Showing papers on "Shift operator published in 2010"
TL;DR: In this article, the problem of boundedness of the supremal operator in weighted L p -spaces on the cone of non-negative nondecreasing functions is reduced to the boundedness problem of the fractional maximal operator M α, 0 ≤ α < n.
Abstract: The problem of boundedness of the fractional maximal operator M α, 0 ≤ α < n, in general local Morrey-type spaces is reduced to the problem of boundedness of the supremal operator in weighted L p -spaces on the cone of non-negative non-decreasing functions. This allows obtaining sharp sufficient conditions for boundedness for all admissible values of the parameters, which, for a certain range of the parameters wider than known before, coincide with the necessary ones.
103 citations
TL;DR: The generalized ordered weighted logarithm averaging (GOWLA) operator is a new aggregation operator that generalizes the ordered weighted geometric averaging (OWGA) operator that adds to the OWGA operator an additional parameter controlling the power to which the arguments are raised.
Abstract: We present the generalized ordered weighted logarithm averaging (GOWLA) operator based on an optimal deviation model It is a new aggregation operator that generalizes the ordered weighted geometric averaging (OWGA) operator This operator adds to the OWGA operator an additional parameter controlling the power to which the arguments are raised We further generalize the GOWLA operator and obtain the generalized ordered weighted hybrid logarithm averaging (GOWHLA) operator We next introduce a nonlinear objective programming model for determining GOWHLA weights and an approach to group decision making based on the GOWHLA operator Finally, we present a numerical example to illustrate the new approach in human resource management problem © 2010 Wiley Periodicals, Inc
86 citations
TL;DR: An alternative expression for the length operator in loop quantum gravity is presented in this paper, which is background independent, symmetric, positive semidefinite, and well defined on the kinematical Hilbert space.
Abstract: An alternative expression for the length operator in loop quantum gravity is presented. The operator is background independent, symmetric, positive semidefinite, and well defined on the kinematical Hilbert space. The expression for the regularized length operator can moreover be understood both from a simple geometrical perspective as the average of a formula relating the length to area, volume and flux operators, and also consistently as the result of direct substitution of the densitized triad operator with the functional derivative operator into the regularized expression of the length. Both these derivations are discussed, and the origin of an undetermined overall factor in each case is also elucidated.
64 citations
TL;DR: In this paper, the elementary operator L, acting on the Hilbert-Schmidt Class C 2 (H ), given by L (T ) = ATB, with A and B bounded operators on H, was considered.
42 citations
TL;DR: In this paper, the action of the discrete maximal operator on Newtonian, Holder and BMO spaces on metric measure spaces equipped with a doubling measure and a Poincare inequality is studied.
Abstract: We study the action of the so-called discrete maximal operator on Newtonian, Holder and BMO spaces on metric measure spaces equipped with a doubling measure and a Poincare inequality. The discrete maximal operator has better regularity properties than the standard Hardy-Littlewood maximal operator and hence is a more flexible tool in this context.
41 citations
01 Nov 2010
TL;DR: In this article, a continuous kernel function for Hermitian operators is defined, which is a generalization of the Lowner kernel function, and it is shown that it is positive definite if and only if whenever.
Abstract: Let be a real continuous function on an interval, and consider the operator function defined for Hermitian operators . We will show that if is increasing w.r.t. the operator order, then for the operator function is convex. Let and be functions defined on an interval . Suppose is non-decreasing and is increasing. Then we will define the continuous kernel function by , which is a generalization of the Lowner kernel function. We will see that it is positive definite if and only if whenever for Hermitian operators , and we give a method to construct a large number of infinitely divisible kernel functions.
39 citations
TL;DR: The theory of Euler integration is extended from the class of constructible functions to that of “tame” R-valued functions (definable with respect to an o-minimal structure) and the corresponding integral operator has some unusual defects but has a compelling Morse-theoretic interpretation.
Abstract: We extend the theory of Euler integration from the class of constructible functions to that of “tame” R-valued functions (definable with respect to an o-minimal structure). The corresponding integral operator has some unusual defects (it is not a linear operator); however, it has a compelling Morse-theoretic interpretation. In addition, it is an advantageous setting in which to integrate in applications to diffused and noisy data in sensor networks.
