Nonzero Operator Research Articles

A Nichtnegativstellensatz for polynomials in noncommuting variables

Let S⋃{f} be a set of symmetric polynomials in noncommuting variables. If f satisfies a polynomial identity Σihi*fhi = 1 + Σigi*sigi for some si ∈ S ⋃ {1}, then f is obviously nowhere negative semidefinite on the class of tuples of nonzero operators defined by the system of inequalities s ≥ 0 (s ∈ S). We prove the converse under the additional assumption that the quadratic module generated by S is Archimedean.

Modulus hyperinvariant subspaces for quasinilpotent operators at a non-zero positive vector on lp-Spaces

In 1993, Y. A. Abramovich, C. D. Aliprantis and O. Burkinshaw showed that every continuous operator with modulus on an lp-space (1 ≤ p < ∞) whose modulus commutes with a non-zero positive operator T on lp that is quasinilpotent at a non-zero positive vector x0 has a non-trivial invariant closed subspace. In this paper, it is proved that if \({\mathcal{C}} \neq \{0\}\) is a collection of continuous operators with moduli on lp that is finitely modulus-quasinilpotent at a non-zero positive vector x0 then \({\mathcal{C}}\) and its right modulus sub-commutant \({\mathcal{C}}_{m}^{\prime}\) have a common non-trivial invariant closed subspace. In particular, all continuous operators with moduli on lp whose moduli commute with a non-zero positive operator I on lp that is quasinilpotent at a non-zero positive vector x0 have a common non-trivial invariant closed subspace, so that all positive operators on lp which commute with a non-zero positive operator S on lp that is quasinilpotent at a non-zero positive vector x0 have a common non-trivial invariant closed subspace.

Some remarks on a paper by Liu and van Rooij

Complementing the work of T.-S. Liu and A.C.M. van Rooij we show that the existence of non-zero translation invariant operators between certain function spaces on a locally compact group implies its amenability.

Minimal rank and reflexivity of operator spaces

Let S be an n-dimensional space of linear operators between the linear spaces U and V over an algebraically closed field F. Improving results of Larson, Ding, and Li and Pan we show the following. Theorem. Let S 1 ,.., S n be a basis of S. Assume that every nonzero operator in S has rank larger than n. Then a linear operator T: U → V belongs to S if and only if for every u ∈ U, Tu is a linear combination of S 1 u,..., S n u.

On operators satisfying T∗∣ T2∣ T ⩾ T∗∣ T∣ 2T

Let T be a bounded linear operator on a complex Hilbert space H . In this paper we introduce the class, denoted QA , of operators satisfying T ∗∣ T 2∣ T ⩾ T ∗∣ T∣ 2 T and we prove basic structural properties of these operators. Using these results, we also prove that if E is the Riesz idempotent for a non-zero isolated point λ ∘ of the spectrum of T ∈ QA , then E is self-adjoint, and we give a necessary and sufficient condition for T ⊗ S to be in QA when T and S are both non-zero operators.

A theorem on reflexive large rank operator spaces

If every nonzero operator in an n n -dimensional operator space S \mathbb {S} has rank ⩾ 2 n \geqslant 2n , then S \mathbb {S} is reflexive.

Integral Equations and Operator Theory

A complex number λ is an extended eigenvalue of an operator A if there is a nonzero operator X such that AX = λ XA. We characterize the set of extended eigenvalues, which we call extended point spectrum, for operators acting on finite dimensional spaces, finite rank operators, Jordan blocks, and C0 contractions. We also describe the relationship between the extended eigenvalues of an operator A and its powers. As an application, we show that the commutant of an operator A coincides with that of A n , n ≥ 2, n ∈ N if the extended point spectrum of A does not contain any n–th root of unity other than 1. The converse is also true if either A or A* has trivial kernel.

Strictly Singular and Cosingular Multiplications

AbstractLet L(X) be the space of bounded linear operators on the Banach space X. We study the strict singularity and cosingularity of the two-sidedmultiplication operators S ↦ ASB on L(X), where A, B ∈ L(X) are fixed bounded operators and X is a classical Banach space. Let 1 &lt; p &lt; ∞and p ≠ 2. Our main result establishes that the multiplication S ↦ ASB is strictly singular on L(Lp(0, 1)) if and only if the non-zero operators A, B ∈ L(Lp(0, 1)) are strictly singular. We also discuss the case where X is a L1- or a L∞-space, as well as several other relevant examples.

