Showing papers on "Nuclear operator published in 1986"
31 Dec 1986
TL;DR: In this paper, the authors present operators and basic applications for factorization through a Hilbert space Type and cotype, and apply them to Grothendieck's conjecture and the volume ratio method.
Abstract: Absolutely summing operators and basic applications Factorization through a Hilbert space Type and cotype. Kwapien's theorem The ""abstract"" version of Grothendieck's theorem Grothendieck's theorem Banach spaces satisfying Grothendieck's theorem Applications of the volume ratio method Banach lattices $C^*$-algebras Counterexamples to Grothendieck's conjecture.
580 citations
TL;DR: In this article, it was shown that a positive compact (ideal-) irreducible operator T in a Banach lattice has spectral radius r (T)>0, i.e., T is not quasi-nilpotent.
Abstract: The purpose of the present paper is to show that a positive compact (ideal-) irreducible operator T in a Banach lattice (of dimension greater than one) has spectral radius r (T)>0 , i.e., T is not quasi-nilpotent. It was shown in [6] that there exist positive irreducible operators which are quasi-nilpotent and the question under what conditions a positive irreducible operator T satisfies r ( T ) > 0 has been studied extensively (see e.g. [6] and [7], Sect. V.6). In particular we mention the Ando-Krieger theorem (see e.g. [9], Theorem 136.9), which states that a positive irreducible kernel operator T in a Banach function space has r (T)>0. Recently in [8], some special situations are discussed in which a compact positive irreducible operator has a strictly positive spectral radius. Let L be a (real or complex) Banach lattice. By 2~~ we denote the Banach space of all bounded linear operators in L. As usual, we write S__< T in ~ ( L ) whenever Su
145 citations
TL;DR: In this article, the authors studied the Weyl transforms of the tempered distributions on the phase space and that of the star-exponentials which gave the spectrum in this process of quantization.
54 citations
TL;DR: In this paper, Lipschitz-continuous nonlinear maps in finite-dimensional Banach and Hilbert spaces are characterized quantitatively in terms of certain functionals, which are used to assess qualitative properties such as invertibility and enable a generalization of some well-known matrix results directly to nonlinear operators.
Abstract: We consider Lipschitz-continuous nonlinear maps in finite-dimensional Banach and Hilbert spaces. Boundedness and monotonicity of the operator are characterized quantitatively in terms of certain functionals. These functionals are used to assess qualitative properties such as invertibility, and also enable a generalization of some well-known matrix results directly to nonlinear operators. Closely related to the numerical range of a matrix, the Gerschgorin domain is introduced for nonlinear operators. This point set in the complex plane is always convex and contains the spectrum of the operator's Jacobian matrices. Finally, we focus on nonlinear operators in Hilbert space and hint at some generalizations of the von Neumann spectral theory.
34 citations
TL;DR: In this article, the extremal structure of the dual unit balls of various operator spaces is studied and applications to the duality of operator spaces and differentiability properties of the norm in operator spaces are given.
Abstract: We study the extremal structure of the dual unit balls of various operator spaces. Mainly, we show that the classes of [w*-] strongly exposed, [w*-] exposed, and denting points in the dual unit balls of spaces of compact operators between Banach spacesX andY are completely — and in a canonical way — determined by the corresponding classes of points in the unit balls of the (bi-)duals of the factor spacesX andY. Applications to the duality of operator spaces and differentiability properties of the norm in operator spaces are given.
25 citations
21 citations
TL;DR: In this article, the authors prove the stability of the index and the semicontinuity of the dimensions of the cohomology groups of semi-Fredholm complexes of Banach spaces and closed linear operators with respect to perturbations of the operators and of the underlying spaces.
17 citations
TL;DR: In this article, it was shown that the trace class norm of the Hankel operator in terms of the symbol function cannot be estimated for since the first two terms in the coefficient sequence have been removed by differentiation.
