Some results for operators on a model space

We investigate some problems for truncated Toeplitz operators. Namely, the solvability of the Riccati operator equation on the set of all truncated Toeplitz operators on the model space K _θ = H²ΘθH² is studied. We study in terms of Berezin symbols invertibility of model operators. We also prove some results for the Berezin number of the truncated Toeplitz operators. Moreover, we study some property for H²-functions in terms of noncyclicity of co-analytic Toeplitz operators and hypercyclicity of model operators.     

Toeplitz operators on Bergman spaces

Let ? be an element in \(H^\infty (D) + C(\overline D )\) such that ?^* is locally sectorial. In this paper it is shown that the Toeplitz operator defined on the Bergman spaceA ² (D) is Fredholm. Also, it is proved that ifS is an operator onA ²(D) which commutes with the Toeplitz operatorT _? whose symbol ? is a finite Blaschke product, thenS H ^∞ (D) is contained inH ^∞ (D). Moreover, some spectral properties of Toeplitz operators are discussed, and it is shown that the spectrum of a class of Toeplitz operators defined on the Bergman spaceA ² (D), is not connected.     

Derived space of L p (G, H)

Let G denote a locally compact abelian group and H a separable Hilbert space. Let L _p (G, H), 1 ≤ p < ∞, be the space of H-valued measurable functions which are in the usual L _p space. Motivated by the work of Helgason [1], Figa-Talamanca [11] and Bachelis [2, 3], we have defined the derived space of the Banach space L _p (G, H) and have studied its properties. Similar to the scalar case, we prove that if G is a noncompact, locally compact abelian group, then L _p ⁰ (G, H) = {0} holds for 1 ≤ p < 2. Let G be a compact abelian group and Γ be its dual group. Let S _p (G, H) be the L ₁(G) Banach module of functions in L _p (G, H) having unconditionally convergent Fourier series in L _p -norm. We show that S _p (G, H) coincides with the derived space L _p ⁰ (G, H), as in the scalar valued case. We also show that if G is compact and abelian, then L _p ⁰ (G, H) = L ₂(G, H) holds for 1 ≤ p ≤ 2. Thus, if F ∈ L _p (G, H), 1 ≤ p < 2 and F has an unconditionally convergent Fourier series in L _p -norm, then F ∈ L ₂(G, H). Let Ω be the set of all functions on Γ taking only the values 1, ?1 and Ω* be the set of all complex-valued functions on Γ having absolute value 1. As an application of the derived space L _p ⁰ (G, H), we prove the following main result of this paper. Let G be a compact abelian group and F be an H-valued function on the dual group Γ such that $$ \sum \omega (\gamma )F(\gamma )\gamma $$ is a Fourier-Stieltjes series of some measure µ ∈ M(G, H) for every scalar function ω such that |ω(γ)| = 1. Then F ∈ l ²(Γ, H).     

Hankel and Toeplitz operators in hardy spaces

In this paper one considers some general theorems of the theory of Hankel and Toeplitz operators in spaces of analytic functions. Under natural restrictions on the spaces X, it is shown that the symbols of the Toeplitz operators, acting in X, are bounded. One describes completely the symbols of the Hankel and Toeplitz operators, acting from H^p into ¯H_q (into H^q) for 0 < p, q < .Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 141, pp. 165–175, 1985.The author expresses his gratitude to A. L. Vol'berg and S. V. Kislyakov for useful discussions.     

From microscopic theory to macroscopic theory——symmetries and order parameters of rigid molecules

Density functional theory is used to describe the phase behaviors of rigid molecules. The construction of the kernel function is discussed. Excluded-volume potential is calculated for two types of molecules with C _2v symmetry. Molecular symmetries lead to the symmetries of the kernel function and the density function, enabling a reduction of configuration space. By approximating the kernel function with a polynomial, the system can be fully characterized by some moments corresponding to the form of the kernel function. The symmetries of the kernel function determine the form of the polynomial, while the coefficients are determined by the temperature and molecular parameters. The analysis of the impact of coefficients helps us to choose independent variables in the moments as order parameters. Combining the analysis and some simulation results, we propose a minimal set of order parameters for bent-core molecules.     