Irreducible representations for toroidal Lie-algebras

Let G be a simple finite-dimensional Lie-algebra over the complex numbers C . The universal central extension of G ⊗ C [ t 1 ± 1 , … , t n ± 1 ] is denoted by τ 0 . We add degree derivations d 1 , … , d n to τ 0 and denote the resulting Lie-algebra by τ which we call a toroidal Lie-algebra. For n ⩾ 2 it is known that the center of τ 0 is infinite dimensional. This infinite center, which is only an abelian ideal in τ , does not act as scalars on any irreducible representation of τ . In this paper, we prove that the study of irreducible representation of τ with finite-dimensional weight spaces is reduced to the study of irreducible representation for τ 0 ⊕ C d n with finite-dimensional weight spaces on which the center acts as scalars. In the process we prove an interesting result for n ⩾ 2 . Let τ ¯ be the quotient of τ by the non-zero degree central operators. Then τ ¯ does not admit representations with finite dimensional weight spaces where the zero degree center acts non-trivially.

Minimum-error discrimination among mirror-symmetric mixed quantum states

We provide an exact solution of finding the optimum measurement strategy for distinguishing three mixed quantum states with mirror symmetry. The signal states reside in two-dimensional Hilbert space. By mirror symmetry, two of the three states have the same prior probabilities and transform into each other under the mirror symmetry. We find that the optimum measurement strategy may consist of three, two, or a single nonzero probability operator measure elements depending upon the prior probabilities of the quantum signals.

A Note on the Range of Toeplitz Operators

A well known lemma attributed to Coburn states that a Toeplitz operator with non-trivial kernel acting on the Hardy space must have dense range. We show that the range of a non-zero Toeplitz operator with non-trivial kernel must contain all polynomials and state this in a precise form.

ON OPERATORS WITH AN ABSOLUTE VALUE CONDITION

Let A denote the class of bounded linear Hilbert space operators with the property that jA 2 j ‚ jAj 2 . In this paper we show that A-operators are flnitely ascensive and that, for non-zero operators A and B, A › B is in A if and only if A and B are in A. Also, it is shown that if A is an operator such that p(A) is in A for a non-trivial polynomial p, then Weyl's theorem holds for f(A), where f is a function analytic on an open neighborhood of the spectrum of A.

Three family Z3 orbifold trinification, MSSM and doublet–triplet splitting problem

A Z3 orbifold compactification of E8×E8′ heterotic string is considered toward a trinification SU(3)3 with three light families. The GUT scale VEVs of the SU(2)W×U(1)Y×SU(3)c singlet chiral fields in two sets of the trinification spectrum allow an acceptable symmetry breaking pattern toward MSSM. We show that a doublet–triplet splitting is related to the absence of a ΔB nonzero operator.

On a framework of scattering for dissipative systems

In this paper we study the existence of scattering solutions for some dissipative systems which contain elastic wave with dissipative boundary conditions in a half space of R3 (cf. Dermenjian-Guillot [1]). First we give a framework based on the idea of Simon [18] and apply it to elastic wave mentioned above. In applying the abstract framework, we shall use the Mellin transformation (cf. Perry [14]) as a key tool. Let H be separable Hilbert space with inner 〈· ·〉H. The norm is denoted by ‖ · ‖H. Let { ( )} ≧0 and { 0( )} ∈R be a contraction semigroup in H and a unitary group in H0, respectively. We denote the generator of ( ) and 0( ) by and 0, respectively ( ( ) = − and 0( ) = − 0 ). We make the following assumptions on and 0. (A1) σ( 0) = σ ( 0) = R or [0 ∞). (A2) ( − )−1 − ( 0 − )−1 defined as a form is extended to a compact operator in H. (A3) There exist non-zero projection operators in H, + and −, such that + + − = and

ON INVARIANT SUBSPACES FOR POWER-BOUNDED OPERATORS OF CLASS $C_{1\cdot}$

We prove that if $T$ is a power-bounded operator of class $C_{*\cdot}$ on a Hilbert space which commutes with a nonzero quasinilpotent operator, then $T$ has a nontrivial invariant subspace. Connections with the questions of convergence of $T^n$ to $0$ in the strong operator topology and of cyclicity of power-bounded operators of class $C_{1\cdot}$ are discussed.

Extended eigenvalues and the Volterra operator

In this paper we consider the integral Volterra operator on the space L^2(0,1). We say that a complex number \lambda is an extended eigenvalue ofV if there exists a nonzero operator X satisfying the equation XV=\lambda VX. We show that the set of extended eigenvalues of V is precisely the interval (0,\infty ) and the corresponding eigenvectors may be chosen to be integral operators as well.

