Multiplication Operators Research Articles

A formula for symmetry recursion operators from non-variational symmetries of partial differential equations

An explicit formula to find symmetry recursion operators for partial differential equations (PDEs) is obtained from new results connecting variational integrating factors and non-variational symmetries. The formula is special case of a general formula that produces a pre-symplectic operator from a non-gradient adjoint-symmetry. These formulas are illustrated by several examples of linear PDEs and integrable nonlinear PDEs. Additionally, a classification of quasilinear second-order PDEs admitting a multiplicative symmetry recursion operator through the first formula is presented.

THE IONESCU–WAINGER MULTIPLIER THEOREM AND THE ADELES

The Ionescu–Wainger multiplier theorem establishes good L p bounds for Fourier multiplier operators localized to major arcs; it has become an indispensible tool in discrete harmonic analysis. We give a simplified proof of this theorem with more explicit constants (removing logarithmic losses that were present in previous versions of the theorem), and give a more general variant involving adelic Fourier multipliers. We also establish a closely related adelic sampling theorem that shows that ℓ p ( Z d ) norms of functions with Fourier transform supported on major arcs are comparable to the L p ( A Z d ) norm of their adelic counterparts.

Some properties of 2-orthogonal polynomials associated with inverse Gaussian distribution

In this work, we discuss several desirable properties of the inverse Gaussian (IG) family involving orthogonal polynomials. In particular, we discuss the cumulant-generating function and associated polynomials. A particularly important property is that the associated polynomials satisfy a four-term recurrence relation. Then we state and calculate the famous creation and annihilation operators of the Heisenberg-Weyl Lie algebra that acting on IG polynomials Pn as the multiplicative operator X and the derivative one D on the monomials xn . Finally, I used the Maple computer algebra system to program and Calculate the six first IG polynomials and their roots numerically.

Maximal operators associated with bilinear multipliers of limited decay

Results analogous to those proved by Rubio de Francia [28] are obtained for a class of maximal functions formed by dilations of bilinear multiplier operators of limited decay. We focus our attention on L2 × L2 → L1 estimates. We discuss two applications: the boundedness of the bilinear maximal Bochner-Riesz operator and of the bilinear spherical maximal operator. For the latter we improve the known results in [1] by reducing the dimension restriction from n ≥ 8 to n ≥ 4.

Random Walks on Graphs and Approximation of L2-Invariants

In this work, we interpret right multiplication operators $R_{w}: l^{2}(G) \rightarrow l^{2}(G)$ , $w \in \mathbb {C}[G]$ as random walk operators on certain labelled graphs we employ that are analogous to Cayley graphs. Applying a generalization of the graph convergence defined by R. I. Grigorchuk and A. Żuk to these graphs gives a new interpretation and proof of a special case of W. Luck’s famous Theorem on the Approximation of L2-Betti numbers for countable residually finite groups by means of exhausting towers of finite-index subgroups. In particular, using this interpretation, the theorem follows naturally from standard theorems in probability theory concerning the weak convergence of probability measures that are characterized by their moments. This paper is mainly a direct adaptation of the ideas of Grigorchuk, Zuk and Luck to this setting. We aim to explain how these ideas are related and give a short exposition of them.

Bound states of Schrödinger-type operators on one and two dimensional lattices

We study the spectral properties of the Schrödinger-type operatorHˆμ:=Hˆ0+μVˆ,μ≥0, associated to a one-particle system in d-dimensional lattice Zd, d=1,2, where the non-perturbed operator Hˆ0 is a self-adjoint convolution-type operator generated by a Hopping matrix eˆ:Zd→C and the potential Vˆ is the multiplication operator by vˆ:Zd→R. Under certain regularity assumption on eˆ and a decay assumption on vˆ, we establish the existence or non-existence and also the finiteness of eigenvalues of Hˆμ. Moreover, in the case of existence we study the asymptotics of eigenvalues of Hˆμ as μ↘0.

A trace formula for higher order ordinary differential operators

We obtain a first-order trace formula for a higher order differential operator on a closed interval in the case where the perturbation operator is the operator of multiplication by a finite complex-valued charge. For operators of even orders , the result contains a term of new type, previously unknown. Bibliography: 15 titles.

