Uniformly Conditioned Bases of Spectral Subspaces

A condition number of an ordered basis of a finite-dimensional normed space is defined in an intrinsic manner. This concept is extended to a sequence of bases of finite-dimensional normed spaces, and is used to determine uniform conditioning of such a sequence. We address the problem of finding a sequence of uniformly conditioned bases of spectral subspaces of operators of the form T _n  = S _n  + U _n , where S _n is a finite-rank operator on a Banach space and U _n is an operator which satisfies an invariance condition with respect to S _n . This problem is reduced to constructing a sequence of uniformly conditioned bases of spectral subspaces of operators on ? ^n×1. The applicability of these considerations in practical as well as theoretical aspects of spectral approximation is pointed out.     

Analytic operator functions with compact spectrum. I. Spectral nodes,linearization and equivalence

This paper arose from an attempt to classify analytic operator functions modulo equivalence in terms of their linearizations and to use the linearization as a tool to obtain spectral factorizations. In this first part spectral linearizations and spectral nodes are introduced to provide a general framework to deal with problems concerning the uniqueness of a linearization and the existence of analytic divisors. Two analytic operator functionsW ₁(.) andW ₂(.) with compact spectrum are shown to have similar spectral linearizations if and only if for some Banach spaceZ the functionsW ₁(.) I _Z andW ₂(.) I _Z are equivalent. In parts II and III of this paper spectral nodes will be used intensively to deal with a number of factorization problems. In particular, in part III for Hilbert spaces and bounded domains a full solution of the inverse problem will be given, which will be used to construct spectral factorizations explicitly and to solve the problem of spectrum displacement.Research supported by the Netherlands Organization for the Advancement of Pure Research (Z.W.O.).This paper was written while the third author was a senior visiting fellow at the Vrije Universiteit at Amsterdam.     

Some spectral properties of the streaming operator with general boundary conditions

This paper deals with the spectral study of the streaming operator with general boundary conditions defined by means of a boundary operator K. We study the positivity and the irreducibility of the generated semigroup proved in [M. Boulanouar, L’opérateur d’Advection: existence d’un C ₀-semi-groupe (I), Transp. Theory Stat. Phys. 31, 2002, 153–167], in the case ‖K‖ ⩾ 1. We also give some spectral properties of the streaming operator and we characterize the type of the generated semigroup in terms of the solution of a characteristic equation.     

Time Asymptotic Behaviour for a One-Velocity Transport Operator with Maxwell Boundary Condition

This paper is concerned with the spectral analysis of a one-velocity transport operator with Maxwell boundary condition in L ₁-space. After a detailed spectral analysis it is shown that the associated Cauchy problem is governed by a C ₀-semigroup. Next, we discuss the irreducibility of the transport semigroup. In particular, we show that the transport semigroup is irreducible. Finally, a spectral decomposition of the solutions into an asymptotic term and a transient one which will be estimated for smooth initial data is given.     

Nonself-Adjoint Operators with Almost Hermitian Spectrum: Weak Annihilators

We consider nonself-adjoint nondissipative trace class additive perturbations L=A+iV of a bounded self-adjoint operator A in a Hilbert space ,H. The main goal is to study the properties of the singular spectral subspace N _i ⁰ of L corresponding to part of the real singular spectrum and playing a special role in spectral theory of nonself-adjoint nondissipative operators.To some extent, the properties of N _i ⁰ resemble those of the singular spectral subspace of a self-adjoint operator. Namely, we prove that L and the adjoint operator ,L ^* are weakly annihilated by some scalar bounded outer analytic functions if and only if both of them satisfy the condition N _i ⁰ =H. This is a generalization of the well-known Cayley identity to nonself-adjoint operators of the above-mentioned class.     

