. We also analyze the cone spectrum of continuous order-preserving homogeneous maps on finite dimensional closed cones. In particular, we prove that if C is a polyhedral cone with m faces, then σC(f) contains at most m - 1 elements, and this upper bound is sharp for each polyhedral cone. Moreover, for each nonpolyhedral cone there exist maps whose cone spectrum is infinite. [ABSTRACT FROM AUTHOR]

Let A be an operator algebra on a Hilbert space. We say that an element G ? A is an all-derivable point of A if every derivable linear mapping ? at G (i.e. ?(ST) = ?(S)T + S?(T) for any S, T ∊ algN with ST = G) is a derivation. Suppose that N is a nontrivial complete nest on a Hilbert space H. We show in this paper that G ∊ algN is an all-derivable point if and only if G ≠ 0. [ABSTRACT FROM AUTHOR]

In 2006, C. L. Jiang and Z. Y. Wang posed an open problem: If T is a strongly irreducible operator, is A'(T)/rad.A_(T) commutative? They conjectured that the answer is positive. In this paper, to negatively answer their problem, a counterexample is given. [ABSTRACT FROM AUTHOR]

In this paper we study the existence of strict and classical solutions for a class of abstract non-autonomous integro-differential equations in Banach spaces. Some applications of the abstract results are considered. [ABSTRACT FROM AUTHOR]

The infinitesimal earthquake theorem gives a one-to-one correspondence between Thurston bounded earthquake measures and normalized Zygmund bounded functions. In this paper, we provide an intrinsic proof of a theorem given in an earlier paper by the author; that is, we show that the cross-ratio norm of a Zygmund bounded function is equivalent to the Thurston norm of the earthquake measure in the correspondence. [ABSTRACT FROM AUTHOR]

Let ${\cal A}$ be the disk algebra. In this paper we address the following question: Under what conditions on the points $z_{k,n} \in \mathbf{ D}$ do there exist operators $L_n :{\cal A} \to {\cal A}$ such that \[ L_n f = \sum^{m_n}_{k=1} f(z_{k,n})l_{k,n},\qquad f, l_{k,n} \in {\cal A}, \] and $L_nf \to f$, $n \to \infty$, for every $f \in {\cal A}$? Here the convergence is understood in the sense of $sup$ norm in $\cal A$. Our first result shows that if $z_{k,n}$ satisfy Carleson condition, then there exists a function $f \in {\cal A}$ such that $L_nf \not\to f$, $n \to \infty$. This is a non-trivial generalization of results of Somorjai (1980) and Partington (1997). It also provides a partial converse to a result of Totik (1984). The second result of this paper shows that if $L_n$ are required to be projections, then for any choice of $z_{k,n}$ the operators $L_n$ do not converge to the identity operator. This theorem generalizes the famous theorem of Faber and implies that the disk algebra does not have an interpolating basis. [ABSTRACT FROM AUTHOR]

We apply Rossi's half-plane version of Borel's theorem to study the zero distribution of linear combinations of \mathcal {A}-entire functions (Theorem 1.2). This provides a unified way to study linear q-difference, difference and differential operators (with entire coefficients) preserving subsets of \mathcal {A}-entire functions, and hence obtain several analogous results for the Hermite-Poulain theorem to linear finite (q-)difference operators with polynomial coefficients. The method also produces a result on the existence of infinitely many non-real zeros of some differential polynomials of functions in certain sub-classes of \mathcal {A}-entire functions. [ABSTRACT FROM AUTHOR]

Let (\Omega,\Sigma,\mu) be a finite measure space and \mathcal {N}(X,Y) stand for the Banach space of all nuclear operators between Banach spaces X and Y, equipped with the nuclear norm \|\cdot \|_{nuc}. It is shown that if X^* has the Radon-Nikodym property, then a bounded linear operator T:L^\infty (\mu,X)\rightarrow Y is a \sigma-order continuous nuclear operator between the Banach spaces L^\infty (\mu,X) and Y if and only if its representing measure m_T has a Radon-Nikodym derivative in L^1(\mu,\mathcal {N}(X,Y)) and if and only if T has a Bochner kernel in L^1(\mu,\mathcal {N}(X,Y)). [ABSTRACT FROM AUTHOR]