Embedding and Volterra integral operators from Dirichlet-Morrey spaces into general function spaces

In this paper, the boundedness and compactness of embedding from Dirichlet-Morrey spaces into tent spaces are investigated. As an application, we characterize the boundedness of the Volterra integral operator from to general function spaces . Meanwhile, the operator and the multiplication operator from to are studied. Furthermore, the essential norm of and from to are also considered.

A Description of Commutant of Multiplication Operators on Bergman Spaces of the Polydisk

A Description of Commutant of Multiplication Operators on Bergman Spaces of the Polydisk

Operators on Orlicz-slice spaces and Orlicz-slice Hardy spaces

We establish the mapping properties of the singular integral operators, the Fourier integral operators and the geometric maximal operators on the Orlicz-slice spaces by using extrapolation. By extending the extrapolation theory to the Hardy Orlicz-slice spaces, we also obtain the mapping properties of the maximal Bochner-Riesz means, the parametric Marcinkiewicz integrals and the multiplier operators on the Hardy Orlicz-slice spaces.

Higher order local Dirichlet integrals and de Branges-Rovnyak spaces

We investigate expansive Hilbert space operators T that are finite rank perturbations of isometric operators. If the spectrum of T is contained in the closed unit disc D‾, then such operators are of the form T=U⊕R, where U is isometric and R is unitarily equivalent to the operator of multiplication by the variable z on a de Branges-Rovnyak space H(B). In fact, the space H(B) is defined in terms of a rational operator-valued Schur function B. In the case when dim⁡ker⁡T⁎=1, then H(B) can be taken to be a space of scalar-valued analytic functions in D, and the function B has a mate a defined by |B|2+|a|2=1 a.e. on ∂D. We show the mate a of a rational B is of the form a(z)=a(0)p(z)q(z), where p and q are appropriately derived from the characteristic polynomials of two associated operators. If T is a 2m-isometric expansive operator, then all zeros of p lie in the unit circle, and we completely describe the spaces H(B) by use of what we call the local Dirichlet integral of order m at the point w∈∂D.

The Quantization of Gravity: Quantization of the Hamilton Equations

We quantize the Hamilton equations instead of the Hamilton condition. The resulting equation has the simple form −Δu=0 in a fiber bundle, where the Laplacian is the Laplacian of the Wheeler–DeWitt metric provided n≠4. Using then separation of variables, the solutions u can be expressed as products of temporal and spatial eigenfunctions, where the spatial eigenfunctions are eigenfunctions of the Laplacian in the symmetric space SL(n,R)/SO(n). Since one can define a Schwartz space and tempered distributions in SL(n,R)/SO(n) as well as a Fourier transform, Fourier quantization can be applied such that the spatial eigenfunctions are transformed to Dirac measures and the spatial Laplacian to a multiplication operator.

Self-Similar Cauchy Problems and Generalized Mittag-Leffler Functions

Abstract By observing that the fractional Caputo derivative of order α ∈ (0, 1) can be expressed in terms of a multiplicative convolution operator, we introduce and study a class of such operators which also have the same self-similarity property as the Caputo derivative. We proceed by identifying a subclass which is in bijection with the set of Bernstein functions and we provide several representations of their eigenfunctions, expressed in terms of the corresponding Bernstein function, that generalize the Mittag-Leffler function. Each eigenfunction turns out to be the Laplace transform of the right-inverse of a non-decreasing self-similar Markov process associated via the so-called Lamperti mapping to this Bernstein function. Resorting to spectral theoretical arguments, we investigate the generalized Cauchy problems, defined with these self-similar multiplicative convolution operators. In particular, we provide both a stochastic representation, expressed in terms of these inverse processes, and an explicit representation, given in terms of the generalized Mittag-Leffler functions, of the solution of these self-similar Cauchy problems. This work could be seen as an-in depth analysis of a specific class, the one with the self-similarity property, of the general inverse of increasing Markov processes introduced in [15].