Fredholm-Volterra integral equation with singular kernel

The purpose of this paper is to obtain the solution of Fredholm-Volterra integral equation with singular kernel in the space L₂(?1, 1) × C(0,T), 0 ≤t ≤T < ∞, under certain conditions. The numerical method is used to solve the Fredholm integral equation of the second kind with weak singular kernel using the Toeplitz matrices. Also, the error estimate is computed and some numerical examples are computed using the MathCad package.     

Microlocal kernel of pseudodifferential operators at a hyperbolic fixed point

We study the microlocal kernel of h-pseudodifferential operators Oph(p)−z, where z belongs to some neighborhood of size O(h) of a critical value of its principal symbol p₀(x,ξ). We suppose that this critical value corresponds to a hyperbolic fixed point of the Hamiltonian flow Hp₀. First we describe propagation of singularities at such a hyperbolic fixed point, both in the analytic and in the C^∞ category. In both cases, we show that the null solution is the only element of this microlocal kernel which vanishes on the stable incoming manifold, but for energies z in some discrete set. For energies z out of this set, we build the element of the microlocal kernel with given data on the incoming manifold. We describe completely the operator which associate the value of this null solution on the outgoing manifold to the initial data on the incoming one. In particular it appears to be a semiclassical Fourier integral operator associated to some natural canonical relation.     

Bargaining set theory and stability in coalition governments

Various bargaining set theories are compared as predictors of coalition government portfolio distribution. While the kernel and B₁-bargaining set are known to exist in voting games with side payments, it is argued that the kernel, and thus B₁, are poor predictors. The B₂-bargaining set, a subset of B₁, when it exists is shown to be a good payoff predictor in a fractionalized and depolarized parliamentary situation (Finland: 1945ndash;1971). Moreover this predictor provides some explanation for the formation of surplus (winning but not minimal) coalitions.     

Extremal functions as divisors for kernels of Toeplitz operators

In this paper, an extremal function of a Banach space of analytic functions in the unit disk (not all functions vanishing at 0) is a function solving the extremal problem for functions f of norm 1. We study extremal functions of kernels of Toeplitz operators on Hardy spaces H^p, 1<p<∞. Such kernels are special cases of so-called nearly invariant subspaces with respect to the backward shift, for which Hitt proved that when p=2, extremal functions act as isometric divisors. We show that the extremal function is still a contractive divisor when p<2 and an expansive divisor when p>2 (modulo p-dependent multiplicative constants). We give examples showing that the extremal function may fail to be a contractive divisor when p>2 and also fail to be an expansive divisor when p<2. We discuss to what extent these results characterize the Toeplitz operators via invariant subspaces for the backward shift.     

The influence of minimal subgroups of focal subgroups on the structure of finite groups

A subgroup H of a finite group G is said to be complemented in G if there exists a subgroup K of G such that G=HK and H∩K=1. In this paper, it is proved that a finite group G is p-nilpotent provided p is the smallest prime number dividing the order of G and every minimal subgroup of the p-focal subgroup of G is complemented in N_G(P), where P is a Sylow p-subgroup of G. As some applications, some interesting results related with complemented minimal subgroups of focal subgroups are obtained.     

Toeplitz operators on Herglotz wave functions

The space of Herglotz wave functions in R² consists of all the solutions of the Helmholtz equation that can be represented as the Fourier transform in R² of a measure supported in the circle and with density in L²(S¹). This space has a structure of a Hilbert space with reproducing kernel. The purpose of this article is to study Toeplitz operators with nonnegative radial symbols, defined on this space. We study the symbols defining bounded and compact Toeplitz operators as well as the Toeplitz operators belonging to the Schatten classes sp.     

