Showing papers on "Nuclear operator published in 1996"

Introduction to spectral theory : with applications to Schrödinger operators

Abstract: 1 The Spectrum of Linear Operators and Hilbert Spaces.- 2 The Geometry of a Hilbert Space and Its Subspaces.- 3 Exponential Decay of Eigenfunctions.- 4 Operators on Hilbert Spaces.- 5 Self-Adjoint Operators.- 6 Riesz Projections and Isolated Points of the Spectrum.- 7 The Essential Spectrum: Weyl's Criterion.- 8 Self-Adjointness: Part 1. The Kato Inequality.- 9 Compact Operators.- 10 Locally Compact Operators and Their Application to Schrodinger Operators.- 11 Semiclassical Analysis of Schrodinger Operators I: The Harmonic Approximation.- 12 Semiclassical Analysis of Schrodinger Operators II: The Splitting of Eigenvalues.- 13 Self-Adjointness: Part 2. The Kato-Rellich Theorem 131.- 14 Relatively Compact Operators and the Weyl Theorem.- 15 Perturbation Theory: Relatively Bounded Perturbations.- 16 Theory of Quantum Resonances I: The Aguilar-Balslev-Combes-Simon Theorem.- 17 Spectral Deformation Theory.- 18 Spectral Deformation of Schrodinger Operators.- 19 The General Theory of Spectral Stability.- 20 Theory of Quantum Resonances II: The Shape Resonance Model.- 21 Quantum Nontrapping Estimates.- 22 Theory of Quantum Resonances III: Resonance Width.- 23 Other Topics in the Theory of Quantum Resonances.- Appendix 1. Introduction to Banach Spaces.- A1.1 Linear Vector Spaces and Norms.- A1.2 Elementary Topology in Normed Vector Spaces.- A1.3 Banach Spaces.- A1.4 Compactness.- 1. Density results.- 2. The Holder Inequality.- 3. The Minkowski Inequality.- 4. Lebesgue Dominated Convergence.- Appendix 3. Linear Operators on Banach Spaces.- A3.1 Linear Operators.- A3.2 Continuity and Boundedness of Linear Operators.- A3.3 The Graph of an Operator and Closure.- A3.4 Inverses of Linear Operators.- A3.5 Different Topologies on L(X).- Appendix 4. The Fourier Transform, Sobolev Spaces, and Convolutions.- A4.1 Fourier Transform.- A4.2 Sobolev Spaces.- A4.3 Convolutions.- References.

The operator Hilbert space OH, complex interpolation, and tensor norms

Abstract: Introduction The operator Hilbert space Complex interpolation The $oh$ tensor product Weights on partially ordered vector spaces $(2,w)$-summing operators The gamma-norms and their dual norms Operators factoring through $OH$ Factorization through a Hilbertian operator space On the "local theory" of operator spaces Open questions References.

Projection algorithms and monotone operators

Abstract: This thesis consists of two parts. In Part I, projection algorithms for solving convex feasibility problems in Hilbert space are studied. Powerful techniques from Convex Analysis are employed within a very general framework that covers and extends many well-known results. Ostensibly different looking conditions sufficient for linear convergence are shown to be special instances of regularity--a concept new in this context. Numerous examples, including subgradient algorithms, are presented. Several notions of monotonicity of operators on Banach spaces are analyzed in Part II. Utilizing Convex and Functional Analysis, it is shown that for a bounded linear positive semi-definite operator, all these "monotonicities" coincide with the monotonicity of the conjugate operator. Moreover, monotonicity of the conjugate operator is automatic in many classical Banach spaces but not in spaces containing a complemented copy of the space of absolutely convergent sequences.

On the structure of tensor products of $l_p$-spaces.

Abstract: A Banach space X is prime if every infinite-dimensional complemented subspace contains a further subspace which is isomorphic to X. A Banach space X is said to be primary if whenever X — Y © Z, X is isomorphic to either Y or Z. The classical examples of prime spaces are the spaces ipj 1 < p < oc. Many spaces derived from the £p-spaces in various ways are primary (see for example [AEO] and [CL]). The primarity of B(H) was shown by Blower [B] in 1990, and Arias [A] has recently developed further techniques which are used to prove the primarity of Cι, the space of trace class operators (this was first shown by Arazy [Arl, Ar2]). It has become clear that these techniques are not naturally confined to a Hubert space context; in the present paper we wish to extend the results to a variety of tensor products and operator spaces of ^-spaces (and in some cases £p-spaces). We also include some related results. Some of the intermediate propositions (on factoring operators through the identity) may actually be true for a wider class of Banach spaces (those with unconditional bases which have nontrivial lower and upper estimates). In fact, the combinatorial aspects of the factorization can be applied quite generally, and may have other applications. The proofs of primarity, however, rely on Pelczyήski's decomposition method which is not so readily extended. We have thus kept mainly to the case of injective and projective tensor products of tv spaces throughout. The results we obtain apply to the growing study of polynomials on Banach spaces since polynomials may be considered as symmetric multilinear operators with an equivalent norm (see [FJ], [M],

