Nonzero Operator Research Articles

Homotopic groups of photon propagators in isotropic, bi-isotropic and Faraday materials

With the use of an evolutionary approach the branches of solutions of the tensor dispersion photon equations in isotropic, bi-isotropic and Faraday media are investigated. It is shown that the homotopic groups of photon propagators with the generators being the refractive index operators correspond to these solutions. The latter operators are divided into two groups. These are operators with non-zero trace and traceless operators. The traceless operators correspond to meeting photons and are represented by infinite sets of isometries and involutions of the three-dimensional space. The analysis of polarization states is carried out for photons meeting in non-absorbing and absorbing isotropic media and curvatures, torsions and Darboux-Lagrange vectors of the spiral lines related to such states are found.

On the infinite product of operators in Hilbert space

We give a necessary and sufficient condition for a certain set of infinite products of linear operators to be zero. We shall investigate also the case when this set of infinite products converges to a non-zero operator. The main device in these results is a weighted version of the König Lemma for infinite trees in graph theory.

The Fourier binest algebra

The Fourier binest algebra is defined as the intersection of the Volterra nest algebra on L2(ℝ) with its conjugate by the Fourier transform. Despite the absence of nonzero finite rank operators this algebra is equal to the closure in the weak operator topology of the Hilbert–Schmidt bianalytic pseudo-differential operators. The (non-distributive) invariant subspace lattice is determined as an augmentation of the Volterra and analytic nests (the Fourier binest) by a continuum of nests associated with the unimodular functions exp(−isx2/2) for s>0. This multinest is the reflexive closure of the Fourier binest and, as a topological space with the weak operator topology, it is shown to be homeomorphic to the unit disc. Using this identification the unitary automorphism group of the algebra is determined as the semi-direct product ℝ2×κℝ for the action κt(λ, μ) =(etλ, e−t μ).

On a pattern of reflexive operator spaces

A linear subspace M M is a separating subspace for an operator space S S if the only member of S S annihilating M M is 0. It is proved in this paper that if S S has a strictly separating vector x x and a separating subspace M M satisfying S x ∩ [ S M ] = { 0 } Sx \cap [SM] = \{0\} , then S S is reflexive. Applying this to finite dimensional S S leads to more results on reflexivity. For example, if dim S = n S = n , and every nonzero operator in S S has rank > n 2 > n^{2} , then S S is reflexive.

Realisations of W 3 symmetry

We perform a systematic investigation of free-scalar realisations of the Zamolodchikov W 3 algebra in which the operator product of two spin-three generators contains a non-zero operator of spin four which has vanishing norm. This generalises earlier work where such an operator was required to be absent. By allowing this spin-four null operator we obtain several realisations of the W 3 algebra both in terms of two scalars as well as in terms of an arbitrary number n of free scalars. Our analysis is complete for the case of two-scalar realisations.

On the supremally regulated complex stability radius as a real robustness measure

The main result of this paper is the identification of a class of real systems for which the supremal Hinrichsen-Pritchard complex stability radius is equal to the supremal Hinrichsen-Pritchard real stability radius. For this class (which includes all planar linear systems with non-zero input operator) the remarkable differences, which are known to exist between these two measures of robustness in the absence of control, can be eliminated by state feedback. The respective differences and equivalences are illustrated by examples of single-input uncertain planar control systems.

Hilbert-Schmidt Hankel Operators on the Bergman Space

We show that there are no nonzero Hilbert-Schmidt Hankel operators on the Bergman space of the open unit ball in ${{\mathbf {C}}^n}$ with antiholomorphic symbols when $n \geq 2$.

Hilbert-Schmidt Hankel operators on the Bergman space

We show that there are no nonzero Hilbert-Schmidt Hankel operators on the Bergman space of the open unit ball in C n {{\mathbf {C}}^n} with antiholomorphic symbols when n ≥ 2 n \geq 2 .

Résolubilité globale d'opérateurs différentiels invariants sur certains groupes de Lie

We study the existence of global fundamental solutions of bi-invariant linear differential operators on the direct product G = H × K, where H and K are Lie groups, K compact. Using the partial Fourier transform on K, we prove that a bi-invariant operator P on G admits a fundamental solution on U × K (with U open subset of H) if and only if its partial Fourier coefficients satisfy a condition of slow growth and each one admits a fundamental solution on U. Hence we deduce an explicit necessary condition for the existence of a global solution for P on G. We also give a sufficient condition for the existence of a fundamental solution of P on G in the case where the group H is solvable and simply connected. Using Rouvière's method, based on L 2 inequalities, we show that a differential operator satisfying this condition has a fundamental solution on every relatively compact open subset of G (i.e., a semiglobal solution). It follows that P has a fundamental solution on G: use of the notion of P-convexity enables us to obtain global solutions from semiglobal solutions. We also prove the existence of a global fundamental solution for every nonzero bi-invariant operator on a simply connected solvable Lie group.

