A lifting theorem and uniform algebras

Let A be a uniform algebra on X and P a set of all probability measures on X. For each µ in P, H2 (µ) is the closure of A in L2 (µ) and Tt is a Toeplitz operator on H2 (µ) for a continuous function cf> on X. In this paper we study the invertibility and the spectrum of Tip = L EB Tt. We show that if Tip is invertible then the index of cf> is zero and if the converse is true for an arbitrary continuous function cf> then A is a Dirichlet algebra on X. Moreover we study the spectrum of Tip.

Weighted norm inequalities for pseudo-pseudodifferential operators defined by amplitudes

We prove in this paper one weight norm inequalities for some positive Bergman-type operators.

We derive an asymptotic formula for the argument of a Blaschke product in the upper half-plane with purely imaginary zeros. We then use this formula to find conditions for the quotient of two such Blaschke products to be continuous on the real line. These results are applied to certain Hankel and Toeplitz operators arising in the Cauchy problem for the Korteweg-de Vries equation. Our main theorems include certain compactness conditions for Hankel operators and invertibility conditions for Toeplitz operators with oscillating symbols containing such quotients. As a by-product we obtain a well-posedness result for the Korteweg-de Vries equation.

In this chapter we introduce shift-invariant algebras, the main objects of this book. These are uniform algebras on a compact connected group G, consisting of continuous functions on G, whose spectrum is contained in a semigroup S of the dual group Ĝ. If Ĝ is a subgroup of ℝ, and S ⊂ ℝ+, then the maximal ideal space of the corresponding shift-invariant algebra is the G-disc, or, big disc over G. In this chapter we describe two important models of shift-invariant algebras, namely, by the means of almost periodic functions on ℝ, and by the means of H∞-functions on the unit circle. The set of automorphisms, and the peak groups of shift-invariant algebras are also characterized. Extensions on G-discs and groups of several classical theorems of Complex Analysis, such as Radó’s theorem for null-sets and the Riemann theorem for removable singularities of analytic functions, are stated and proved. It is shown that these extensions hold for some semigroups, while in general they fail. In principle we state all results for general shift-invariant algebras AS, though they apply automatically to the particular cases of algebras APS of almost periodic functions, and of H S ∞ -algebras. Asymptotically almost periodic functions combine the properties of classical almost periodic functions on ℝ, and of continuous functions on ℝ that vanish at infinity.

Uniform algebras that can be expressed as inductive limits of standard simpler algebras are of particular interest. For instance, some G-disc algebras are inductive limits of sequences of disc algebras, connected with finite Blaschke products, called also Blaschke inductive limit algebras. Here we show, among others, that only G-disc algebras, and the spaces \( H^\infty (\mathbb{D}_G ) \) with G ⊂ ℚ, can be expressed as limits of countable inductive sequences of algebras of type \( A(\mathbb{D}) \) and H∞ correspondingly. We study also inductive limits of sequences of spaces of type H∞ and prove corresponding corona theorems. Further, we establish relationships between Bourgain algebras of coordinate algebras, and the Bourgain algebra of their inductive limit, and also between H∞-spaces on \( \mathbb{D} \), as coordinate algebras in an inductive sequence, and their inductive limit. While we state all results for general shift-invariant algebras AS, they apply automatically to the particular cases of algebras APS of almost periodic functions, and of H S ∞ -algebras.

In this article we define two-wavelet localization operators corresponding to an irreducible and square-integrable representation of a locally compact Hausdorff group on a Hilbert space. The group structure admitting an irreducible and square-integrable representation which is related to β-Stockwell transform, that we shall use in this article β ∈ R have been introduced in Boggiatto et al. [P. Boggiatto, C. Fernandez, and A. Galbis, A group representation related to the Stockwell transform, Indiana Univ. Math. J. 58(5) (2009), pp. 2277–2296]. The Schatten–von Neumann norm inequalities of these two-wavelet localization operators are established. The traces and the trace class norm inequalities of the trace class two-wavelet localization operators are given.

We completely classify all Toeplitz and Hankel operators which commute; namely, we prove that that a non-trivial Hankel operator and a non-trivial Toeplitz operator commute if and only if the Hankel operator has symbolzψ, where ψ is the symbol of the Toeplitz operator, and ψ is an affine function of the characteristic function of certain “anti-symmetric” sets of the unit circle.