Abstract: Peller [4, 5] has proved that a Hankel operator S on the Hardy space H2 is in the trace class if and only if with h analytic on the open unit disc Dand with its second derivative belonging to the Bergman space L1a. This theorem does not include an estimate for the trace class norm ∥S∥1, of the operator in terms of the symbol function. In fact it is clear that cannot give an estimate for since the first two terms in the coefficient sequence of the Hankel operator have been removed by differentiation.
14 citations
TL;DR: In this paper, the authors give summability results for the eigenvalues of certain types of compact operators that are then applied to study integral operators and prove that they can be used to obtain integral operators.
13 citations
Book•
01 Jan 1986
TL;DR: The theory of decomposable operators has been studied extensively in the literature, e.g. in this article, where the invariant subspace problem has been considered and a characterization of generalized zero of negative type of functions of the class N has been given.
Abstract: Some topics in the theory of decomposable operators.- Korovkin closures in Banach algebras.- On the theory of the class % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqamaaBa % aaleaacqGH1ecWdaWgaaadbaGaaGimaaqabaaaleqaaaaa!397A!$${A_{ {\aleph _0}}}$$ with applications to invariant subspaces and the Bergman shift operator.- Examples of chains of invariant subspaces.- Isometric dilations of commuting contractions. IV.- Recent results on reflexivity of operators and algebras of operators.- Schur analysis for matrices with finite number of negative squares.- Uniform algebras, Hankel operators and invariant subspaces.- Contractions generiques.- Hilbert modules over function algebras.- Differential equation of an inner function. II.- On a class of unitary operators in Krein space.- Naimark dilations, state-space generators and transmission lines.- Contractions being weakly similar to unitaries.- A characterization of generalized zeros of negative type of functions of the class N?.- Skew-symmetric operators and isometries on a real Banach space with a hyperorthogonal basis.- On the operator equation ax + (ax)* ? ?x= b.- On the inverse problem of a dissipative scattering theory. I.- Some spectral properties of an operator.- Hyponormal operators and eigendistributions.- The invariant subspace problem: a description with further applications of a combinatorial proof.- Some results on cohomology with Borel cochains, with applications to group action on operator algebras.- Spectral mapping theorems for analytic functional calculi.- Prediction theory and choice sequences: an alternate approach.- Idael properties of order bounded operators on ordered Banach spaces which are not Banach lattices.- More about pseudodifferential operators on bounded domains.
TL;DR: In this paper, the authors prove a trace theorem for anisotropic weighted Sobolev spaces in a cube Q naturally associated to a class of degenerate elliptic operators, where the fundamental property of this class is the existence of a suitable metric d which is natural for the operators.
Abstract: In this paper, we prove a trace theorem for anisotropic weighted Sobolev spaces in a cube Q naturally associated to a class of degenerate elliptic operators. The fundamental property of this class is the existence of a suitable metric d which is “natural” for the operators. The basic tool of the proof is a representation formula obtained via suitable non-euclidean translations closely fitting the geometry of the d-balls. In a more particular situation, we construct a right inverse of the trace operator and we describe the compatibility conditions on the edges of Q.
TL;DR: In this paper, the authors generalize Zagier's results to the case of the Hilbert modular forms for the congruence subgroup Γ 0 (M) and show that the second L-functions attached to the Hilbert cusp forms are special values of the trace of the Hecke operators.
Abstract: In this paper, we consider 1) the explicit formula of the trace of the Hecke operators acting on the space of the Hilbert cusp forms, and 2) the special values of the second L-functions attached to the Hilbert cusp forms. The method is that of Zagier[12], and the results of this paper are the generalization of his results to the case of the Hilbert modular forms for the congruence subgroup Γ0 (M).
01 Jan 1986
TL;DR: In this paper, it was shown that when E' has the Radon-Nikodym property, every weakly compact operator on C(S, E) can be lifted to a weaker operator on K( K, E).