Fully nonlinear elliptic equations and the dini condition

Abstract

We investigate transmission problems with strongly Lipschitz interfaces for the Dirac equation by establishing spectral estimates on an associated boundary singular integral operator, the rotation operator. Using Rellich estimates we obtain angular spectral estimates on both the essential and full spectrum for general bi-oblique transmission problems. Specializing to the normal transmission problem, we investigate transmission problems for Maxwell's equations using a nilpotent exterior/interior derivativeoperator. The fundamental commutation properties for this operator with the two basic reflection operators are proved. We show how the L ₂spectral estimates are inherited for the domain of the exterior/interior derivative operator and prove some complementary eigenvalue estimates. Finally we use a general algebraic theorem to prove a regularity property needed for Maxwell's equations.     

Spectrum Problems for Singular Integral Operators with Carleman Shift

In this paper we are concerned with the complete spectral analysis for the operator 𝒯 = 𝒳𝒮𝒰 in the space L_p(𝕋) (𝕋 denoting the unit circle), where 𝒳 is the characteristic function of some arc of 𝕋, 𝒮 is the singular integral operator with Cauchy kernel and 𝒰 is a Carleman shift operator which satisfies the relations 𝒰² = I and 𝒮𝒰 = ±𝒰𝒮, where the sign + or — is taken in dependence on whether 𝒰 is a shift operator on L_p(𝕋) preserving or changing the orientation of 𝕋. This includes the identification of the Fredholm and essential parts of the spectrum of the operator 𝒯, the determination of the defect numbers of 𝒯 — λI, for λ in the Fredholm part of the spectrum, as well as a formula for the resolvent operator.     

Algebraic Equivalence and Homology Classes of Real Algebraic Cycles

In this note we study the spectral properties of a multiplication operator in the space L_p(X)^m which is given by an m by m matrix of measurable functions. Our particular interest is directed to the eigenvalues and the isolated spectral points which turn out to be eigenvalues. We apply these results in order to investigate the spectrum of an ordinary differential operator with so called “floating singularities”.     

Spectre de l'opérateur de transfert en dimension 1

We study spectral properties of a transfer operator ℳΦ(x)=∑_ω g _ω(x)Φ(ψ_ω x) acting on functions of bounded variation. Using a symmetrical integral, we first obtain bounds on its spectral and essential spectral radii. We then consider the dynamical determinant Det^#(Id +zℳ). Our main theorem generalizes to discontinuous weights the result of Baladi and Ruelle (for continuous weights) on the link between zeroes of the sharp determinant and eigenvalues of the transfer operator. The proof is based on regularizing the weights and uses a (new) spectral result giving the surjectivity of some applications between eigenspaces of operators. Received: 8 May 2001     

One-parameter orthogonality relations for basic hypergeometric series

The second order hypergeometric q-difference operator is studied for the value c = −q. For certain parameter regimes the corresponding recurrence relation can be related to a symmetric operator on the Hilbert space ℓ²( ). The operator has deficiency indices (1, 1) and we describe as explicitly as possible the spectral resolutions of the self-adjoint extensions. This gives rise to one-parameter orthogonality relations for sums of two ₂₁-series. In particular, we find that the Ismail-Zhang q-analogue of the exponential function satisfies certain orthogonality relations.     

Generalized slant Toeplitz operators on H2

For an integer k ≥ 2, k^th‐order slant Toeplitz operator U_φ [1] with symbol φ in L^∞(??), where ?? is the unit circle in the complex plane, is an operator whose representing matrixM = (α_ij ) is given by α_ij = 〈φ, z^ki–j〉, where 〈. , .〉 is the usual inner product in L²(??). The operator V_φ denotes the compression of U_φ to H²(??) (Hardy space). Algebraic and spectral properties of the operator V_φ are discussed. It is proved that spectral radius of V_φ equals the spectral radius of U_φ, if φ is analytic or co‐analytic, and if T_φ is invertible then the spectrum of V_φ contains a closed disc and the interior of the disc consists of eigenvalues of infinite multiplicities. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Research problem: on the norms of infinite Cauchy–Toeplitz-plus-Cauchy–Hankel matrices

We give a conjecture and research problem related to ? _p operator and spectral norms of the matrix T _n ?+?H _n .     

