Multiplication Operators Research Articles

Nonparametric estimation of the fragmentation kernel based on a partial differential equation stationary distribution approximation

AbstractWe consider a stochastic individual‐based model in continuous time to describe a size‐structured population for cell divisions. This model is motivated by the detection of cellular aging in biology. We here address the problem of nonparametric estimation of the kernel ruling the divisions based on the eigenvalue problem related to the asymptotic behavior in large population. This inverse problem involves a multiplicative deconvolution operator. Using Fourier techniques we derive a nonparametric estimator whose consistency is studied. The main difficulty comes from the nonstandard equations connecting the Fourier transforms of the kernel and the parameters of the model. A numerical study is carried out and we pay special attention to the derivation of bandwidths by using resampling.

Multiplicator type operators and approximation of periodic functions of one variable by trigonometric polynomials

The norms of the images of multiplier type operators generated by an arbitrary generator are estimated in terms of the best approximations of univariate periodic functions by trigonometric polynomials in the -spaces, . As corollaries, estimates for the quality of approximation by Fourier means, an inverse theorem of approximation theory, comparison theorems, an analogue of the Marchaud inequality for generalized moduli of smoothness defined by a periodic generator, as well as some constructive sufficient conditions for generalized smoothness and Bernstein type inequalities for generalized derivatives of trigonometric polynomials are obtained. Bibliography: 49 titles.

Gelfand transforms and boundary representations of complete Nevanlinna–Pick quotients

The main objects under study are quotients of multiplier algebras of certain complete Nevanlinna–Pick spaces, examples of which include the Drury–Arveson space on the ball and the Dirichlet space on the disc. We are particularly interested in the non-commutative Choquet boundaries for these quotients. Arveson’s notion of hyperrigidity is shown to be detectable through the essential normality of some natural multiplication operators, thus extending previously known results on the Arveson–Douglas conjecture. We also highlight how the non-commutative Choquet boundaries of these quotients are intertwined with their Gelfand transforms being completely isometric. Finally, we isolate analytic and topological conditions on the so-called supports of the underlying ideals that clarify the nature of the non-commutative Choquet boundaries.

*-Controlled frames in Hilbert C*-modules

The main purpose of this paper is to introduce the notion of *-T-controlled frames in Hilbert [Formula: see text]-modules. We present some results of frames in the view of *-T-controlled frames in Hilbert [Formula: see text]-modules. Also, we define *-w-frames and give their relations to *-T-controlled frames and multiplier operators.

Multiplication Operators on Orlicz Generalized Difference (sss)

In this article, we inspect the sufficient conditions on the Orlicz generalized difference sequence space to be premodular Banach (sss). We look at some topological and geometrical structures of the multiplication operators described on Orlicz generalized difference prequasi normed (sss).

The essential norm of multiplication operators acting on Orlicz sequence spaces

We calculate the measure of non-compactness or the essential norm of the multiplication operator Mu acting on Orlicz sequence spaces lφ. As a consequence of our result, we obtain a known criteria for the compactness of multiplication operator acting on lφ.

Operator-valued Fourier–Haar multipliers on vector-valued $${L^1}$$ spaces II: a characterisation of finite dimensionality

A necessary and sufficient condition is given to ensure the boundedness of Fourier–Haar multiplier operators from $$L^1 ([0, 1], X)$$ to $$L^1 ([0, 1], Y)$$ , where X is an arbitrary finite dimensional Banach space and Y is an arbitrary Banach space. The Fourier–Haar multiplier sequences come not from $${\mathbb {R}}$$ , as in the classical case, but from the space of bounded operators from the Banach space X to the Banach space Y. Moreover, it is shown that this condition characterises the finite dimensionality of the Banach space X.

The commutant and invariant subspaces for dual truncated Toeplitz operators

Dual truncated Toeplitz operators on the orthogonal complement of the model space $$K_u^2(=H^2 \ominus uH^2)$$ with u nonconstant inner function are defined to be the compression of multiplication operators to the orthogonal complement of $$K_u^2$$ in $$L^2$$ . In this paper, we give a complete characterization of the commutant of dual truncated Toeplitz operator $$D_z$$ , and we even obtain the commutant of all dual truncated Toeplitz operators with bounded analytic symbols. Moreover, we describe the nontrival invariant subspaces of $$D_z$$ .

