scholarly journals Finite Hilbert transforms and compactness

1992 ◽  
Vol 46 (3) ◽  
pp. 475-478
Author(s):  
Susumu Okada
Keyword(s):  
Banach Space ◽  
Positive Integer ◽  
Linear Operator ◽  
Hilbert Transform ◽  
Hilbert Transforms ◽  
Finite Hilbert Transform ◽  
Strictly Singular

It is shown that for the finite Hilbert transform Tp on the Banach space Lp(]–1, 1[), 1 < p < ∞, the linear operator is not strictly singular whenever n is a positive integer.

scholarly journals Diagonalization of the finite Hilbert transform on two adjacent intervals: the Riemann–Hilbert approach

2020 ◽  
Vol 10 (3) ◽  
Author(s):  
Marco Bertola ◽  
Elliot Blackstone ◽  
Alexander Katsevich ◽  
Alexander Tovbis
Keyword(s):  
Error Estimates ◽  
Hilbert Transform ◽  
Linear Operators ◽  
Hilbert Transforms ◽  
Finite Hilbert Transform ◽  
Novel Approach ◽  
Interior Problem

Abstract In this paper we study the spectra of bounded self-adjoint linear operators that are related to finite Hilbert transforms $$\mathcal {H}_L:L^2([b_L,0])\rightarrow L^2([0,b_R])$$ H L : L 2 ( [ b L , 0 ] ) → L 2 ( [ 0 , b R ] ) and $$\mathcal {H}_R:L^2([0,b_R])\rightarrow L^2([b_L,0])$$ H R : L 2 ( [ 0 , b R ] ) → L 2 ( [ b L , 0 ] ) . These operators arise when one studies the interior problem of tomography. The diagonalization of $$\mathcal {H}_R,\mathcal {H}_L$$ H R , H L has been previously obtained, but only asymptotically when $$b_L\ne -b_R$$ b L ≠ - b R . We implement a novel approach based on the method of matrix Riemann–Hilbert problems (RHP) which diagonalizes $$\mathcal {H}_R,\mathcal {H}_L$$ H R , H L explicitly. We also find the asymptotics of the solution to a related RHP and obtain error estimates.

scholarly journals The Hilbert Transform and Orthogonal Martingales in Banach Spaces

2019 ◽  
Author(s):  
Adam Osękowski ◽  
Ivan Yaroslavtsev
Keyword(s):  
Banach Space ◽  
Singular Integral ◽  
Hilbert Transform ◽  
Real Line ◽  
Harmonic Functions ◽  
Integral Operators ◽  
Continuous Functions ◽  
Hilbert Transforms ◽  
Discrete Hilbert Transform ◽  
The Hilbert Transform

Abstract Let $X$ be a given Banach space, and let $M$ and $N$ be two orthogonal $X$-valued local martingales such that $N$ is weakly differentially subordinate to $M$. The paper contains the proof of the estimate $\mathbb E \Psi (N_t) \leq C_{\Phi ,\Psi ,X} \mathbb E \Phi (M_t)$, $t\geq 0$, where $\Phi , \Psi :X \to \mathbb R_+$ are convex continuous functions and the least admissible constant $C_{\Phi ,\Psi ,X}$ coincides with the $\Phi ,\Psi $-norm of the periodic Hilbert transform. As a corollary, it is shown that the $\Phi ,\Psi $-norms of the periodic Hilbert transform, the Hilbert transform on the real line, and the discrete Hilbert transform are the same if $\Phi $ is symmetric. We also prove that under certain natural assumptions on $\Phi $ and $\Psi $, the condition $C_{\Phi ,\Psi ,X}<\infty $ yields the UMD property of the space $X$. As an application, we provide comparison of $L^p$-norms of the periodic Hilbert transform to Wiener and Paley–Walsh decoupling constants. We also study the norms of the periodic, nonperiodic, and discrete Hilbert transforms and present the corresponding estimates in the context of differentially subordinate harmonic functions and more general singular integral operators.

Projections on Tree-Like banach Spaces

1985 ◽  
Vol 37 (5) ◽  
pp. 908-920
Author(s):  
A. D. Andrew
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Banach Spaces ◽  
Bounded Linear Operator ◽  
Unconditional Basis ◽  
Reflexive Banach Space ◽  
Separable Space ◽  
Bounded Linear ◽  
Tree Space

1. In this paper, we investigate the ranges of projections on certain Banach spaces of functions defined on a diadic tree. The notion of a “tree-like” Banach space is due to James 4], who used it to construct the separable space JT which has nonseparable dual and yet does not contain l1. This idea has proved useful. In [3], Hagler constructed a hereditarily c0 tree space, HT, and Schechtman [6] constructed, for each 1 ≦ p ≦ ∞, a reflexive Banach space, STp with a 1-unconditional basis which does not contain lp yet is uniformly isomorphic to for each n.In [1] we showed that if U is a bounded linear operator on JT, then there exists a subspace W ⊂ JT, isomorphic to JT such that either U or (1 — U) acts as an isomorphism on W and UW or (1 — U)W is complemented in JT. In this paper, we establish this result for the Hagler and Schechtman tree spaces.

