Showing papers on "Multiplication operator published in 2008"
TL;DR: In this paper, an integral-type operator on the space of holomorphic functions on the unit ball is introduced, which is an extension of the product of composition and integrates integral operators on unit disks.
Abstract: We introduce an integral-type operator, denoted by 𝑃 𝑔 𝜑 ,
on the space of holomorphic functions on the unit ball 𝔹 ⊂ ℂ 𝑛 , which is an extension of the product of composition and
integral operators on the unit disk. The operator norm of
𝑃 𝑔 𝜑 from the weighted Bergman space 𝐴 𝑝 𝛼 ( 𝔹 ) to the
Bloch-type space ℬ 𝜇 ( 𝔹 ) or the little Bloch-type
space ℬ 𝜇 , 0 ( 𝔹 ) is calculated. The compactness
of the operator is characterized in terms of inducing functions
𝑔 and 𝜑 . Upper and lower bounds for the essential norm
of the operator 𝑃 𝑔 𝜑 ∶ 𝐴 𝑝 𝛼 ( 𝔹 ) → ℬ 𝜇 ( 𝔹 ) ,
when 𝑝 > 1 , are also given.
76 citations
TL;DR: In this article, the authors introduced the notion of an F-quadratic stochastic operator and proved that any trajectory of such an operator exponentially converges to a fixed point.
Abstract: In this paper, we introduce the notion of an F-quadratic stochastic operator. It is shown that each F-quadratic operator has a unique fixed point. Besides, it is proved that any trajectory of an F-quadratic stochastic operator exponentially rapidly converges to this fixed point.
70 citations
TL;DR: In this paper, an operator norm localization property and its applications to the coarse Novikov conjecture in operator K-theory was studied. But it was shown that a sequence of expanding graphs does not possess the operator norm localisation property.
54 citations
TL;DR: In this article, the basic properties of bounded and compact weighted composition operators on the Hilbert Hardy space on the open unit disk are summarized and used to study composition operator on Hardy-Smirnov spaces.
Abstract: Operators on function spaces acting by composition to the right with a fixed selfmap φ of some set are called composition operators of symbol φ. A weighted composition operator is an operator equal to a composition operator followed by a multiplication operator. We summarize the basic properties of bounded and compact weighted composition operators on the Hilbert Hardy space on the open unit disk and use them to study composition operators on Hardy–Smirnov spaces.
46 citations
TL;DR: In this article, the authors studied the spectral properties of a Laplace operator associated with a graph in question and showed that the closure of the operator is selfadjoint in the Hilbert space.
Abstract: We study the operator theory associated with such infinite graphs $G$ as occur in electrical networks, in fractals, in statistical mechanics, and even in internet search engines. Our emphasis is on the determination of spectral data for a natural Laplace operator associated with the graph in question. This operator $\Delta$ will depend not only on $G$, but also on a prescribed positive real valued function $c$ defined on the edges in $G$. In electrical network models, this function $c$ will determine a conductance number for each edge. We show that the corresponding Laplace operator $\Delta$ is automatically essential selfadjoint. By this we mean that $\Delta$ is defined on the dense subspace $\mathcal{D}$ (of all the real valued functions on the set of vertices $G^{0}$ with finite support) in the Hilbert space $l^{2}% (G^{0})$. The conclusion is that the closure of the operator $\Delta$ is selfadjoint in $l^{2}(G^{0})$, and so in particular that it has a unique spectral resolution, determined by a projection valued measure on the Borel subsets of the infinite half-line. We prove that generically our graph Laplace operator $\Delta=\Delta_{c}$ will have continuous spectrum. For a given infinite graph $G$ with conductance function $c$, we set up a system of finite graphs with periodic boundary conditions such the finite spectra, for an ascending family of finite graphs, will have the Laplace operator for $G$ as its limit.
42 citations
TL;DR: In this paper, a representation theorem for local operator spaces is proposed, which extends Ruan's representation theorem on operator spaces, and local operator systems which are locally convex versions of the operator systems are shown to have unique multinormed C * -algebra structure with respect to the original matrix topology.
39 citations
TL;DR: In this paper, the authors studied the solvability of the adjointable operator AXB ∗ - BX ∗ A ∗ = C in the general setting of adjoint operators between Hilbert C ∗-modules, and proposed necessary and sufficient conditions for the existence of a solution to this equation.
37 citations
TL;DR: In this paper, it was shown that every orthogonal projection operator P(M) (0≠M∈N) is an all-derivable point of algN for the strong operator topology.
33 citations
TL;DR: In this paper, the authors classify real hypersurfaces of complex projective space C P m, m ⩾ 3, with D -recurrent structure Jacobi operator and apply this result to prove the non-existence of such hypersurface with recurrent structure JacobI operator.
