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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]

A recent paper of A. Connes and W.D. van Suijlekom [Comm. Math. Phys. 383 (2021), pp. 2021–2067] identifies the operator system of n\times n Toeplitz matrices with the dual of the space of all trigonometric polynomials of degree less than n. The present paper examines this identification in somewhat more detail by showing explicitly that the Connes–van Suijlekom isomorphism is a unital complete order isomorphism of operator systems. Applications include two special results in matrix analysis: (i) that every positive linear map of the n × n complex matrices is completely positive when restricted to the operator subsystem of Toeplitz matrices and (ii) that every linear unital isometry of the n\times n Toeplitz matrices into the algebra of all n\times n complex matrices is a unitary similarity transformation. An operator systems approach to Toeplitz matrices yields new insights into the positivity of block Toeplitz matrices, which are viewed herein as elements of tensor product spaces of an arbitrary operator system with the operator system of n\times n complex Toeplitz matrices. In particular, it is shown that min and max positivity are distinct if the blocks themselves are Toeplitz matrices, and that the maximally entangled Toeplitz matrix ξn generates an extremal ray in the cone of all continuous n × n Toeplitz-matrix valued functions ƒ on the unit circle S1 whose Fourier coefficients Ƕ(k) vanish for |k| ≥ n. Lastly, it is noted that all positive Toeplitz matrices over nuclear C*-algebras are approximately separable. [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]

This paper is concerned with the algebraic and analytic q-de Rham complexes attached to linear q-difference operators with Laurent polynomial coefficients over the field of complex numbers. There is a natural morphism from the former to the latter complex. Whether or not it is a quasi-isomorphism, i.e. , whether or not the induced morphisms on the corresponding cohomology spaces are isomorphisms, is the basic question considered in the present paper. We study this question following three distinct approaches. The first one is based on duality, and leads to a direct connection between the problem considered in the present paper and the convergence of formal series solutions of q-difference equations. The second approach is sheaf theoretic, based on growth considerations. The third one relies on the local analytic theory of q-difference equations. The paper ends with an extension of our results to variants of the above q-de Rham complexes when certain q-spirals of poles are allowed. Our study includes the case |q| = 1. [ABSTRACT FROM AUTHOR]

We discuss various questions of the following kind: for a continuous map X \to Y from a compact metric space to a simplicial complex, can one guarantee the existence of a fiber large in the sense of Urysohn width? The d-width measures how well a space can be approximated by a d-dimensional complex. The results of this paper include the following. Any piecewise linear map f: [0,1]^{m+2} \to Y^m from the unit euclidean (m+2)-cube to an m-polyhedron must have a fiber of 1-width at least \frac {1}{2\beta m +m^2 + m + 1}, where \beta = \sup _{y\in Y} rkH_1(f^{-1}(y)) measures the topological complexity of the map. There exists a piecewise smooth map X^{3m+1} \to \mathbb {R}^m, with X a riemannian (3m+1)-manifold of large 3m-width, and with all fibers being topological (2m+1)-balls of arbitrarily small (m+1)-width. [ABSTRACT FROM AUTHOR]

Let X be a separable infinite dimensional (real or complex) Banach space. Hájek and Smith, in 2010, constructed a linear bounded operator T on X such that AT ≔ {x ∈ X : ||Tnx|| → ∞} is not dense and has nonempty interior, whenever X admits a symmetric basis. Augé, in 2012, extended the previous construction to each separable Banach space, introducing the notion of wild operators. This work is divided in two parts. In the first part we define and explore the notion of asymptotically separated sets on Banach spaces, giving several examples whenever the ambient space has either finite or infinite dimension. We show how these sets can be used to construct operators with non-intuitive dynamics. Specifically, operators T : X → X for which the set AT and the set of recurrent points form a partition of the space. Secondly, we investigate geometric and spectral properties of operators with wild dynamic and we provide a wild operator whose spectrum is equal to the closed unit disk. We end this paper giving some comments about the norm closure of the set of wild operators. [ABSTRACT FROM AUTHOR]

In this paper we study the existence of periodic solutions to the partial functional differential equation {dy(t)/dt = By(t) + L(yt) + ƒ(t,yt), ∀ t ≥ 0, y0 = φ ∈ CB where B: Y \rightarrow Y is a Hille-Yosida operator on a Banach space Y. For CB ≔ {φ ∈ C([−r,0];Y): φ (0) ∈ D(B), yt ∈ CB is defined by yt(θ) = y(t + θ), θ∈ [−r,0], L: CB → Y is a bounded linear operator, and f:\mathbb {R}\times CB → Y is a continuous map and is T-periodic in the time variable t. Sufficient conditions on B, L and ƒ(t,yt) are given to ensure the existence of T-periodic solutions. The results then are applied to establish the existence of periodic solutions in a reaction-diffusion equation with time delay and the diffusive Nicholson's blowflies equation. [ABSTRACT FROM AUTHOR]