39 citations
TL;DR: In this article, the authors consider the inverse problem of reconstructing the convolution operator from the spectrum and prove the global solvability of this nonlinear integral equation with a singularity.
Abstract: We consider the sum of the Sturm-Liouville operator and a convolution operator. We study the inverse problem of reconstructing the convolution operator from the spectrum. This problem is reduced to a nonlinear integral equation with a singularity. We prove the global solvability of this nonlinear equation, which permits one to show that the asymptotics of the spectrum is a necessary and sufficient condition for the solvability of the inverse problem. The proof is constructive.
34 citations
TL;DR: In this paper, the Srivastava-Attiya integral operator is used to define analytic and spiral-like functions of complex order involving the integral operator and some inclusion relations along with integral preserving properties of these classes.
Abstract: In this paper, we introduce and study certain subclasses of analytic and spiral-like functions of complex order involving the Srivastava–Attiya integral operator. We also investigate some inclusion relations along with integral preserving properties of these classes.
30 citations
TL;DR: In this article, the authors studied a Sturm-Liouville operator with eigenparameter-dependent boundary conditions and transmission conditions at two interior points, and proved that the operator is self-adjoint in an appropriate space H, discuss completeness of its eigenfunctions in H, and obtain its Green function.
Abstract: In this paper, we study a Sturm–Liouville operator with eigenparameter-dependent boundary conditions and transmission conditions at two interior points. By establishing a new operator A associated with the problem, we prove that the operator A is self-adjoint in an appropriate space H, discuss completeness of its eigenfunctions in H, and obtain its Green function. Copyright © 2010 John Wiley & Sons, Ltd.
29 citations
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TL;DR: In this paper, a reduced operator system S ⊆ B(H) is defined as a system whose boundary ideal is at most a constant in the Choquet boundary of the Hilbert space.
Abstract: We classify operator systems S ⊆ B(H ) that act on finite dimensional Hilbert spaces H by making use of the noncommutative Choquet boundary. S is said to be reduced when its boundary ideal is {0}. In the category of operator systems, that property functions as semisimplicity does in the category of complex Banach algebras. We construct explicit examples of reduced operator systems using sequences of “parameterizing maps” � k : C r → B(Hk), k = 1 ,...,N . We show that every reduced operator system is isomorphic to one of these, and that two sequences give rise to isomorphic operator systems if and only if they are “unitarily equivalent” parameterizing sequences. Finally, we construct nonreduced operator systems S that have a given boundary ideal K and a given reduced image in C ∗ (S)/K, and show that these constructed examples exhaust the possibilities.
TL;DR: In this paper, the authors deal with analysis and synthesis problems of spatially interconnected systems where communicated information may get lost between subsystems and propose a procedure of designing distributed dynamic output feedback controllers within the linear matrix inequality (LMI) framework.
TL;DR: In this paper, the authors derive some criteria for univalence of a general integral operator for analytic functions in the open unit disk and prove that they are not univalent.
TL;DR: In this paper, the spectral properties of a 3 × 3 block operator matrix with unbounded entries and domain consisting of vectors which satisfy certain relations between their components are studied, and the essential spectra of this operator are determined.
Abstract: In this paper we study spectral properties of a 3 × 3 block operator matrix with unbounded entries and with domain consisting of vectors which satisfy certain relations between their components. It is shown that, under certain conditions, this block operator matrix defines a closed operator, and the essential spectra of this operator are determined. These results are applied to a three-group transport equation.
TL;DR: In this article, the position of the spectrum of a self-adjoint operator L with respect to its spectrum of an anti-commuting operator was investigated and norm bounds on the operator angles between maximal uniformly definite reducing subspaces of the unperturbed operator A and the perturbed operator L were obtained in terms of the norm of V and the distances between pairs of disjoint spectral sets associated with the operator L and/or the operator A.
Abstract: Given a self-adjoint involution J on a Hilbert space H, we consider a J-self-adjoint operator L=A+V on H where A is a possibly unbounded self-adjoint operator commuting with J and V a bounded J-self-adjoint operator anti-commuting with J. We establish optimal estimates on the position of the spectrum of L with respect to the spectrum of A and we obtain norm bounds on the operator angles between maximal uniformly definite reducing subspaces of the unperturbed operator A and the perturbed operator L. All the bounds are given in terms of the norm of V and the distances between pairs of disjoint spectral sets associated with the operator L and/or the operator A. As an example, the quantum harmonic oscillator under a PT-symmetric perturbation is discussed. The sharp norm bounds obtained for the operator angles generalize the celebrated Davis-Kahan trigonometric theorems to the case of J-self-adjoint perturbations.