Abstract: Let K and S be compact Hausdorff spaces and 8 a continuous function from K onto S. Then for any Banach space E the map / -» / ° 9 isometrically embeds C(S, £) as a closed subspace of C(K, E). In this note we prove that when E' has the Radon-Nikodym property, every weakly compact operator on C(S, E) can be lifted to a weakly compact operator on C( K, E). As a consequence, we prove that the compact dispersed spaces K are characterized by the fact that C(K, E) has the Dunford-Pettis property whenever E has.
TL;DR: In this article, the stability of non-semi-Fredholm operators in arbitrary Banach spaces was studied under compact perturbations, and the authors derived a curious characterization of separable Banach space.
TL;DR: In this paper, the spectral properties of Riesz operators on Banach lattices were established and applied to the spectral analysis of convolution operators on convolutional lattices.
Abstract: The r-asymptotically quasi finite rank operators were introduced in [10]. For regular operators on Banach lattices, these operators are the order theoretic analogue of Riesz operators on Banach spaces. We establish their basic properties and apply these in the spectral analysis of convolution operators.
TL;DR: In this paper, it is shown that if a locally convex Hausdorff space X inherits any completeness properties that the space of continuous linear operators, L (X), in X, may have (for the topology of pointwise convergence in X), this is false if sequential completeness is substituted for quasicompleteness.
Abstract: Whereas a locally convex Hausdorff space X inherits any completeness properties that the space of continuous linear operators, L (X), in X, may have (for the topology of pointwise convergence in X), this is not so in the converse situation and is the problem discussed here. The barrelledness of X in its Mackey topology plays an important role: if L (X) is quasicomplete, then X is barrelled for its Mackey topology. Consequently, for Mackey spaces X is turns out that L (X) is quasicomplete if and only if X is quasicomplete and barrelled: this is false if sequential completeness is substituted for quasicompleteness. Furthermore, there exist non-barrelled spaces X for which X and L (X) are quasicomplete (sequentially complete). Hence, although barrelledness is a sufficient condition for completeness of L (X) in various senses, it is certainly not necessary.
TL;DR: This chapter estimates the nuclear norm ∥T∥ 1 for certain elementary Hankel operators T and shows that a non-negative decreasing sequence {a n } is the coefficient sequence of a nuclear operator if the sequence is also convex, but can fail to be the coefficient sequences of anuclear operator in the absence of convexity.
Abstract: Publisher Summary This chapter discusses some nuclear Hankel operators. The chapter establishes certain sufficient conditions for nuclearity directly in terms of the coefficient sequence {an}. The chapter estimates the nuclear norm ∥T∥1 for certain elementary Hankel operators T. As a corollary, the chapter shows that a non-negative decreasing sequence {an} is the coefficient sequence of a nuclear operator if the sequence is also convex, but can fail to be the coefficient sequence of a nuclear operator in the absence of convexity. The coefficient sequence {an} is a sequence of complex numbers, and sufficient conditions are obtained to support that it is the coefficient sequence of a nuclear Hankel operator.
01 Feb 1986
TL;DR: In this paper, a complete characterization of the commutant of a class of cyclic subnormal operators closely related to the unilateral shift is given, where the authors define a measure IL such that the analytic bounded point evaluations of p2(tL) n LX(yt) is the set of bounded analytic functions on the open unit disk D in the complex plane.