The symmetrized bidisc has been a rich field of holomorphic function theory and operator theory. A certain well-known reproducing kernel Hilbert space of holomorphic functions on the symmetrized bidisc resembles the Hardy space of the unit disc in several aspects. This space is known as the Hardy space of the symmetrized bidisc. We introduce the study of those operators on the Hardy space of the symmetrized bidisc that are analogous to Toeplitz operators on the Hardy space of the unit disc. More explicitly, we first study multiplication operators on a bigger space (an $L^2$-space) and then study compressions of these multiplication operators to the Hardy space of the symmetrized bidisc and prove the following major results. (1) Theorem I analyzes the Hardy space of the symmetrized bidisc, not just as a Hilbert space, but as a Hilbert module over the polynomial ring and finds three isomorphic copies of it as $\mathbb D^2$-contractive Hilbert modules. (2) Theorem II provides an algebraic, Brown and Halmos-type characterization of Toeplitz operators. (3) Theorem III gives several characterizations of an analytic Toeplitz operator. (4) Theorem IV characterizes asymptotic Toeplitz operators. (5) Theorem V is a commutant lifting theorem. (6) Theorem VI yields an algebraic characterization of dual Toeplitz operators. Every section from Section 2 to Section 7 contains a theorem each, the main result of that section.

Given function Ω on ℝn , we define the fractional maximal operator and the fractional integral operator by and respectively, where 0 < α < n. In this paper we study the weighted norm inequalities of MΩα and TΩα for appropriate α, s and A(p, q) weights in the case that Ω∈ Ls(Sn-1)(s> 1), homogeneous of degree zero.

Introduces tensor-product neural networks, composed of a layer of univariate neurons followed by a net of polynomial post-processing. We look at the general approximation problem by these networks observing in particular their relationship to the Stone-Weierstrass theorem for uniform function algebras. The implementation of the post-processing as a two-layer network with logarithmic and exponential neurons leads to potentially important 'generalised' product networks which, however, require a complex approximation theory of the Mu/spl uml/ntz-Szasz-Ehrenpreis type. A backpropagation algorithm for product networks is presented and used in three computational experiments. In particular, approximation by a sigmoid product network is compared to that of a single-layer radial basis network and a multiple-layer sigmoid network.

We extend our recent results concerning norm inequalities on cones to include the case when 0 > p > 1 0 > p > 1 .

The square root of the spectral radius of the Hankel plus Toeplitz operator has been shown to be the achievable performance of the mixed-sensitivity H/sup infinity / design. The computation of the spectral radius is the bottleneck in the synthesis of the H/sup infinity / controller. In this paper, the spectral properties of the Hankel plus Toeplitz operator are investigated. A finite procedure for computing the spectral radius of the Hankel plus Toeplitz operator is proposed.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

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 Hp into ¯Hq (into Hq) for 0 < p, q < ∞.

The paper deals with the invertibility of Toeplitz plus Hankel operators T(a)+H(b) acting on classical Hardy spaces on the unit circle T. It is supposed that the generating functions a and b satisfy the condition a(t)a(1/t) = b(t)b(1/t), t ∈ T. Special attention is paid to the case of piecewise continuous generating functions. In some cases the dimensions of null spaces of the operator T(a) + H(b) and its adjoint are described. Fredholm properties of Toeplitz plus Hankel operatorsT(a)+H(b) with piecewise continuous generating functions a and b have been studied for many years. These operators are considered in various Banach and Hilbert spaces, and the results ob- tained show that the structure of the algebras generated by such operators is much more complicated than the structure of the algebras generated by one-dimensional singular integral operators with piecewise continuous coefficients defined on closed smooth curves. Moreover, calculating the index of Toeplitz plus Hankel operators, one encounters even more difficult problems, and the difficulti es grow if one attempts to study invertibility or one-sided invertibility of such operators. It is worth noting that, in general, Fredholm operators T(a) +H(b) are not one-sided invertible. Nev- ertheless, a few works where one-sided invertibility was discussed, have appeared in literature recently. They are mainly concerned with two special cases of Toeplitz plus Hankel operators—viz. with the operators having the form M(a) := T(a) +H(a) or the form f