Operators between different weighted Fréchet and (LB)-spaces of analytic functions

We study some classical operators defined on the weighted Bergman Frechet space $A^p_{\alpha+}$ (resp. weighted Bergman (LB)-space $A^p_{\alpha-}$) arising as the projective limit (resp. inductive limit) of the standard weighted Bergman spaces into the growth Frechet space $H^\infty_{\alpha+}$ (resp. growth (LB)-space $H^\infty_{\alpha-}$), which is the projective limit (resp. inductive limit) of the growth Banach spaces. We show that, for an analytic self map $\varphi$ of the unit disc $\mathbb{D}$, the continuities of the weighted composition operator $W_{g,\varphi}$, the Volterra integral operator $T_g$, and the pointwise multiplication operator $M_g$ defined via the identical symbol function are characterized by the same condition determined by the symbol's state of belonging to a Bloch-type space. These results have consequences related to the invertibility of $W_{g,\varphi}$ acting on a weighted Bergman Frechet or (LB)-space. Some results concerning eigenvalues of such composition operators $C_\varphi$ are presented.

Certain Classes of Operators on Some Weighted Hyperbolic Function Spaces

In this paper, some classes of concerned multiplication operators consisting of analytic and hyperbolic functions are defined and considered. Furthermore, some properties such as boundedness and compactness of the new operators are discussed. Finally, a general class of weighted hyperbolic Bloch functions is characterized by metric spaces.

Cohen summing multilinear multiplication operators

We give an example of an operator solving in the negative a question of Q. Bu and Z. Shi. Suggested by this example, we find the necessary and sufficient conditions for the multilinear multiplication operator MV:lp1(X1)×⋅⋅⋅×lpk(Xk)→lq(Y) defined byMV((xn1)n∈N,...,(xnk)n∈N)=(Vn(xn1,...,xnk))n∈Nto be Cohen p-summing.

Unitary equivalence of multiplication operators on the Bergman spaces of polygons

AbstractIn this paper, we will show that the unitary equivalence of two multiplication operators on the Bergman spaces on polygons depends on the geometry of the polygon.

Uniform and Lq-ensemble reachability of parameter-dependent linear systems

In this paper, we consider families of linear systems (linear ensembles) defined by matrix pairs (A(θ),B(θ)) depending on a parameter θ∈P that is varying over a compact subset P of the complex plane. In particular, we investigate the existence of open-loop controls which are independent of the parameter θ∈P and steer a given family of initial states x0(θ) arbitrarily close to a desired family of terminal states f(θ) in finite time. Here, the maps θ↦x0(θ) and θ↦f(θ) are assumed to lie in a common appropriately chosen Banach space Xn(P) of Cn-valued functions. If this task is solvable for all initial and terminal states, the pair (A(θ),B(θ)) is called (completely) ensemble controllable with respect to Xn(P).Using a well-known infinite-dimensional version of the Kalman rank condition for systems on Banach spaces, we derive sufficient conditions for cascade and parallel connections of linear ensembles. Moreover, we prove an abstract decomposition theorem which results from a spectral splitting of the matrix family A(θ). Based on these findings as well as on cyclicity conditions for multiplication operators and approximation theory, we obtain necessary and sufficient conditions for ensemble controllability (reachability) with respect to the space of continuous functions and the space of Lq-functions. In the last section, results on averaged controllability (reachability) for linear families (A(θ),B(θ),C(θ)) are presented.

Multiplier and Composition Operator Between Several Holomorphic Function Spaces in $$\mathbf {C^{n}}$$

Multiplier and Composition Operator Between Several Holomorphic Function Spaces in $$\mathbf {C^{n}}$$

Weighted composition operators on spaces of functions with derivative in a Bergman space

Following the idea of the paper by Contreras and Hernandez-Diaz, we first give the characterization of the boundedness of the weighted composition operator $$W_{\varphi ,\phi }$$ on $$B^p$$ . Then, we investigate the boundedness and the compactness of composition operators $$C_{\varphi}$$ on $$B^p$$ and study the spectrum of the multiplication operator $$M_{\phi}$$ on $$B^p$$ , as well. Finally, motivated by the paper by Cuckovic and Paudyal, we describe the relationships between the invariant subspaces of $$M_{z}$$ on $$B^p_{0}$$ and T on $$A^p$$ , where T is the sum of multiplication operator and Volterra operator. Moreover, we provide some Beurling-type invariant subspaces of $$M_{z}$$ on $$B^p$$ and $$B^p_0$$ , respectively.