Each non-zero convolution operator on the entire functions admits a continuous linear right inverse

Each non-zero convolution operator on the entire functions admits a continuous linear right inverse

An Operator-Theoretic Formulation of Asynchronous Exponential Growth

A strongly continuous semigroup of bounded linear operators $T(t)$, $t \geqslant 0$, in the Banach space $X$ has asynchronous exponential growth with intrinsic growth constant ${\lambda _0}$ provided that there is a nonzero finite rank operator ${P_0}$ in $X$ such that ${\lim _{t \to \infty }}{e^{ - {\lambda _0}t}}T(t) = {P_0}$. Necessary and sufficient conditions are established for $T(t)$, $t \geqslant 0$, to have asynchronous exponential growth. Applications are made to a maturity-time model of cell population growth and a transition probability model of cell population growth.

Division by Holomorphic Functions and Convolution Equations in Infinite Dimension

Let $E$ be a complex complete dual nuclear locally convex space (i.e. its strong dual is nuclear), $\Omega$ a connected open set in $E$ and $\mathcal {E}(\Omega )$ the space of the ${C^\infty }$ functions on $\Omega$ (in the real sense). Then we show that any element of $\mathcal {E}’(\Omega )$ may be divided by any nonzero holomorphic function on $\Omega$ with the quotient as an element of $\mathcal {E}’(\Omega )$. This result has for standard consequence a new proof of the surjectivity of any nonzero convolution operator on the space $\operatorname {Exp} (E’)$ of entire functions of exponential type on the dual $E’$ of $E$. As an application of the above division result and of a result of ${C^\infty }$ solvability of the $\overline \partial$ equation in strong duals of nuclear Fréchet spaces we study the solutions of the homogeneous convolution equations in $\operatorname {Exp} (E’)$ in terms of the zero set of their characteristic functions.

Convolution equations in spaces of infinite-dimensional entire functions of exponential and related types

We prove results of existence and approximation of the solutions of the convolution equations in spaces of entire functions of exponential type on infinite dimensional spaces. In particular we obtain: let E be a complex, quasi-complete and dual nuclear locally convex space and Ω \Omega a convex balanced open subset of E; let H ( Ω ) \mathcal {H} (\Omega ) be the space of the holomorphic functions on Ω \Omega , equipped with the compact open topology and H ′ ( Ω ) \mathcal {H}’(\Omega ) its strong dual; let F H ′ ( Ω ) \mathcal {F} \mathcal {H}’(\Omega ) denote the image of H ′ ( Ω ) \mathcal {H}’(\Omega ) through the Fourier-Borel transform F \mathcal {F} ; equip this space F H ′ ( Ω ) \mathcal {F} \mathcal {H}’(\Omega ) with the image of the topology of H ′ ( Ω ) \mathcal {H}’(\Omega ) via the map F \mathcal {F} . Then, “every nonzero convolution operator on F H ′ ( Ω ) \mathcal {F} \mathcal {H}’(\Omega ) is surjective” and “every solution of the homogeneous equation is limit of exponential-polynomial solutions". Our results are more generally valid when E is a Schwartz bornological vector space with the approximation property. Previous results in Fréchet-Schwartz and Silva spaces are thus extended to domains that are not Fréchet or D.F.-spaces.

Linear operators on 𝐿_{𝑝} for 0

If 0 > p > 1 0\, > \,p\, > \,1 we classify completely the linear operators T : L p → X T:\,{L_p}\, \to \,X where X is a p-convex symmetric quasi-Banach function space. We also show that if T : L p → L 0 T:\,{L_p}\, \to \,{L_0} is a nonzero linear operator, then for p > q ⩽ 2 p\, > \,q\, \leqslant \,2 there is a subspace Z of L p {L_p} , isomorphic to L q {L_q} , such that the restriction of T to Z is an isomorphism. On the other hand, we show that if p > q > ∞ p\, > \,q\, > \,\infty , the Lorentz space L ( p , q ) L(p,\,q) is a quotient of L p {L_p} which contains no copy of l p {l_p} .

On density of algebras with minimal invariant operator ranges

Let A \mathfrak {A} be an arbitrary subalgebra of B ( H ) \mathcal {B}(\mathcal {H}) and let M \mathfrak {M} be a dense operator range invariant under A \mathfrak {A} such that every nonzero operator range invariant under A \mathfrak {A} contains M \mathfrak {M} . Then the closure of A \mathfrak {A} in the strong operator topology is B ( H ) \mathcal {B}(\mathcal {H}) .