Abstract: An m-measure is defined to be a measure IL such that the analytic bounded point evaluations of p2(ft) iS the open unit disk D in the complex plane, and the weak* closure of the analytic polynomials in Lf(tL) is the set of bounded analytic functions on D. A complete characterization of p2(tL) n LX(yt), the commutant of the cyclic subnormal operator of multiplication by 7 on p2 ( t), is then obtained. In this paper a complete characterization is given of the commutant of a class of cyclic subnormal operators closely related to the unilateral shift. An operator S on a Hilbert space X is subnormal if there is a Hilbert space Xr containing X#' and a normal operator N on X such that N(,#') c X#' and S = NjY (the restriction of N to k'). The weak* topology on B(-') is the topology which B(.Xk) has as the Banach space dual of the trace class operators [4]. A measure ts is always a compactly supported, positive regular Borel measure on the complex plane, C. If S is a cyclic subnormal operator, then there exists a measure yt such that S is unitarily equivalent to S , the operator of multiplication by z on p2(ft) = the closure of the analytic polynomials in L2(yt) [4]. Yoshino's Theorem [4] states that the map from p2(.t) n Ll([) onto { SI,}' = the commutant of Sl,, given by f (S,,) = multiplication by 4, is an isometric isomorphism and a weak* homeomorphism. For functions f, g in L2(tt), (f, g) -ff gdet, If1112 = ((f f)1/)2 and Ilf II,, denotes the ti-essential supremum of f. Let m denote normalized arc length measure on aD, the boundary of the open unit disk. Thus Sr, is the unilateral shift. A class of measures with many of the properties of m will be defined after some notation is set. If ,t is a measure, then B(,u), the set of bounded point evaluations of p2(t), consists of those X in C for which the linear functional p p(X) has a bounded extension from the polynomials to p2(n4). Equivalently, X E B([t) if and only if there exists k. in p2(t,) such that p(X) = (p, kx) for all polynomials p. Bj(p4) the set of analytic bounded point evaluations of P2(11), is the largest open subset of B(i) such that the function
01 Jan 1986
TL;DR: In this paper, the authors present an operator on the space l 1 of a weighted shift operator, which can be chosen in such a way that, though T has no invariant subspaces, T2 does.
Abstract: The paper falls conveniently into three sections: first, the motivation for the definitions made in the original proof [1] (together with a clear outline of how the proof proceeds); second, a description of how we obtain an operator on the space l1; and third, an account of the spectrum of our operator on l1, of how it is related to a weighted shift operator, and of how it can be chosen in such a way that, though T has no invariant subspaces, T2 does.
TL;DR: In this article, the authors present certain aspects of the theory of spectral operators in Grothendieck spaces with the Dunford-Pettis property, briefly, GDP-spaces, thereby elaborating on the recent note [ 10 ].
Abstract: The purpose of this note is to present certain aspects of the theory of spectral operators in Grothendieck spaces with the Dunford-Pettis property, briefly, GDP-spaces, thereby elaborating on the recent note [ 10 ]. For example, the sum and product of commuting spectral operators in such spaces are again spectral operators (cf. Proposition 2.1) and a continuous linear operator is spectral if and only if it has finite spectrum (cf. Proposition 2.2). Accordingly, if a spectral operator is of finite type, then its spectrum consists entirely of eigenvalues. Furthermore, it turns out that there are no unbounded spectral operators in such spaces (cf. Proposition 2.4). As a simple application of these results we are able to determine which multiplication operators in certain function spaces are spectral operators.
Journal Article•
TL;DR: In this article, the basic facts of random elements and random operators acting in Hilbert spaces are given, and reduction methods are developed for several models for a measurement scheme, including methods for a model with a random operator.
Abstract: The basic facts are given in the theory of random elements and random operators acting in Hilbert spaces. Reduction methods are developed for several models for a measurement scheme, including methods for a model with a random operator.Bibliography: 6 titles.
TL;DR: In this paper, the notion of trace and determinant was extended to a wider set of compact operators than the set of trace class operators, and the trace and the determinant were expressed via the eigenvalues of the operator.
Abstract: In the present paper we extend the notion of trace and determinant for wider sets of compact operators than the set of trace class operators. We express the trace and the determinant via the eigenvalues of the operator. As an application, we obtain a spectral expression for detL(λ) where
$$L(\lambda ) = I - \sum\limits_{j = 1}^n {\lambda ^j \varepsilon H_j ,with} H_j S_1 ,j \geqq 1$$