We prove that the rank of a non-trivial co-doubly commuting submodule is 2. More precisely, let ϕ, ψ ∈ H∞(D) be two inner functions. If Qϕ = H²(D)/ϕH²(D) and Qψ = H²(D)/ψH²(D), then rank (Qϕ × Qψ)⊥ = 2. An immediate consequence is the following: Let S be a co-doubly commuting submodule of H²(D2). Then rank S = 1 if and only if S = ΦH²(D2) for some one variable inner function Φ ∈ H∞(D²). This answers a question posed by R. G. Douglas and R. Yang [Integral Equations Operator Theory 38(2000), pp207-221]. [ABSTRACT FROM AUTHOR]

We study comparison formulas for ζ-regularized determinants of self-adjoint extensions of the Laplacian on flat conical surfaces of genus g ≥ 2. The cases of trivial and non-trivial holonomy of the metric turn out to differ significantly. [ABSTRACT FROM AUTHOR]

We study the initial value problem (*) {CΔαu(n)= Au(n +1),n ∈ N0; u(0) = u0 ∈ X, when A is a closed linear operator with domain D(A) defined on a Banach space X. We introduce a method based on the Poisson distribution to show existence and qualitative properties of solutions for the problem (*), using operator-theoretical conditions on A. We show how several properties for fractional differences, including their own definition, are connected with the continuous case by means of sampling using the Poisson distribution. We prove necessary conditions for stability of solutions, that are only based on the spectral properties of the operator A in the case of Hilbert spaces. [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 note we give a simple sufficient condition for an affine iterated function system to admit an invariant affine subspace persistently with respect to changes in the translation parameters. This yields further examples of tuples of contracting linear maps which do not satisfy the conclusions of Falconer's theorem on the Hausdorff dimension of almost every self-affine set. We also obtain new examples of iterated function systems of similarity transformations which cannot satisfy the open set condition for any choice of translation parameters, and resolve a related question of Peres and Solomyak. [ABSTRACT FROM AUTHOR]

We prove that if Banach lattices E and F are reflexive and each positive linear operator from E to F is compact then {\mathcal L}^r(E;F), the space of all regular linear operators from E to F, is reflexive. Conversely, if E^\ast or F has the bounded regular approximation property then the reflexivity of {\mathcal L}^r(E;F) implies that each positive linear operator from E to F is compact. Analogously we also study the reflexivity for the space of regular multilinear operators on Banach lattices. [ABSTRACT FROM AUTHOR]

Given a closed semi-algebraic set X \subset \mathbb {R}^n and a continuous semi-algebraic mapping G \colon X \to \mathbb {R}^m, it will be shown that there exists an open dense semi-algebraic subset \mathscr {U} of L(\mathbb {R}^n, \mathbb {R}^m), the space of all linear mappings from \mathbb {R}^n to \mathbb {R}^m, such that for all F \in \mathscr {U}, the image (F + G)(X) is a closed (semi-algebraic) set in \mathbb {R}^m. To do this, we study the tangent cone at infinity C_\infty X and the set E_\infty X \subset C_\infty X of (unit) exceptional directions at infinity of X. Specifically we show that the set E_\infty X is nowhere dense in C_\infty X \cap \mathbb {S}^{n - 1}. [ABSTRACT FROM AUTHOR]

We prove that the projective tensor product of two C(K) spaces, where K is an infinite countable metric compact space, is c0-saturated and is therefore a subprojective space. This completes some recent work on subprojectivity of projective tensor products involving C(K) spaces by T. Oikhberg and E. Spinu. [ABSTRACT FROM AUTHOR]

In a case study on asymptotics of spectral quantities of Schrödinger operators in fractional Sobolev spaces on the circle we show how a nonlinear version of the Riesz-Thorin theorem on the interpolation of linear operators can be applied. [ABSTRACT FROM AUTHOR]

An algebra of bounded linear operators on a Banach space is said to be strongly compact provided that its unit ball is precompact in the strong operator topology, and a bounded linear operator on a Banach space is said to be strongly compact provided that the algebra with identity generated by the operator is strongly compact. Our interest in this notion stems from the work of Lomonosov on the existence of invariant subspaces. We consider a local spectral condition that is sufficient for a bounded linear operator on a Banach space to be strongly compact. This condition is then applied to describe a large class of strongly compact, injective bilateral weighted shifts on Hilbert spaces, extending earlier work of Fernàndez-Valles and the first author. Further applications are also derived; for instance, a strongly compact, invertible bilateral weighted shift is constructed in such a way that its inverse fails to be a strongly compact operator. [ABSTRACT FROM AUTHOR]