New results in the perturbation theory of maximal monotone and -accretive operators in Banach spaces

Abstract: Let X be a real Banach space and G a bounded, open and convex subset of X. The solvability of the fixed point problem (*) Tx + Cx 3 x in D(T) n G is considered, where T . X D D(T) -* 2X is a possibly discontinuous m-dissipative operator and C: G -X is completely continuous. It is assumed that X is uniformly convex, D(T) n G $& 0 and (T + C)(D(T) n OG) C G. A result of Browder, concerning single-valued operators T that are either uniformly continuous or continuous with X* uniformly convex, is extended to the present case. Browder's method cannot be applied in this setting, even in the single-valued case, because there is no class of permissible homeomorphisms. Let IF {= 7Z+ -+ R-; /3(r) -O 0 as r -* oo}. The effect of a weak boundary condition of the type (u + Cx, x) > -/3(11xfl)flxl12 on the range of operators T + C is studied for m-accretive and maximal monotone operators T. Here, ,B E F, x C D(T) with sufficiently large norm and u C Tx. Various new eigenvalue results are given involving the solvability of Tx + ACx 3 0 with respect to (A, x) E (0, oo) x D(T). Several results do not require the continuity of the operator C. Four open problems are also given, the solution of which would improve upon certain results of the paper.

C0-groups andC0-semigroups of linear operators on hereditarily indecomposable Banach spaces

Abstract: A Banach space X is called hereditarily indecomposable (briefly, H.I3 if, whenever Y and Z are closed, infinite dimensional subspaces of X and ~5 > 0, then there exist unit vectors y ~ Y and z ~ Z such that II Y z < 3. This is equivalent to the following property: whenever Y and Z are closed, infinite dimensional subspaces of X satisfying Yc~ Z = {0}, then Y + Z is non-closed. Examples of H.I. spaces were recently exhibited by Gowers and Maurey [9], where it was also shown that bounded linear operators T in such spaces are somewhat special, e.g. there is a unique point 2 r in the spectrum e(T) of T such that T 2 r I is strictly singular, [9; w This is further exemplified in the recent articles [8], [22]. In this note we consider the class of (closed) operators T on H.I. spaces which generate Co-groups or Co-semigroups. One of the main results is that generators of Co-groups are always bounded operators. Moreover, if the group is of polynomial growth, then ( T 2 ~ I ) k is compact for some k ~ N qsee Theorems 2.3 and 3.2), and ( T ~rI1 k is a finite rank operator iff a(T) is a finite set tsee Proposition 3.3). For the generator T of a uniformly bounded Co-grou p, the H.I. property places a severe (and somewhat curious~ restriction on or(T); namely, there is a finite set H ~ iN. such that ~(T) is contained in the rational span of H (cf. Proposition 3.4). There are also analogues of these results for discrete groups. For the case of Co-semigroups m H.I. spaces the situation changes somewhat. Firstly, the generator T need not be bounded; see Example 2.4. However. a(T) is always either a convergent sequence in tI2~ = G w {oo} or a finite set (possibly empty); see Proposition 2.2. If ~(T) ~ ~ is a bounded, infinite set, then T e L(X); see Proposition 1.3. Most importantly, the generator T of any Co-semigroup satisfies the spectral mapping theorem, i.e. e t~(r) = cr(e tr) \\ {0} ; see Proposition 2.5. As for the case of Co-groups, if (Z D(Tt\"t is the generator of a uniformly bounded Co-semigrou p, then a(T) c~ ilR cannot contain an infinite, rationally independent set (cf. Remark 2). If, in addition, the H.I. space is reflexive and or(T)c~ iN is an infinite set, then ~r(T)\\ iN is necessarily finite, T is bounded and T )~rI is compact; see Proposition 4.2.

Topics in Pseudo-Differential Operators

Abstract: Introduction Order and true order of linear operators in some vector spaces Asymptotic expansions of linear operators in some vector spaces Pseudo-differential operators in the spaces 1,s(Rn) Pseudo-differential operators in F(Lp) and in Hs(Rn) spaces Alternative representation formulas for operators G(x,D) and G(x,D) Kohn-Nirenberg homogeneous and C[infinity]-symbols and their associated operators Compactness of the operator A(x,D) - A(x,D) in the space-1(L1(Rn))(A(x,D) =A(x,D)-modulo the compact operators Gohbergis Lemma and applications References Index of symbols Subject index

Linear differential operators with unbounded operator coefficients and semigroups of bounded operators

Abstract: Associated with a family of evolution operators in a complex Banach space is a linear unbounded operator, which is studied with the aid of a semigroup of difference operators and a difference operator in a sequence space. Some formulas for the spectra of the linear operators in question (in particular, for abstract hyperbolic differential operators) and the spectrum mapping theorem for the semigroup of difference operators are obtained.

