Hyponormality of Toeplitz Operators with Polynomial Symbols: An Extremal Case

If T_φ is a hyponormal Toeplitz operator with polynomial symbol φ = ḡ + f (f, g ∈ H^∞ (𝕋 )) such that g divides f, and if ψ := then where μ is the leading coefficient of ψ and 𝒵(ψ) denotes the set of zeros of ψ. In this paper we present a necessary and sufficient condition for T_φ to be hyponormal when φ enjoys an extremal case in the above inequality, that is, equality holds in the above inequality.     

Invertible weighted composition operators

A weighted composition operator Cψ,φ takes an analytic map f on the open unit disc of the complex plane to the analytic map ψ⋅f°φ where φ is an analytic map of the open unit disc into itself and ψ is an analytic map on the open unit disc. This paper studies the invertibility of such operators. The two maps ψ and φ are characterized when Cψ,φ acts on the Hardy-Hilbert space of the unit disc H²(D). Depending upon the nature of the fixed points of φ spectra are then investigated.     

Subnormal Toeplitz operators and the kernels of their self-commutators

In this note we give a connection between subnormal Toeplitz operators and the kernels of their self-commutators. This is closely related to P.R. Halmos's Problem 5: Is every subnormal Toeplitz operator either normal or analytic? Our main theorem is as follows: If φ∈L^∞ is such that φ and are of bounded type (that is, they are quotients of two analytic functions on the open unit disk) and if the kernel of the self-commutator of Tφ is invariant for Tφ then Tφ is either normal or analytic.     

Commutativity of kth‐order slant Toeplitz operators

In this paper, commutativity of k^th‐order slant Toeplitz operators are discussed. We show that commutativity and essential commutativity of two slant Toeplitz operators are the same. Also, we study k^th‐order slant Toeplitz operators on the Bergman space L²_a(D) and give some commuting properties, algebraic and spectral properties of k^th‐order slant Toeplitz operators on the Bergman space (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Functions of bounded boundary rotation

Classes of functionsU _k, which generalize starlike functions in the same manner that the classV _k of functions with boundary rotation bounded bykπ generalizes convex functions, are defined. The radius of univalence and starlikeness is determined. The behavior off _α(z) = ∫ ₀ ^z [f'(t)]^α dt is determined for various classes of functions. It is shown that the image of |z|<1 underV _kfunctions contains the disc of radius 1/k centered at the origin, andV _k functions are continuous in |z|≦1 with the exception of at most [k/2+1] points on |z|=1.     

Starlikeness and subordination properties of a linear operator

In this paper, we introduce a new class of p-valent analytic functions defined by using a linear operator L_k^α. For functions in this class H_k^α(p,λh) we estimate the coefficients. Furthermore, some subordination properties related to the operator L_k^α are also derived.     

Rank one perturbations and singular integral operators

We consider rank one perturbations Aα=A+α(⋅,φ)φ of a self-adjoint operator A with cyclic vector φ∈H−1(A) on a Hilbert space H. The spectral representation of the perturbed operator Aα is given by a singular integral operator of special form. Such operators exhibit what we call ‘rigidity’ and are connected with two weight estimates for the Hilbert transform. Also, some results about two weight estimates of Cauchy (Hilbert) transforms are proved. In particular, it is proved that the regularized Cauchy transforms Tε are uniformly (in ε) bounded operators from L²(μ) to L²(μα), where μ and μα are the spectral measures of A and Aα, respectively. As an application, a sufficient condition for Aα to have a pure absolutely continuous spectrum on a closed interval is given in terms of the density of the spectral measure of A with respect to φ. Some examples, like Jacobi matrices and Schrödinger operators with L² potentials are considered.     

The Evolution of the Weyl and Maxwell Fields in Curved Space—Times

The covariant Weyl (spin s = 1/2) and Maxwell (s = 1) equations in certain local charts (u, φ) of a space-time (M, g) are considered. It is shown that the condition g⁰⁰(x) > 0 for all x ε u is necessary and sufficient to rewrite them in a unified manner as evolution equations δ_tφ = L_(s)φ. Here L_(s) is a linear first order differential operator on the pre—Hilbert space (C (U_t, ^2s+1). (…)), where U_t ? IR³ is the image of the coordinate map of the spacelike hyper-surface t = const, and (φ, C) = ?U_t ? *Q d⁽³⁾x with a suitable Hermitian n × n- matrix Q = Q(t,x). The total energy of the spinor field ? with respect to U_t is then simply given by E = 〈?,?〉. In this way inequalities for the energy change rate with respect to time, δ_t|?|² = 2Re (?, L_(s)?) are obtained. As an application, the Kerr—Newman black hole is studied, yielding quantitative estimates for the energy change rate. These estimates especially confirm the energy conservation of the Weyl field and the well—known superradiance of electromagnetic waves.     

A Class of Operators on Lh 2

Let D be the open unit disc in ? and let L_h ² be the space of quadratic integrable harmonic functions defined on D. Let \(\varphi: {\bar D}\rightarrow {\rm C}\) be a function in L^∞(D) with the property that φ(b) = lim_x→b,x?Dφ(x) for all b ? ?D. Define the operator C_φ in L_h ² as follows: C_φf = Q(φ·f),f ? L_h ², where Q is the orthogonal projection from L² (D) on L_h ². The following results are proved. If φ¦_?D ≡ 0, then C_φ is a compact linear operator and if φ¦_?D vanishes nowhere, then C_φ is a Fredholm operator.     