TL;DR: In this paper, the Darboux integrability of a differential-difference equation with unknown t(n, x) depending on the continuous and discrete variables x and n is studied.
Abstract: A differential-difference equation with unknown t(n, x) depending on the continuous and discrete variables x and n is studied. We call an equation of such kind Darboux integrable if there exist two functions (called integrals) F and I of a finite number of dynamical variables such that DxF = 0 and DI = I, where Dx is the operator of total differentiation with respect to x and D is the shift operator: Dp(n) = p(n + 1). It is proved that the integrals can be brought to some canonical form. A method of construction of an explicit formula for a general solution to Darboux-integrable chains is discussed and such solutions are found for a class of chains.
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TL;DR: In this paper, the authors give a characterization of the two-weight inequality for a simple vector-valued operator for 1 < p < ∞, where p is the number of vertices in the vector.
Abstract: We give a characterization of the two-weight inequality for a simple vector-valued operator. Special cases of our result have been considered before in the form of the weighted Carleson embedding theorem, the dyadic positive operators of Nazarov, Treil, and Volberg in the square integrable case, and Lacey, Sawyer, Uriarte-Tuero in the L^p case. The main technique of this paper is a Sawyer-style argument and the characterization is for 1 < p < \infty. We are unaware of instances where this operator has been given attention in the two-weight setting before.
TL;DR: The convergence of a time discretisation with variable time steps is shown for a class of doubly nonlinear evolution equations of second order and proves existence of a weak solution.
Abstract: The convergence of a time discretisation with variable time steps is shown for a class of doubly nonlinear evolution equations of second order. This also proves existence of a weak solution. The operator acting on the zero-order term is assumed to be the sum of a linear, bounded, symmetric, strongly positive operator and a nonlinear operator that fulfils a certain growth and a Holder-type continuity condition. The operator acting on the first-order time derivative is a nonlinear hemicontinuous operator that fulfils a certain growth condition and is (up to some shift) monotone and coercive.
TL;DR: In this paper, an expansion of the unitary evolution operator, associated with a given Schrodinger equation, in terms of a finite product of explicit unitary operators is proposed, which can be truncated at the desired level of approximation, as shown in the given examples.
Abstract: We propose an expansion of the unitary evolution operator, associated with a given Schr\"odinger equation, in terms of a finite product of explicit unitary operators. In this manner, this unitary expansion can be truncated at the desired level of approximation, as shown in the given examples.
TL;DR: In this article, the existence of a sequence of eigenfunctions for the 1-Laplace operator was verified by showing that the corresponding variational problem has a series of critical points.
Abstract: The paper verifies the existence of a sequence of eigenfunctions for the 1-Laplace operator by
showing that the corresponding variational problem has a sequence of critical points. Since the
functionals entering the variational problem are not differentiable, critical points are defined by
means of the weak slope.
TL;DR: The lattice vertex operator V_L associated to a positive definite even lattice L has an automorphism of order 2 lifted from the -1 isometry of L as discussed by the authors.
Abstract: The lattice vertex operator V_L associated to a positive definite even lattice L has an automorphism of order 2 lifted from -1 isometry of L. It is established that the fixed point vertex operator algebra V_L^+ is rational.
TL;DR: In this article, a necessary and sufficient condition for a unit vector f to be a stationary vector of the generalized eigenvalue problem Tf = λAf is obtained. And a theorem of Williams is generalized to obtain a translatable radius of an operator in the direction of another operator.
Abstract: One of the couple of translatable radii of an operator in the direction of another operator introduced in earlier work [PAUL, K.: Translatable radii of an operator in the direction of another operator, Scientae Mathematicae 2 (1999), 119–122] is studied in details. A necessary and sufficient condition for a unit vector f to be a stationary vector of the generalized eigenvalue problem Tf = λAf is obtained. Finally a theorem of Williams ([WILLIAMS, J. P.: Finite operators, Proc. Amer. Math. Soc. 26 (1970), 129–136]) is generalized to obtain a translatable radius of an operator in the direction of another operator.