Abstract: We classify real hypersurfaces of complex projective space C P m , m ⩾ 3 , with D -recurrent structure Jacobi operator and apply this result to prove the non-existence of such hypersurfaces with recurrent structure Jacobi operator.
32 citations
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TL;DR: In this paper, the authors presented a four-loop anomalous dimension of the SU(2) sector Konishi operator in N = 4 SYM, as an example of "wrapping" corrections to the known result for long operators.
Abstract: We present a calculation of the four-loop anomalous dimension of the SU(2) sector Konishi operator in N = 4 SYM, as an example of \wrapping" corrections to the known result for long operators We use the known dilatation operator at four loops acting on long operator, and just calculate those diagrams which are aected by the change from operator length L > 4 to L = 4 We nd that the answer involves a [5], so it has trancendentality degree ve Our result diers from previous proposals and calculations We also discuss some ideas for extending this analysis to determine nite size corrections for operators of arbitrary length in the SU(2) sector
31 citations
01 Nov 2008
TL;DR: In this paper, the authors prove the following results without using the spectral theorem: they prove all the above results and give examples to illustrate all of the above points without using spectral theorem.
Abstract: Let H
1, H
2 be Hilbert spaces and T be a closed linear operator defined on a dense subspace D(T) in H
1 and taking values in H
2. In this article we prove the following results:
We prove all the above results without using the spectral theorem. Also, we give examples to illustrate all the above results.
11 Jun 2008
TL;DR: It is shown that if the trajectory of the solution of an operator differential equation starts inside the operator algebra, it will remain inside for all times and is solvable and equivalent to the standard LQR problem without the information constraint.
Abstract: A canonical decentralized optimal control problem with quadratic cost criteria can be cast as an LQR problem in which the stabilizing controller is restricted to lie in a constraint set. We characterize a wide class of systems and constraint sets for which the canonical problem is tractable. We employ the notion of operator algebras to study the structural properties of the canonical problem. Examples of some widely used operator algebras in the context of distributed control include the subspace of infinite and finite dimensional spatially decaying operators, lower (or upper) triangular matrices, and circulant matrices. For a given operator algebra, we prove that if the trajectory of the solution of an operator differential equation starts inside the operator algebra, it will remain inside for all times. Using this result, we show that if the constraint set is an operator algebra, the canonical problem is solvable and equivalent to the standard LQR problem without the information constraint.
TL;DR: In this article, an operator theoretic approach to orthogonal rational functions based on the identification of a suitable matrix representation of the multiplication operator associated with the corresponding orthogonality measure is presented.
TL;DR: In this paper, the unconditional basis property in the space of square integrable functions for a Schrodinger operator defined on a finite interval was studied and it was shown that anti-a priori estimates without a positive power on the right-hand side on the inequality are necessary and sufficient conditions for unconditional basis properties for an arbitrary choice of associated functions.
Abstract: We consider a one-dimensional Schrodinger operator defined on a finite interval. We show that anti-a priori estimates without a positive power on the right-hand side on the inequality are necessary and sufficient conditions for the unconditional basis property in the space of square integrable functions for an arbitrary choice of associated functions. We suggest a new definition of associated functions of linear operators in a Hilbert space.
TL;DR: In this paper, it was proved that the closure of the operator (A ∗ A + α I ) − 1 A ∗, with the domain D ( A ∆ ), where α > 0 is a constant, is a linear bounded everywhere defined operator with norm ≤ 1 2 α ǫ.
Journal Article•
TL;DR: In this article, the authors proposed continuous interval argument OWH (C-OWH) operator for aggregating continuous interval arguments on the basis of ordered weighted harmonic averaging operator and discussed some properties of these operators.
01 Jan 2008
TL;DR: In this article, it was shown that the multiplication operator on the Bergman space is unitarily equivalent to a weighted unilateral shift operator of finite multiplicity if and only if its symbol is a constant multiple of the N-th power of a Mobius transform.
Abstract: In this paper we show that the multiplication operator on the Bergman space is unitarily equivalent to a weighted unilateral shift operator of finite multiplicity if and only if its symbol is a constant multiple of the N-th power of a Mobius transform.
TL;DR: In this paper, the multiplication and composition operators induced by operator valued maps on Bochner spaces were studied and their closedness, compactness, and spectrum was discussed, and the closedness and compactness of the operators were discussed.
Abstract: In this paper we study the multiplication and composition operators induced by operator valued maps on Bochner spaces (Lorentz-Bochner and rearrangement invariant-Bochner) and discuss their closedness, compactness and spectrum
TL;DR: In this paper, the relation between the quaternion H-type group and the boundary of the unit ball on the two-dimensional quaternionic space was studied. And the precise form of Cauchy-Szego kernel and the orthogonal projection operator was obtained.