We show how a number of well-known uncertainty principles for the Fourier transform, such as the Heisenberg uncertainty principle, the Donoho-Stark uncertainty principle, and Meshulam's nonabelian uncertainty principle, have little to do with the structure of the Fourier transform itself. Rather, all of these results follow from very weak properties of the Fourier transform (shared by numerous linear operators), namely that it is bounded as an operator L1 → L∞, and that it is unitary. Using a single, simple proof template, and only these (or weaker) properties, we obtain some new proofs and many generalizations of these basic uncertainty principles, to new operators and to new settings, in a completely unified way. Together with our general overview, this paper can also serve as a survey of the many facets of the phenomena known as uncertainty principles. [ABSTRACT FROM AUTHOR]

A new proof is presented of a theorem of L. Gurvits [LANL Unclassified Technical Report (2001), LAUR–01–2030], which states that the cone of positive block-Toeplitz matrices with matrix entries has no entangled elements. The proof of the Gurvits separation theorem is achieved by making use of the structure of the operator system dual of the operator system C(S^1)^{(n)} of n\times n Toeplitz matrices over the complex field, and by determining precisely the structure of the generators of the extremal rays of the positive cones of the operator systems C(S^1)^{(n)}\otimes _{\text {min}}\mathcal {B}(\mathcal {H}) and C(S^1)_{(n)}\otimes _{\text {min}}\mathcal {B}(\mathcal {H}), where \mathcal {H} is an arbitrary Hilbert space and C(S^1)_{(n)} is the operator system dual of C(S^1)^{(n)}. Our approach also has the advantage of providing some new information concerning positive Toeplitz matrices whose entries are from \mathcal {B}(\mathcal {H}) when \mathcal {H} has infinite dimension. In particular, we prove that normal positive linear maps \psi on \mathcal {B}(\mathcal {H}) are partially completely positive in the sense that \psi ^{(n)}(x) is positive whenever x is a positive n\times n Toeplitz matrix with entries from \mathcal {B}(\mathcal {H}). We also establish a certain factorisation theorem for positive Toeplitz matrices (of operators), showing an equivalence between the Gurvits approach to separation and an earlier approach of T. Ando [Acta Sci. Math. (Szeged) 31 (1970), pp. 319–334] to universality. [ABSTRACT FROM AUTHOR]

This paper presents a posteriori error estimates for conforming numerical approximations of eigenvalue clusters of second-order self-adjoint elliptic linear operators with compact resolvent. Given a cluster of eigenvalues, we estimate the error in the sum of the eigenvalues, as well as the error in the eigenvectors represented through the density matrix, i.e., the orthogonal projector on the associated eigenspace. This allows us to deal with degenerate (multiple) eigenvalues within the framework. All the bounds are valid under the only assumption that the cluster is separated from the surrounding smaller and larger eigenvalues; we show how this assumption can be numerically checked. Our bounds are guaranteed and converge with the same speed as the exact errors. They can be turned into fully computable bounds as soon as an estimate on the dual norm of the residual is available, which is presented in two particular cases: the Laplace eigenvalue problem discretized with conforming finite elements, and a Schrödinger operator with periodic boundary conditions of the form −Δ + V discretized with planewaves. For these two cases, numerical illustrations are provided on a set of test problems. [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]

The class of Hölder-Brascamp-Lieb inequalities is a collection of multilinear inequalities generalizing a convolution inequality of Young and the Loomis-Whitney inequalities. The full range of exponents was classified in a paper of Bennett, Carbery, Christ, and Tao [Math. Res. Lett. 17 (2010), no. 4, 647-666]. In a setting similar to that of Ivanisvili and Volberg [J. Lond. Mat. Sci (2) (2015), no. 3, 657-674], we introduce a notion of size for these inequalities which generalizes Lp norms. Under this new setup, we then determine necessary and sufficient conditions for a generalized Hölder-Brascamp-Lieb-type inequality to hold and establish sufficient conditions for maximizers to exist when the underlying linear maps match those of the convolution inequality of Young. [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]

The theory of quaternionic operators has applications in several different fields, such as quantum mechanics, fractional evolution problems, and quaternionic Schur analysis, just to name a few. The main difference between complex and quaternionic operator theory is based on the definition of a spectrum. In fact, in quaternionic operator theory the classical notion of a resolvent operator and the one of a spectrum need to be replaced by the two S-resolvent operators and the S-spectrum. This is a consequence of the noncommutativity of the quaternionic setting. Indeed, the S-spectrum of a quaternionic linear operator T is given by the noninvertibility of a second order operator. This presents new challenges which make our approach to perturbation theory of quaternionic operators different from the classical case. In this paper we study the problem of perturbation of a quaternionic normal operator in a Hilbert space by making use of the concepts of S-spectrum and of slice hyperholomorphicity of the S-resolvent operators. For this new setting we prove results on the perturbation of quaternionic normal operators by operators belonging to a Schatten class and give conditions which guarantee the existence of a nontrivial hyperinvariant subspace of a quaternionic linear operator. [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]