TL;DR: The case of the Cauchy-Riemann operator for Cl0,3 and the Fueter operator is studied in this article, and the relation between the two operators and the Dirac operator is established.
Abstract: For a positive integer n let Cl0,n be the universal Clifford algebra with the signature (0,n). The name Clifford analysis is usually referred to the function theories for functions in the kernels of the two operators: the (Cliffordian) Cauchy–Riemann operator and the Dirac operator. For n=2, Cl0,2 becomes the skew-field of Hamilton's quaternions for which the two operators are widely known: the Moisil–Theodoresco and the Fueter operators. We establish the precise relations between the Moisil–Theodoresco operator and the Dirac operator for Cl0,3. It turns out that the case of the Cauchy–Riemann operator for Cl0,3 and the Fueter operator is more sophisticated, and we describe the peculiarities emerging here. Copyright © 2009 John Wiley & Sons, Ltd.
TL;DR: For translation invariant maps defined on the von Neumann algebra V N(G) associated with a discrete group G, the authors proved an operator space version of Maurey's theorem, which claims that every absolutely (p, 1)-summing map on C(K) is automatically q-summing for q > p.
Abstract: We prove an operator space version of Maurey’s theorem, which claims that every absolutely (p, 1)-summing map on C(K) is automatically absolutely q-summing for q > p. Our results imply in particular that every completely bounded map from B(H) with values in Pisier’s operator space OH is completely p-summing for p > 2. This fails for p = 2. As applications, we obtain eigenvalue estimates for translation invariant maps defined on the von Neumann algebra V
N(G) associated with a discrete group G. We also develop a notion of cotype which is compatible with factorization results on noncommutative L
p
spaces.
TL;DR: A variant of the Jensen-Mercer operator inequality for superquadratic functions, which is a refinement of the Danish mathematician's type, is proved and used to refine some comparison inequalities between operator power and quasi-arithmetic means of Mercer's type.
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TL;DR: In this paper, the concepts of stable and unstable C-symmetry are introduced in the extension theory framework and the main results are the following: if A is a J-self-adjoint extension of S, then either R or C = C, and if A has a real spectrum, then A has stable C symmetry.
Abstract: AoaN J-self-adjoint extensions of the Phillips symmetric operator S are studied. The concepts of stable and unstable C-symmetry are introduced in the extension theory framework. The main results are the following: if A is a J-self-adjoint extension of S, then either �(A) = R or �(A) = C; if A has a real spectrum, then A has a stable C-symmetry and A is similar to a self-adjoint operator; there are no J-self-adjoint extensions of the Phillips operator with unstable C-symmetry.
TL;DR: In this paper, the authors obtained a characterization of an inequality for Riemann-Liouville operator involving suprema in case of nonincreasing weights, and showed that the inequality is equivalent to the inequality for nonincreasing weight.
Abstract: In the paper we obtain a characterization of an inequality for Riemann-Liouville operator involving suprema in case of nonincreasing weights.
TL;DR: In this article, the boundedness of high order Riesz-Bessel transformations generated by generalized shift operator in weighted Lp,ω,γ-spaces with general weights is proved.
Abstract: In this study, the boundedness of the high order Riesz-Bessel transformations generated by generalized shift operator in weighted Lp,ω,γ-spaces with general weights is proved.
TL;DR: In this paper, the classical sampling theorem for bandlimited functions has been generalized to apply to so-called bandlimited operators, that is, to operators with band-limited Kohn-Nirenberg symbols.
TL;DR: In this paper, a linear operator of meromorphic p-valent functions was proposed and defined and a preliminary concept of subordination was introduced to give sharp proofs for certain sufficient conditions of the linear operator aforementioned.
Abstract: Problem statement: By means of the Hadamard product (or convolution), a linear operator was introduced. This operator was motivated by many authors namely Srivastava, Swaminathan, Owa and many others. The operator was indeed needed to create new ideas in the area of geometric function theory. Approach: The linear operator of meromorphic p-valent functions was proposed and defined. The preliminary concept of subordination was introduced to give sharp proofs for certain sufficient conditions of the linear operator aforementioned. Results: Having the operator, subordination theorems established by using standard concept of subordination and reduced to well-known results studied by various authors. The operator was then applied for fractional calculus and obtained new subordination theorem. Conclusion: Therefore, interesting operators could be obtained with some earlier results and standard methods.