A Note on the L^p Integrability of a Class of Bochner\u2013Riesz Kernels

For a general compact variety Gamma of arbitrary codimension, one can consider the L^p mapping properties of the Bochner–Riesz multiplier mΓ,α(ζ)=dist(ζ,Γ)αϕ(ζ)\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} m_{\\Gamma , \\alpha }(\\zeta ) \\ = \\ \\mathrm{dist}(\\zeta , \\Gamma )^{\\alpha } \\phi (\\zeta ) \\end{aligned}$$\\end{document}where alpha > 0 and phi is an appropriate smooth cutoff function. Even for the sphere Gamma = {{mathbb {S}}}^{N-1}, the exact L^p boundedness range remains a central open problem in Euclidean harmonic analysis. In this paper we consider the L^p integrability of the Bochner–Riesz convolution kernel for a particular class of varieties (of any codimension). For a subclass of these varieties the range of L^p integrability of the kernels differs substantially from the L^p boundedness range of the corresponding Bochner–Riesz multiplier operator.

Multiplication operators between different Wiener-type variation spaces

In this article, we characterize all of the multipliers u which define continuous, invertible, finite and closed range, compact, and Fredholm multiplication operators $$M_{u}$$ acting on two different spaces of functions of bounded p-variation in the sense of Wiener.

Operators acting in sequence spaces generated by Dual Banach spaces of discrete Cesàro spaces

The dual spaces $d(p), 1 \lt p \lt \infty ,$ of the discrete Cesaro (Banach) spaces $\mathrm{ces} (q)$, $1 \lt q \lt \infty ,$ were studied by G. Bennett, A. Jagers and others. These (reflexive) dual Banach spaces induce the non-normable Frechet spaces $d (p+) := \bigcap_{r \gt p} d (r), $ for $1 \leq p \lt \infty ,$ and the (LB)-spaces $d (p-) := \bigcup_{ 1 \lt r \lt p } d (r), $ for $ 1 \lt p \leq \infty ,$ recently introduced and investigated in [11]. Here a detailed study is made of various aspects, such as the spectrum, continuity, compactness, mean ergodicity and supercyclicity of the Cesaro operator, multiplication operators and inclusion operators when they act on (and between) such spaces.

Semiring optimizations: dynamic elision of expressions with identity and absorbing elements

This paper describes a compiler optimization to eliminates dynamic occurrences of expressions in the format a ← a ⊕ b ⊗ c . The operation ⊕ must admit an identity element z , such that a ⊕ z = a . Also, z must be the absorbing element of ⊗, such that b ⊗ z = z ⊗ c = z . Semirings where ⊕ is the additive operator and ⊗ is the multiplicative operator meet this contract. This pattern is common in high-performance benchmarks—its canonical representative being the multiply-add operation a ← a + b × c . However, several other expressions involving arithmetic and logic operations satisfy the required algebra. We show that the runtime elimination of such assignments can be implemented in a performance-safe way via online profiling. The elimination of dynamic redundancies involving identity and absorbing elements in 35 programs of the LLVM test suite that present semiring patterns brings an average speedup of 1.19x (total optimized time over total unoptimized time) on top of clang -O3. When projected onto the entire test suite (259 programs) the optimization leads to a speedup of 1.025x. Once added onto clang, semiring optimizations approximates it to TACO, a specialized tensor compiler.

Estimates for [formula omitted]-widths of sets of smooth functions on complex spheres

In this work we investigate n-widths of multiplier operators Λ∗ and Λ, defined for functions on the complex sphere Ωd of ℂd, associated with sequences of multipliers of the type {λm,n∗}m,n∈N, λm,n∗=λ(m+n) and {λm,n}m,n∈N, λm,n=λ(max{m,n}), respectively, for a bounded function λ defined on [0,∞). If the operators Λ∗ and Λ are bounded from Lp(Ωd) into Lq(Ωd), 1≤p,q≤∞, and Up is the closed unit ball of Lp(Ωd), we study lower and upper estimates for the n-widths of Kolmogorov, linear, of Gelfand and of Bernstein, of the sets Λ∗Up and ΛUp in Lq(Ωd). As application we obtain, in particular, estimates for the Kolmogorov n-width of classes of Sobolev, of finitely differentiable, infinitely differentiable and analytic functions on the complex sphere, in Lq(Ωd), which are order sharp in various important situations.