We work over an algebraically closed field of characteristic zero. The purpose of this paper is to characterize a nondegenerate projective variety X with a linear projection which induces a nonbirational map to its image. As an application, for smooth X of degree d and codimension e, we prove the "semiampleness" of the (d - e + 1)th twist of the ideal sheaf. This improves a linear bound of the regularity of smooth projective varieties by Bayer-Mumford-Bertram-Ein-Lazarsfeld, and gives an asymptotic regularity bound. [ABSTRACT FROM AUTHOR]

A power series being given as the solution of a linear differential equation with appropriate initial conditions, minimization consists in finding a non-trivial linear differential equation of minimal order having this power series as a solution. This problem exists in both homogeneous and inhomogeneous variants; it is distinct from, but related to, the classical problem of factorization of differential operators. Recently, minimization has found applications in Transcendental Number Theory, more specifically in the computation of non-zero algebraic points where Siegel's E-functions take algebraic values. We present algorithms and implementations for these questions, and discuss examples and experiments. [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]

We study the differences of composition operators acting on weighted Bergman spaces over the upper half-plane. In this setting not all composition operators are bounded and none are compact. The idea of joint pullback measure is used to give a Carleson measure characterization of when the difference of two composition operators is bounded or compact. Alternate characterizations, not using Carleson measures, are also given for certain large classes of the inducing maps for the operators. The relationship between angular derivatives and compact differences of composition operators is also explored, which, in particular, reveals a new phenomenon due to the upper half-plane not being bounded. Our results produce a variety of examples of distinct composition operators whose difference is compact, including examples when the individual operators are not bounded. The paper closes with a characterization of when the difference of composition operators is Hilbert- Schmidt. [ABSTRACT FROM AUTHOR]

The matrix Sylvester equation for congruence, or T-Sylvester equation, has recently attracted considerable attention as a consequence of its close relation to palindromic eigenvalue problems. The theory concerning TSylvester equations is rather well understood, and there are stable and efficient numerical algorithms which solve these equations for small- to medium-sized matrices. However, developing numerical algorithms for solving large-scale TSylvester equations still remains an open problem. In this paper, we present several projection algorithms based on different Krylov spaces for solving this problem when the right-hand side of the T-Sylvester equation is a low-rank matrix. The new algorithms have been extensively tested, and the reported numerical results show that they work very well in practice, offering clear guidance on which algorithm is the most convenient in each situation. [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]

The semi-invertible version of Oseledets' multiplicative ergodic theorem providing a decomposition of the underlying state space of a random linear dynamical system into fast and slow spaces is deduced for a strongly measurable cocycle on a separable Banach space. This work is a much shorter means of obtaining this general version of the theorem, using measurable growth estimates on subspaces for linear operators combined with a modified version of Kingman's subadditive ergodic theorem. [ABSTRACT FROM AUTHOR]

We continue the study of low dimensional linear representations of mapping class groups of surfaces initiated by Franks–Handel [Proc. Amer. Math. So. 141 (2013), pp. 2951–2962] and Korkmaz [ Low-dimensional linear representations of mapping class groups , preprint, arXiv:1104.4816v2 (2011)]. We consider (2g+1)-dimensional complex linear representations of the pure mapping class groups of compact orientable surfaces of genus g. We give a complete classification of such representations for g \geq 7 up to conjugation, in terms of certain twisted 1-cohomology groups of the mapping class groups. A new ingredient is to use the computation of a related twisted 1-cohomology group by Morita [Ann. Inst. Fourier (Grenoble) 39 (1989), pp. 777–810]. The classification result implies in particular that there are no irreducible linear representations of dimension 2g+1 for g \geq 7, which marks a contrast with the case g=2. [ABSTRACT FROM AUTHOR]

We study the existence of the stochastic flow associated to a linear stochastic evolution equation \begin{equation*} \operatorname {d} X= AX\operatorname {d} t +\sum _{k} B_k X\operatorname {d} W_k, \end{equation*} on a Hilbert space. Our first result covers the case where A is the generator of a C_0-semigroup, and (B_k) is a sequence of bounded linear operators such that \sum _k\|B_k\|<+\infty. We also provide sufficient conditions for the existence of stochastic flows in the Schatten classes beyond the space of Hilbert–Schmidt operators. Some new results and examples concerning the so-called commutative case are presented as well. [ABSTRACT FROM AUTHOR]