The Theory of Constructive Signal Analysis

Let h(x) = e^?αxk(x), where and λ0=0. The closure theorem, V_h = L¹(?), is proved for various α and k (V_h is the L¹-closed variety generated by h). The Tauberian condition, |?| > 0, is not used, since generally this condition is difficult to compute directly. The functions h arise naturally in time series and analytic number theory. The technique of proof is constructive and depends on the semigroup {γ_j} generated by {λ_j}. The semigroup theory which consolidates and completes the results herein will be developed separately as “A closure problem for signals in semigroup invariant systems.”     

On infinite-rank singular perturbations of the Schrödinger operator

Schrödinger operators with infinite-rank singular potentials V=Σ _i,j=1 ^∞ b _ij〈φ_j,·〉φ_i are studied under the condition that the singular elements ψ _j are ξ _j(t)-invariant with respect to scaling transformationsin ?³.     

Global integral criteria for composition operators

Let D_α denote the Dirichlet-type space of functions analytic on the unit disk U and Q_α the conformal invariant version of this space. Any analytic self-map φ of U induces a composition operator C_φ acting on D_α, respectively, Q_α by C_φf=f°φ, where f∈D_α, respectively, f∈Q_α. The aim of this paper is to characterize boundedness and compactness of such operators in terms of global area integrals of φ.     

Hypercyclic composition operators on Hv0‐spaces

We consider analytic self‐maps φ on $\mathbf {D}$ and prove that the composition operator C_φ acting on $H_{v}^0$ is hypercyclic if φ is an automorphism or a hyperbolic non‐automorphic symbol with no fixed point. We give examples of weights v and parabolic non‐automorphisms φ on $\mathbf {D}$ which yield non‐hypercyclic composition operators C_φ on $H_{v}^0$.     

Berezin–Toeplitz quantization for lower energy forms

Let M be an arbitrary complex manifold and let L be a Hermitian holomorphic line bundle over M. We introduce the Berezin–Toeplitz quantization of the open set of M where the curvature on L is nondegenerate. In particular, we quantize any manifold admitting a positive line bundle. The quantum spaces are the spectral spaces corresponding to [0,k^?N], where N>1 is fixed, of the Kodaira Laplace operator acting on forms with values in tensor powers L^k. We establish the asymptotic expansion of associated Toeplitz operators and their composition in the semiclassical limit k→∞ and we define the corresponding star-product. If the Kodaira Laplace operator has a certain spectral gap this method yields quantization by means of harmonic forms. As applications, we obtain the Berezin–Toeplitz quantization for semi-positive and big line bundles.     

Modifiable low‐rank approximation to a matrix

A truncated ULV decomposition (TULVD) of an m×n matrix X of rank k is a decomposition of the form X = ULV^T+E, where U and V are left orthogonal matrices, L is a k×k non‐singular lower triangular matrix, and E is an error matrix. Only U,V, L, and ∥E∥_F are stored, but E is not stored. We propose algorithms for updating and downdating the TULVD. To construct these modification algorithms, we also use a refinement algorithm based upon that in (SIAM J. Matrix Anal. Appl. 2005; 27 (1):198–211) that reduces ∥E∥_F, detects rank degeneracy, corrects it, and sharpens the approximation. Copyright © 2009 John Wiley & Sons, Ltd.     

Multivalued Extended Best Φ-Polynomial Approximation Operator

In this article, we consider the best multivalued polynomial approximation operator, defined in an Orlicz Space L^Φ(B), and its extension to L^φ(B), where φ is the derivative of Φ which is not an N-function. Thus, the extension of the best polynomial approximation operator from L¹(B) to L⁰(B) arises as a particular case of this work, taking Φ(x) = x.     

Stability for the Multi-Dimensional Borg-Levinson Theorem with Partial Spectral Data

We prove a stability estimate related to the multi-dimensional Borg-Levinson theorem of determining a potential from spectral data: the Dirichlet eigenvalues λ _k and the normal derivatives ?φ _k /?ν of the eigenfunctions on the boundary of a bounded domain. The estimate is of Hölder type, and we allow finitely many eigenvalues and normal derivatives to be unknown. We also show that if the spectral data is known asymptotically only, up to O(k ^?α) with α ? 1, then we still have Hölder stability.     

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.     

Weakly Compact Composition Operators on Locally Convex Spaces

Let E be a complete, barrelled locally convex space, let V = (v_n) be an increasing sequence of strictly positive, radial, continuous, bounded weights on the unit disc 𝔻 of the complex plane, and let φ be an analytic self map on 𝔻. The composition operators C_φ : f → f ○ φ on the weighted space of holomorphic functions HV (𝔻, E) which map bounded sets into relatively weakly compact subsets are characterized. Our approach requires a study of wedge operators between spaces of continuous linear maps between locally convex spaces which extends results of Saksman and Tylli [31, 32], and a representation of the space HV (𝔻, E) as a space of operators which complements work by Bierstedt , Bonet and Galbis [4] and by Bierstedt and Holtmanns [6].