On pure subnormal operators with finite rank self-commutators and related operator, tuples

Abstract: In this paper, we study the model of a pure subnormal operator with finite rank self-commutator and of the relatedn-tuple of commuting linear bounded operators. We also give some applications of the model to the theory ofn-tuples of commuting operators with trace class self-commutators.

Trace Theorems for Sobolev Spaces of Variable Order of Differentiation

Abstract: We prove a trace theorem and an extension theorem for Sobolev spaces of variable order of differentiation which are defined by pseudodifferential operators.

Spectra, eigenvectors and overlap functions for representation operators of $q$-deformed algebras

Abstract: Operators of representations corresponding to symmetric elements of theq-deformed algebrasUq(su1,1),Uq(so2,1),Uq(so3,1),Uq(son) and representable by Jacobi matrices are studied. Closures of unbounded symmetric operators of representations of the algebrasUq(su1,1) andUq(so2,1) are not selfadjoint operators. For representations of the discrete series their deficiency indices are (1,1). Bounded symmetric operators of these representations are trace class operators or have continuous simple spectra. Eigenvectors of some operators of representations are evaluated explicitly. Coefficients of transition to eigenvectors (overlap coefficients) are given in terms ofq-orthogonal polynomials. It is shown how results on eigenvectors and overlap coefficients can be used for obtaining new results in representation theory ofq-deformed algebras.

Subnormal operators with compact selfcommutator

Abstract: If $S$ is a hyponormal operator, then Putnam's inequality gives an estimate on the norm of the self-commutator $[S^*,S]$, while the Berger-Shaw theorem gives (under appropriate cyclicity hypotheses) a corresponding estimate on the trace of $[S^*,S]$. Of course these results hold when $S$ is subnormal. In the subnormal setting, the author obtains useful estimates on the norm and essential norm of commutators of the form\break $[T_u,S]$, where $T_u$ is a Toeplitz operator with continuous symbol $u$. A consequence is the following compactness condition. If the essential spectrum of $S$ is the boundary of an open set, then $[S^*,S]$ is compact. The author also proves some trace estimates for commutators. His basic method is a careful analysis of positive operator-valued measures. Abstract For an arbitrary subnormal operator we estimate the essential norm and trace of commutators of the form [T u, S], whereT u is a Toeplitz operator with continuous symbol. In particular, we obtain criteria for the compactness of [S *,S]. The trace estimates apply to multiplication operators on Hardy spaces over general domains. (Less)

On the range of the sum of monotone operators in general Banach spaces

Abstract: The purpose of this paper is to generalize the Brezis-Haraux theorem on the range of the sum of monotone operators from a Hilbert space to general Banach spaces. The result obtained provides that the range R(A + BT) is topologically almost equal to the sum R(A) + R(B) where r is a compatible topology in X** x X* as proposed by Gossez. To illustrate the main result we consider some basic properties of densely maximal monotone operators.

Well-bounded operators on nonreflexive Banach spaces

Abstract: Every well-bounded operator on a reflexive Banach space is of type (B), and hence has a nice integral representation with respect to a spectral family of projections. A longstanding open question in the theory of well-bounded operators is whether there are any nonreflexive Banach spaces with this property. In this paper we extend the known results to show that on a very large class of nonreflexive spaces, one can always find a well-bounded operator which is not of type (B). We also prove that on any Banach space, compact well-bounded operators have a simple representation as a combination of disjoint projections.

On the hard and soft extensions of a nonnegative operator

Abstract: In terms of spaces of boundary values, i.e., in a form that, in the case of differential operators, leads immediately to the boundary conditions, we construct the hard and soft extensions of a nonnegative operator in Hilbert space, interpreted as perturbations with a change of the domain of definition of a given positivedefinite operator for which these extensions are assumed known.

Weighted Composition Operators on Hilbert Spaces of Vector-Valued Functions

Abstract: In this paper we investigate criterion for the hyponormality, cohyponormality, and normality of weighted composition operators acting on Hilbert spaces of vector-valued functions.

Notes on unbounded Toeplitz operators in Segal-Bargmann spaces

Abstract: Relations between different extensions of Toeplitz operators Tφ are studied. Additive properties of closed Toeplitz operators are investigated, in particular necessary and sufficient conditions are given and some applications in case of Toeplitz operators with polynomial symbols are indicated.

Hardy classes, integral operators, and duality on spaces of homogeneous type

Abstract: The authors study Hardy spaces, of arbitrary order, on a space of homogeneous type. This extends earlier work that treated only $H^p$ for $p$ near 1. Applications are given to the boundedness of certain singular integral operators, especially on domains in complex space.