This paper has two parts: on Baumslag--Solitar groups and on general G--trees. In the first part we establish bounds for stable commutator length (scl) in Baumslag--Solitar groups. For a certain class of elements, we further show that scl is computable and takes rational values. We also determine exactly which of these elements admit extremal surfaces. In the second part we establish a universal lower bound of 1/12 for scl of suitable elements of any group acting on a tree. This is achieved by constructing efficient quasimorphisms. Calculations in the group BS(2, 3) show that this is the best possible universal bound, thus answering a question of Calegari and Fujiwara. We also establish scl bounds for acylindrical tree actions. Returning to Baumslag--Solitar groups, we show that their scl spectra have a uniform gap: no element has scl in the interval (0,1/12). [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]

The main result of this paper is a bi-parameter T(b) theorem for the case that b is a tensor product of two pseudo-accretive functions. In the proof, we also discuss the L² boundedness of different types of the b-adapted bi-parameter paraproducts. [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 describe a new method of defining C*-algebras from oriented combinatorial data, thereby generalizing the construction of algebras from directed graphs, higher-rank graphs, and ordered groups. We show that only the most elementary notions of concatenation and cancellation of paths are required to define versions of Cuntz-Krieger and Toeplitz-Cuntz- Krieger algebras, and the presentation by generators and relations follows naturally. We give sufficient conditions for the existence of an AF core, hence of the nuclearity of the C*-algebras, and for aperiodicity, which is used to prove the standard uniqueness theorems. [ABSTRACT FROM AUTHOR]

Consider the Vlasov–Poisson–Landau system with Coulomb potential in the weakly collisional regime on a 3-torus, i.e. \begin{align*} \partial _t F(t,x,v) + v_i \partial _{x_i} F(t,x,v) + E_i(t,x) \partial _{v_i} F(t,x,v) = \nu Q(F,F)(t,x,v),\\ E(t,x) = \nabla \Delta ^{-1} (\int _{\mathbb R^3} F(t,x,v)\, \mathrm {d} v - {{\int }\llap {-}}_{\mathbb T^3} \int _{\mathbb R^3} F(t,x,v)\, \mathrm {d} v \, \mathrm {d} x), \end{align*} with \nu \ll 1. We prove that for \epsilon >0 sufficiently small (but independent of \nu), initial data which are O(\epsilon \nu ^{1/3})-Sobolev space perturbations from the global Maxwellians lead to global-in-time solutions which converge to the global Maxwellians as t\to \infty. The solutions exhibit uniform-in-\nu Landau damping and enhanced dissipation. Our main result is analogous to an earlier result of Bedrossian for the Vlasov–Poisson–Fokker–Planck equation with the same threshold. However, unlike in the Fokker–Planck case, the linear operator cannot be inverted explicitly due to the complexity of the Landau collision operator. For this reason, we develop an energy-based framework, which combines Guo's weighted energy method with the hypocoercive energy method and the commuting vector field method. The proof also relies on pointwise resolvent estimates for the linearized density equation. [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]

We show that for any separable reflexive Banach space X and a large class of Banach spaces E, including those with a subsymmetric shrinking basis but also all spaces L_p[0,1] for 1\le p \le \infty, every bounded linear map \mathcal {B}(E)\to \mathcal {B}(X) which is approximately multiplicative is necessarily close in the operator norm to some bounded homomorphism \mathcal {B}(E)\to \mathcal {B}(X). That is, the pair (\mathcal {B}(E),\mathcal {B}(X)) has the AMNM property in the sense of Johnson [J. London Math. Soc. (2) 37 (1988), pp. 294–316]. Previously this was only known for E=X=\ell _p with 1

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]

We study a Bishop-Phelps-Bollob´as version of Lindenstrauss properties A and B. For domain spaces, we study Banach spaces X such that (X, Y) has the Bishop-Phelps-Bollob´as property (BPBp) for every Banach space Y. We show that in this case, there exists a universal function ηX(ε) such that for every Y, the pair (X, Y) has the BPBp with this function. This allows us to prove some necessary isometric conditions for X to have the property. We also prove that if X has this property in every equivalent norm, then X is one-dimensional. For range spaces, we study Banach spaces Y such that (X, Y) has the Bishop-Phelps-Bollob´as property for every Banach space X. In this case, we show that there is a universal function ηY (ε) such that for every X, the pair (X, Y) has the BPBp with this function. This implies that this property of Y is strictly stronger than Lindenstrauss property B. The main tool to get these results is the study of the Bishop-Phelps-Bollob´as property for c0-, ℓ1- and ℓ∞-sums of Banach spaces. [ABSTRACT FROM AUTHOR]