Invariant subspaces and extremum problems in spaces of vector-valued functions

In this article we obtain, for 1 ? p ? ∞, a characterization of the invariant subspaces of spaces of vector-valued L^p functions defined on the unit circle—i.e., of those subspaces invariant under multiplication by e^ix. This result is then applied to extend, to the corresponding Hardy classes of vector-valued functions, the known characterizations of the extreme points of the unit ball in the scalar Hardy classes H¹ and H^∞. Finally, it is shown that the characterization of the closure of the set of extreme points of the unit ball in H¹ changes significantly when we pass from the scalar to the vector case.     

Generating polynomials for matrices with Toeplitz or conjugate-Toeplitz inverses

A complex matrix A = [a_ij] has been called conjugate-Toeplitz if a_ij = c^i?1(a_i?j), where c( ) denotes conjugation. A necessary and sufficient condition is derived for a matrix H to have a conjugate-Toeplitz inverse. The elements of H are generated from the coefficients of certain polynomials. The result is simplified either when H is to have a Toeplitz inverse, or when H is banded and is to have a conjugate-Toeplitz inverse. Each of these consequences of the main theorem is a different generalization of a previous result on banded matrices having a Toeplitz inverse.     

Unitary equivalence of one-parameter groups of Toeplitz and composition operators

The composition operators on H² whose symbols are hyperbolic automorphisms of the unit disk fixing ±1 comprise a one-parameter group and the analytic Toeplitz operators coming from covering maps of annuli centered at the origin whose radii are reciprocals also form a one-parameter group. Using the eigenvectors of the composition operators and of the adjoints of the Toeplitz operators, a direct unitary equivalence is found between the restrictions to zH² of the group of Toeplitz operators and the group of adjoints of these composition operators. On the other hand, it is shown that there is not a unitary equivalence of the groups of Toeplitz operators and the adjoints of the composition operators on the whole of H², but there is a similarity between them.     

On operators commuting with Toeplitz operators modulo the finite rank operators

Let X be a bounded linear operator on the Hardy space H² of the unit disk. We show that if is of finite rank for every inner function θ, then X=T_?+F for some Toeplitz operator T_? and some finite rank operator F on H². This solves a variant of an open question where the compactness replaces the finite rank conditions.     

A smooth nonparametric quantile estimator from right-censored data

Based on randomly right-censored data, a smooth nonparametric estimator of the quantile function of the lifetime distribution is studied. The estimator is defined to be the solution x_n(p) of F_n(x_n(p)) = p, where F_n is the distribution function corresponding to a kernel estimator of the lifetime density. The strong consistency and asymptotic normality of x_n(p) are shown. Data-based selection of the bandwidth required for computing F_n is investigated using bootstrap methods. Illustrative examples are given.     

Boundedness of Hausdorff operators on Lebesgue spaces and Hardy spaces

In this paper, we study the boundedness of the Hausdorff operator H_? on the real line R. First, we start with an easy case by establishing the boundedness of the Hausdorff operator on the Lebesgue space L~p(R)and the Hardy space H~1(R). The key idea is to reformulate H_? as a Calder′on-Zygmund convolution operator,from which its boundedness is proved by verifying the Hrmander condition of the convolution kernel. Secondly,to prove the boundedness on the Hardy space H~p(R) with 0 p 1, we rewrite the Hausdorff operator as a singular integral operator with the non-convolution kernel. This novel reformulation, in combination with the Taibleson-Weiss molecular characterization of H~p(R) spaces, enables us to obtain the desired results. Those results significantly extend the known boundedness of the Hausdorff operator on H~1(R).     

On operators commuting with Toeplitz operators modulo the compact operators

We prove that an operator on H² of the disc commutes modulo the compacts with all analytic Toeplitz operators if and only if it is a compact perturbation of a Toeplitz operator with symbol in H^∞ + C. Consequently, the essential commutant of the whole Toeplitz algebra is the algebra of Toeplitz operators with symbol in QC. The image in the Calkin algebra of the Toeplitz operators with symbol in H^∞ + C is a maximal abelian algebra. These results lead to a characterization of automorphisms of the algebra of compact perturbations of the analytic Toeplitz operators.