Biinvariant operators on nilpotent Lie groups

The purpose of this article is to prove P-convexity for biinvariant differential operators on connected simply connected nilpotent Lie groups. More precisely, we show that for any compact subset K of a connected simply connected nilpotent Lie group N, and for any non-zero biinvariant differential operator P on N, there is a compact subset L ~ K with the property that whenever the support of Pu is contained in L for a C OO function of compact support u on N, then the support of u is contained in L. I am grateful to M. Duflo, to A. Cerezo, and to F. Rouvi6re for several helpful discussions. Solubility properties of biinvariant operators have been considered by several authors. S. Helgason 1,6] proves local solvability of biinvariant operators on semisimple Lie groups. Rais 1,8] proves the existence of a fundamental solution for a biinvariant operator on a connected simply connected nilpotent Lie group. Duflo and Rais 1,4] prove the local solvability of biinvariant operators on a solvable Lie group and Rouvi6re [9] proves semi-global solvability for biinvariant operators on simply connected solvable groups. Finally, Duflo 1,3] proves local solvability of biinvariant operators on any Lie group whatsoever. Semi-global solvability is in general false even for noncompact simple groups as was demonstrated by A. Cerezo and F. Rouvi6re 1,2]. Finally, even local solvability of left invariant operators is frequently false as was shown by L. Hormander, c.f. 1,6-1 and independently by A. Cerezo and F. Rouvi6re I-1]. From our result and that of Rais 1,8-1 or Rouvi6re 1,9-1, we conclude the global solvability of biinvariant operators on simply connected nilpotent Lie groups, i.e. that for any C oO function f and nonzero biinvariant operator P on a simply connected nilpotent Lie group N, there exists a Coo function u on N such that Pu =f. For simply connected abelian Lie groups, this reduces to the theorem of Malgrange and Ehrenpreis that constant coefficient differential operators on R n are globally solvable, c.f. 1,11-1. Thus our Theorem2 can be regarded as a generalization of the Malgrange-Ehrenpreis theorem. Henceforward N will denote a connected simply connected nilpotent Lie group, and 9l its Lie algebra. We write exp: 91--*N for the exponential map of 91

Nonselfadjoint representations of 𝐶*-algebras

The following strengthening of a result of B. A. Barnes is proved: If ϕ \phi is a topologically irreducible representation of a C ∗ {C^ \ast } -algebra A \mathfrak {A} on a Banach space such that ϕ ( A ) \phi (\mathfrak {A}) contains a nonzero finite-rank operator, then ϕ \phi is similar to an irreducible ∗ ^ \ast -representation of A \mathfrak {A} (and is thus automatically continuous).

Convolution operators on 𝐺-holomorphic functions in infinite dimensions

For a complex vector space E, let H G ( E ) {H_G}(E) denote the space of G (Gateaux)-holomorphic functions on E ( f : E → C E\;(f:E \to C is G-holomorphic if the restriction of f to every finite dimensional subspace of E is holomorphic in the usual sense). The most natural topology on H G ( E ) {H_G}(E) is that of uniform convergence on finite dimensional compact subsets of E. A convolution operator A on H G ( E ) {H_G}(E) is a continuous linear mapping A : H G ( E ) → H G ( E ) A:{H_G}(E) \to {H_G}(E) such that A commutes with translations. The concept of a convolution operator generalizes that of a differential operator with constant coefficients. We prove that if A is a convolution operator on H G ( E ) {H_G}(E) , then the kernel of A is the closed linear span of the exponential polynomials contained in the kernel. In addition, we show that any nonzero convolution operator on H G ( E ) {H_G}(E) is a surjective mapping.

Study of the collision operator, ψ(z), for some exactly soluble models

The collision operator, ψ( z), introduced in the study of non-equilibrium statistical mechanics by Prigogine, Balescu, and Résibois is calculated exactly for some simple quantum-mechanical models. The calculations are done first for the discrete-energy spectrum case. Then, the behavior of the solutions is studies in the limit: l (extension of the system) → ∞, where continua appear in the spectra. It is found that the exact collision operators ψ( z, l) posses the property that lim z →0 ψ ( z, l) is either zero or underfined for finite l, while lim z →10+ lim l →∞ ψ ( z, l) is a nonzero operator.

States and invariant maps of operator algebras

Recently a great deal of progress has been made in the study of groups of automorphisms of operator algebras and states invariant under the automorphisms. We are then given a C*-algebra Q!, a group G of automorphisms of GY satisfying certain requirements and want to draw conclusions about the invariant states and the fixed point algebra in LY under G. The present paper is concerned both with this problem and the converse one. In the latter case we are given a state p of a C*-algebra O? and want to find a group G of auto- morphisms of G! with respect to which p is invariant, such that the above mentioned theory is applicable. In some cases we find a natural abelian subalgebra 9 of OZ, which is the fixed points of a group G and a canonical G-invariant projection @ of U onto 9 such that p = (p ) 9) 0 @. Then we can prove theorems on integral decompo- sition of states. The main portion of this paper rests largely on the results of KovPcs and Sziics [13]. Given a von Neumann algebra 9 and a group G of automorphisms (i.e. *-automorphisms) of 9, we say .J% is G- finite if there is a family 5 of normal G-invariant states of 9 such that if A is a nonzero positive operator in 9 then for some p in F, p(A) # 0. We shall for simplicity say that