Irrelevant operators in the two-dimensional Ising model

By using conformal-field theory, we classify the possible irrelevant operators for the Ising model on the square and triangular lattices. We analyze the existing results for the free energy and its derivatives and for the correlation length, showing that they are in agreement with the conformal-field theory predictions. Moreover, these results imply that the nonlinear scaling field of the energy-momentum tensor vanishes at the critical point. Several other peculiar cancellations are explained in terms of a number of general conjectures. We show that all existing results on the square and triangular lattice are consistent with the assumption that only nonzero spin operators are present.

On properties of Sylvester and Lyapunov operators

Sylvester and Lyapunov operators in real and complex matrix spaces are studied, which include as particular cases the operators arising in the theory of linear time-invariant systems. Let M: F m×n→ F p×q be a linear operator, where F= R or F= C . The operator M is elementary if there exist matrices A∈ F p×m and B∈ F q×n , such that M[X]=AXB . Each M can be represented as a sum of minimum number of elementary operators, called the Sylvester index of M . An expression for the Sylvester index of a general linear operator M is given. An important tool here is a special permutation operator V p,m: F pq×mn→ F pm×nq such that the image V p,m(B T ⊗A) of the matrix of a non-zero elementary operator is equal to the rank 1 matrix vec[A] row[B] , where vec[X] and row[X] are the column-wise and row-wise vector representation of the matrix X. The application of V p,m reduces a sum of Kronecker products of matrices to the standard product of two matrices. A linear operator L: F n×n→ F n×n is a Lyapunov operator if ( L[X]) *= L[X *] , where the star denotes transposition in the real case and complex conjugate transposition in the complex case. Characterisations and parametrisations of the sets of real and complex Lyapunov operators are given and their dimensions are found. Relevant Lyapunov indexes for Lyapunov operators are introduced and calculated. Similar results are given also for several classes of Lyapunov-like linear and pseudo-linear operators. The concept of Lyapunov singular values of a Lyapunov operator is introduced and the application of these values to the sensitivity and a posteriori error analysis of Lyapunov equations is discussed.

Invariant fundamental solutions and solvability for $\mathrm{GL} (n, \mathsf{C})/U(p,q)$.

where is the Dirac measure at the origin of G=H. Consider now the reductive symmetric space G=H ˆ GL n;C†=U p; q†. Let aq be a fundamental Cartan subspace for G=H (the `most compact' Cartan subspace) and let Aq be the associated Cartan subset of G=H, identified with a real abelian subgroup of G. For every non-trivial G-invariant differential operator D we let q D† be the differential operator with constant coefficients on Aq defined via the Harish-Chandra isomorphism. We use the Plancherel formula for GL n;C†=U p; q†, obtained by Bopp and Harinck in [4], to construct invariant fundamental solutions for G-invariant differential operators D on G=H for which the differential operator q D† has a fundamental solution, i.e. a distribution Tq on Aq solving the differential equation: q D†Tq ˆ q; where q is the Dirac measure at the origin of Aq. This result is similar to the results obtained by Benabdallah and Rouvie© re for semisimple Lie groups, see [2, Theore© me 1]. Their and our approach can be seen as a generalization of the method used by Ho« rmander to find fundamental solutions for non-zero differential operators with constant coefficients on R, see [7, p.189f].

Triangularizing semigroups of positive operators on an atomic normed Riesz Space

AbstractIn the first part of the paper we prove several results on the existence of invariant closed ideals for semigroups of bounded operators on a normed Riesz space (of dimension greater than 1) possessing an atom. For instance, if S is a multiplicative semigroup of positive operators on such space that are locally quasinilpotent at the same atom, then S has a non-trivial invariant closed ideal. Furthermore, if T is a non-zero positive operator that is quasinilpotent at an atom and if S is a multiplicative semigroup of positive operators such that TS ≤ ST for all S ∈ S, then S and T have a common non-trivial invariant closed ideal. We also give a simple example of a quasinilpotent compact positive operator on the Banach lattice l∞ with no non-trivial invariant band.The second part is devoted to the triangularizability of collections of operators on an atomic normed Riesz space L. For a semigroup S of quasinilpotent, order continuous, positive, bounded operators on L we determine a chain of invariant closed bands. If, in addition, L has order continuous norm, then this chain is maximal in the lattice of all closed subspaces of L.