A note on Nakano generalized difference sequence space

In this paper, we investigate the necessary conditions on any s-type sequence space to form an operator ideal. As a result, we show that the s-type Nakano generalized difference sequence space X fails to generate an operator ideal. We investigate the sufficient conditions on X to be premodular Banach special space of sequences and the constructed prequasi-operator ideal becomes a small, simple, and closed Banach space and has eigenvalues identical with its s-numbers. Finally, we introduce necessary and sufficient conditions on X explaining some topological and geometrical structures of the multiplication operator defined on X.

Sublinear operators on mixed-norm Hardy spaces with variable exponents

In this paper, we define and study the mixed-norm Hardy spaces with variable exponents. We establish some general principles for the mapping properties of sublinear operators on the mixed-norm Hardy spaces with variable exponents. By using these principles, we obtain the mapping properties of the Calderón–Zygmund operators, the oscillatory singular integral operators, the multiplier operators, the Littlewood–Paley functions, the intrinsic square functions, the parametric Marcinkiewicz integrals and the maximal Bochner–Riesz means on the mixed-norm Hardy spaces with variable exponents.

Remarks on complex symmetric Toeplitz operators

ABSTRACT In this paper, we give an alternative characterization of complex symmetry of Toeplitz operators and block Toeplitz operators. In particular, we prove that a block Toeplitz operator is complex symmetric with the conjugation on the vector-valued Hardy space if and only if , where denotes the multiplication operator on with symbol Φ. As some applications, we provide examples of normal (resp., non-normal) complex symmetric block Toeplitz operators.

Smoothness of functions versus smoothness of approximation processes

We provide a comprehensive study of interrelations between different measures of smoothness of functions on various domains and smoothness properties of approximation processes. Two general approaches to this problem have been developed: The first based on geometric properties of Banach spaces and the second on Littlewood–Paley and Hörmander-type multiplier theorems. In particular, we obtain new sharp inequalities for measures of smoothness given by the [Formula: see text]-functionals or moduli of smoothness. As examples of approximation processes we consider best polynomial and spline approximations, Fourier multiplier operators on [Formula: see text], [Formula: see text], [Formula: see text], nonlinear wavelet approximation, etc.

Structure of elementary operators defining m-left invertible, m-selfadjoint and related classes of operators

We use elementary, algebraic properties of left, right multiplication operators to prove some deep structural properties of left m-invertible, m-isometric, m-selfadjoint and other related classes of Banach space operators, often adding value to extant results.

Weighted composition operator on quaternionic Fock space

This paper is concerned with several important properties of weighted composition operator acting on the quaternionic Fock space $${\mathcal {F}}^2({\mathbb {H}})$$ . Complete equivalent characterizations for its boundedness and compactness are established. As corollaries, the descriptions for composition operator and multiplication operator on $${\mathcal {F}}^2({\mathbb {H}})$$ are presented, which can indicate some well-known existing theories in complex Fock space. Finally, as an appendix the closed expression for the kernel function of $${\mathcal {F}}^2({\mathbb {H}})$$ is exhibited, which can deepen the understanding of $${\mathcal {F}}^2({\mathbb {H}})$$ .

Fourier Transform on the Lobachevsky Plane and Operational Calculus

The classical Fourier transform on the line sends the operator of multiplication by $x$ to $i\frac{d}{d\xi}$ and the operator of differentiation $\frac{d}{d x}$ to the multiplication by $-i\xi$. For the Fourier transform on the Lobachevsky plane we establish a similar correspondence for a certain family of differential operators. It appears that differential operators on the Lobachevsky plane correspond to differential-difference operators in the Fourier-image, where shift operators act in the imaginary direction, i.e., a direction transversal to the integration contour in the Plancherel formula.