scholarly journals THE POLYNOMIAL NUMERICAL INDEX OF A BANACH SPACE

2006 ◽  
Vol 49 (1) ◽  
pp. 39-52 ◽  
Author(s):  
Yun Sung Choi ◽  
Domingo Garcia ◽  
Sung Guen Kim ◽  
Manuel Maestre
Keyword(s):  
Banach Space ◽  
Positive Integer ◽  
Banach Spaces ◽  
Compact Space ◽  
Linear Operators ◽  
Homogeneous Polynomials ◽  
Numerical Radius ◽  
Disc Algebra ◽  
Multilinear Maps ◽  
Numerical Index

AbstractIn this paper, we introduce the polynomial numerical index of order $k$ of a Banach space, generalizing to $k$-homogeneous polynomials the ‘classical’ numerical index defined by Lumer in the 1970s for linear operators. We also prove some results. Let $k$ be a positive integer. We then have the following:(i) $n^{(k)}(C(K))=1$ for every scattered compact space $K$.(ii) The inequality $n^{(k)}(E)\geq k^{k/(1-k)}$ for every complex Banach space $E$ and the constant $k^{k/(1-k)}$ is sharp.(iii) The inequalities$$ n^{(k)}(E)\leq n^{(k-1)}(E)\leq\frac{k^{(k+(1/(k-1)))}}{(k-1)^{k-1}}n^{(k)}(E) $$for every Banach space $E$.(iv) The relation between the polynomial numerical index of $c_0$, $l_1$, $l_{\infty}$ sums of Banach spaces and the infimum of the polynomial numerical indices of them.(v) The relation between the polynomial numerical index of the space $C(K,E)$ and the polynomial numerical index of $E$.(vi) The inequality $n^{(k)}(E^{**})\leq n^{(k)}(E)$ for every Banach space $E$.Finally, some results about the numerical radius of multilinear maps and homogeneous polynomials on $C(K)$ and the disc algebra are given.

scholarly journals A note on best approximation and invertibility of operators on uniformly convex Banach spaces

1991 ◽  
Vol 14 (3) ◽  
pp. 611-614 ◽  
Author(s):  
James R. Holub
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Banach Spaces ◽  
Best Approximation ◽  
Bounded Linear Operator ◽  
Convex Banach Space ◽  
Sufficient Condition ◽  
Uniformly Convex ◽  
Bounded Linear ◽  
Uniformly Convex Banach Spaces

It is shown that ifXis a uniformly convex Banach space andSa bounded linear operator onXfor which‖I−S‖=1, thenSis invertible if and only if‖I−12S‖<1. From this it follows that ifSis invertible onXthen either (i)dist(I,[S])<1, or (ii)0is the unique best approximation toIfrom[S], a natural (partial) converse to the well-known sufficient condition for invertibility thatdist(I,[S])<1.

scholarly journals BOUNDED LINEAR OPERATORS ON SPACES IN NORMED DUALITY

2007 ◽  
Vol 49 (1) ◽  
pp. 145-154
Author(s):  
BRUCE A. BARNES
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Integral Operators ◽  
Continuous Functions ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Closed Range ◽  
Bounded Linear ◽  
Spaces Of Continuous Functions

Abstract.LetTbe a bounded linear operator on a Banach spaceW, assumeWandYare in normed duality, and assume thatThas adjointT†relative toY. In this paper, conditions are given that imply that for all λ≠0, λ−Tand λ −T†maintain important standard operator relationships. For example, under the conditions given, λ −Thas closed range if, and only if, λ −T†has closed range.These general results are shown to apply to certain classes of integral operators acting on spaces of continuous functions.

scholarly journals Iterations converging to distinct solutions of some nonlinear operator equations in Banach space

1986 ◽  
Vol 9 (3) ◽  
pp. 583-587
Author(s):  
Ioannis K. Argyros
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Nonlinear Operator ◽  
Operator Equations ◽  
Nonlinear Operator Equations

We examine the solvability of multilinear equations of the formMk(x,x,…,x)−k   times−=y,   k=2,3,…whereMkis ak-linear operator on a Banach spaceXandy∈Xis fixed.

scholarly journals On the p-norm of the truncated n-dimensional Hilbert transform

1991 ◽  
Vol 43 (2) ◽  
pp. 241-250 ◽  
Author(s):  
J.N. Pandey ◽  
O.P. Singh
Keyword(s):  
Linear Operator ◽  
Linear Combination ◽  
Hilbert Transform ◽  
Bounded Linear Operator ◽  
Measurable Subset ◽  
Bounded Linear ◽  
Finite Linear Combination ◽  
The Hilbert Transform ◽  
Finite Linear ◽  
Hilbert Type

It is shown that a bounded linear operator T from Lρ(Rn) to itself which commutes both with translations and dilatations is a finite linear combination of Hilbert-type transforms. Using this we show that the ρ-norm of the Hilbert transform is the same as the ρ-norm of its truncation to any Lebesgue measurable subset of Rn with non-zero measure.

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