On products of unbounded operators

A symmetric operator X^ is attached to each operator X that leaves the domain of a given positive operator A invariant and makes the product AX symmetric. Some spectral properties of X^ are derived from those of X and, as a consequence, various conditions ensuring positivity of products of the form AX ₁ ... X _n are proved. The question of ^-complete positivity of the mapping p → Ap(X ₁,...,X _n) defined on complex polynomials in n variables is investigated. It is shown that the set ω is related to the McIntosh-Pryde joint spectrum of (X ₁,...,X _n) in case all the operators A, X ₁,...,X _n are bounded. Examples illustrating the theme of the paper are included. This revised version was published online in June 2006 with corrections to the Cover Date.     

Building and solving matrix spectral problems

When an infinite dimensional operator T: X → X is approximated with (a slight perturbation of) an operator T_n : X → X of finite rank less than or equal to n, the spectral elements of an auxiliary matrix Z ∈ ℂ^{n ×n} , lead to those of Tn, if they are computed exactly. This contribution covers a general theoretical framework for matrix problems issued from finite rank discretizations and perturbed variants, the stop criterion of the QR method for eigenvalues, the possibility of using the Newton method to compute a Schur form, and the use of Newton method to refine coarse approximate bases of spectral subspaces. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Spectral Shift Function for Certain Block Operator Matrices

Let L₀ be a 2 × 2 diagonal self‐adjoint block operator matrix with entries A and D. If operators B and B* are added to the off diagonal zeros, certain parts of the spectrum of L₀ move to the right and other parts move to the left. In this paper it is shown that, correspondingly, if B is a trace class operator M. G. Krein's spectral shift function is of constant sign on certain intervals.     

Canonical and Noncanonical Recursion Operators in Multidimensions

The hierarchies of evolution equations associated with the spectral operators ?_x?_y ? R?_y ? Q and ?_x?_y ? Q in the plane are considered. In both cases a recursion operator Ф₁₂, which is nonlocal and generates the hierarchy, is obtained. It is shown that only in the first case does the recursion operator satisfy the canonical geometrical scheme in 2 + 1 dimensions proposed by Fokas and Santini. The general procedure proposed allows one to derive, at the same time, the evolution equations associated with a given linear spectral problem and their Backlund transformations (if they exist), with no need to verify by long and tedious computations the algebraic properties of Ф₁₂. Two equations in the first hierarchy can be considered as two different integrable generalizations in the plane of the dispersive long wave equation. All equations in this hierarchy are shown to be both a dimensional reduction of bi-Hamiltonian n × n matrix evolution equations in multidimensions and a generalization in the plane of bi-Hamiltonian n × n matrix evolution equations on the line.     

A Problem in Linear Matrix Approximation

Let A be a normal operator in ??(H), H a complex Hilbert space, and let ? _A = ? {AX - XA:X ∈ ??(H)} be the commutator subspace of ??(H) associated with A. If B in ??(H) commutes with A, then B is orthogonal to ?_A with respect to the spectral norm; i.e., the null operator is an element of best approximation of B in ? _A. This was proved by J. Anderson in 1973 and extended by P. J. Maher with respect to the Schatten p-norm recently. We take a look at their result from a more approximation theoretical point of view in the finite dimensional setting; in particular, we characterize all elements of best approximation of B in R_A and prove that the metric projection of H onto ?_A is continuous.     

Characterization of the spectrum of the Sturm-Liouville operator with irregular boundary conditions

We consider the spectral problem generated by the Sturm-Liouville equation with arbitrary complex-valued potential, q(x), ∈ L ₂(0, π) and irregular boundary conditions. We derive necessary and sufficient conditions for a set of complex numbers to be the spectrum of such an operator.