Abstract: We study the relations between the quaternion H-type group and the boundary of the unit ball on the two-dimensional quaternionic space. The orthogonal projection of the space of square integrable functions defined on quaternion H-type group into its subspace of boundary values of q-holomorphic functions is considered. The precise form of Cauchy-Szego kernel and the orthogonal projection operator is obtained. The fundamental solution for the operator Δλ is found.
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TL;DR: In this paper, the authors give characterizations of unitaries, isometries, unital operator spaces and unital function spaces, operator systems, C*-algebras, and related objects.
Abstract: We give some new characterizations of unitaries, isometries, unital operator spaces, unital function spaces, operator systems, C*-algebras, and related objects. These characterizations only employ the vector space and operator space structure.
01 Dec 2008
TL;DR: A novel approach to time-delay systems consisting of a linear time-invariant (LTI) system and a pure delay and an operator class described by two finite-dimensional matrices gives an asymptotically exact and nonconservative method for stability analysis if the integer N for fast-lifting is taken sufficiently large.
Abstract: This paper gives a novel approach to time-delay systems consisting of a linear time-invariant (LTI) system and a pure delay. The fast-lifting technique, introduced recently in the context of the study of sampled-data systems, is applied to the monodromy operator of a time-delay system to define the fast-lifted monodromy operator, and a stability condition is given in terms of the spectral radius of the latter operator. Then, by investigating the properties of this operator, an operator class described by two finite-dimensional matrices is introduced as candidates for a solution to the associated operator Lyapunov inequality. It is then established that the analysis restricted to such a class gives an asymptotically exact and nonconservative method for stability analysis if the integer N for fast-lifting is taken sufficiently large.
TL;DR: In this article, the authors considered an integral operator for analytic functions in the open unit disk and proved the convexity properties of the integral operator on the class of analytic functions.
Abstract: We consider an integral operator, , for analytic functions, , in the open unit disk, . The object of this paper is to prove the convexity properties for the integral operator , on the class .
TL;DR: This work proves several important inequalities, and investigates the solution of an operator equation by means of the fixed point index in the theory of topological degree and generalized Rothe’s Theorem.
TL;DR: In this article, it was shown that the Cesaro operator is subdecomposable on H1 and on the standard weighted Bergman spaces, α 0, for analytic functions on the unit disc.
TL;DR: In this article, the authors investigate perturbations of the regular spectrum of an upper triangular matrix M C = [A C 0 B ] acting on a Hilbert space H ⊕ K.
TL;DR: In this article, the authors give a short proof of the existence of the play operator on the space of rectifiable curves making use of basic facts of measure theory, and also drop the separability assumptions usually made by other authors.
Abstract: The vector play operator is the solution operator of a class of evolution variational inequalities arising in continuum mechanics. For regular data the existence of solutions is easily obtained from general results from the theory of maximal monotone operators. If the datum is assumed to be a continuous function of bounded variation, then the existence of a weak solution is usually proved by means of a time discretization procedure. In this paper we give a short proof of the existence of the play operator on the space of rectifiable curves making use of basic facts of measure theory. We also drop the separability assumptions usually made by other authors
DOI•
01 Jan 2008TL;DR: In this paper, it was shown that an operator T between reproducing kernel Hilbert spaces is a multiplication operator if and only if it leaves invariant zero sets for all f and z satisfying f(z)=0.
Abstract: In this note, we prove that an operator between reproducing kernel Hilbert spaces is a multiplication operator if and only if it leaves invariant zero sets. To be more precise, it is shown that an operator T between reproducing kernel Hilbert spaces is a multiplication operator if and only if (Tf)(z)=0 holds for all f and z satisfying f(z)=0. As possible applications, we deduce a general reflexivity result for multiplier algebras, and furthermore prove fully vector-valued generalizations of mulitplier lifting results of Beatrous and Burbea.
TL;DR: In this article, the quantum Hadamard operator in continuum state vector space is decomposed into a single-mode squeezing operator and a position-momentum mutual transform operator.
Abstract: We introduce the quantum Hadamard operator in continuum state vector space and find that it can be decomposed into a single-mode squeezing operator and a position-momentum mutual transform operator. The two-mode Hadamard operator in bipartite entangled state representation is also introduced, which involves the two-mode squeezing operator and |η ↔ ξ mutual transformation operator, where |η and ξ are mutual conjugate entangled states. All the discussions are proceeded by virtue of the IWOP technique.
TL;DR: In this article, the space of Riemannian metrics for which the Dirac operator is invertible was studied on a compact spin manifold and the first main result is a surgery theorem stating that such a metric can be extended.
Abstract: On a compact spin manifold we study the space of Riemannian metrics for which the Dirac operator is invertible. The first main result is a surgery theorem stating that such a metric can be extended ...