We prove a generalization of the Jordan canonical form theorem for a class of bounded linear operators on complex separable Hilbert spaces [ABSTRACT FROM AUTHOR]

The purpose of this paper is to establish a sufficient and necessary condition on weight functions which ensures the boundedness of the commutators of weighted Hardy operators (with symbols in BMO$(mathbb {R}^{n})$) on $L^{p}(mathbb {R}^{n})$, where $1

Let $H$ be a complex Hilbert space. The symbol $A# B$ stands for the geometric mean of the positive bounded linear operators $A,B$ on $H$ in the sense of Ando. In this paper we describe the general form of all automorphisms of the set of positive operators with respect to the operation $#$. We prove that if $dim Hgeq 2$, any such transformation is implemented by an invertible bounded linear or conjugate-linear operator on $H$. [ABSTRACT FROM AUTHOR]

Results on factorization (through linear operators) of polynomials and holomorphic mappings between Banach spaces have been obtained in recent years by several authors. In the present paper, we obtain a factorization result for differentiable mappings through compact operators. Namely, we prove that a mapping $f:Xto Y$ between real Banach spaces is differentiable and its derivative $f'$ is a compact mapping with values in the space ${mathcal K}(X,Y)$ of compact operators from $X$ into $Y$ if and only if $f$ may be written in the form $f=gcirc S$, where the intermediate space is normed, $S$ is a precompact operator, and $g$ is a Gâteaux differentiable mapping with some additional properties. We also show that if $f'$ is uniformly continuous on bounded sets and takes values in ${mathcal K}(X,Y)$, then $f'$ is compact if and only if $f$ is weakly uniformly continuous on bounded sets. [ABSTRACT FROM AUTHOR]

Let $(E) \subset (L^2) \subset (E)^*$ be the canonical framework of white noise analysis over the Gel'fand triple $S({\mathbb R}) \subset L^2({\mathbb R}) \subset S^*({\mathbb R})$ and ${\mathcal L} \equiv {\mathcal L}[(E),(E)^*]$ be the space of continuous linear operators from $(E)$ to $(E)^*$. Let $Q$ be a self-adjoint operator in $(L^2)$ with spectral representation $Q = \int_{\mathbb R}\lambdaP_Q(d\lambda)$. In this paper, it is proved that under appropriate conditions upon $Q$, there exists a unique linear mapping $Z:S^*({\mathbb R}) \longmapsto {\mathcal L}$ such that $Z(f)=\int_{\mathbb R}f(\lambda)P_Q(d\lambda)$ for each $f \in S({\mathbb R})$. The mapping is then naturally used to define $\delta(Q)$ as $Z(\delta)$, where $\delta$ is the Dirac $\delta$-function. Finally, properties of the mapping $Z$ are investigated and several results are obtained. [ABSTRACT FROM AUTHOR]

This paper extends the definition of the Cayley transform defined for symmetric linear operators to any linear relation. [ABSTRACT FROM AUTHOR]

In this paper, relationships among the concepts, majorization, range inclusion, and factorization, are studied in a general setting for bounded linear operators. Some applications of these concepts are given. [ABSTRACT FROM AUTHOR]

Of concern are some simple criteria about the convergence of sequences of positive linear operators and functionals in the framework of spaces of bounded functions which are continuous on a given subset of their domain. Among other things some applications concerning the behaviour of the iterates of Bernstein operators defined both on [0,1] and on the d-dimensional simplex and hypercube (d ≥ 1) are discussed. A final section treats the behaviour of integrated arithmetic means with respect to a probability Borel measure on a convex compact subset K of a locally convex space. As a consequence a general weak law of large numbers for sequences of K-valued random variables is derived. [ABSTRACT FROM AUTHOR]

A linear operator in a separable Hilbert space is called a quasinormal one if it is a sum of a normal operator and a compact one. In the paper, bounds for the spectrum of quasinormal operators are established. In addition, the lower estimate for the spectral radius is derived. Under some restrictions, that estimate improves the well-known results. Applications to integral operators and matrices are discussed. Our results are new even in the finite-dimensional case. [ABSTRACT FROM AUTHOR]