Abstract: Preface to Third Edition. Preface to Second Edition. Preface to First Edition. 0 Preliminaries. 0.1 Heat Conduction. 0.2 Diffusion. 0.3 Reaction-Diffusion Problems. 0.4 The Impulse-Momentum Law: The Motion of Rods and Strings. 0.5 Alternative Formulations of Physical Problems. 0.6 Notes on Convergence. 0.7 The Lebesgue Integral. 1 Green s Functions (Intuitive Ideas). 1.1 Introduction and General Comments. 1.2 The Finite Rod. 1.3 The Maximum Principle. 1.4 Examples of Green s Functions. 2 The Theory of Distributions. 2.1 Basic Ideas, Definitions, and Examples. 2.2 Convergence of Sequences and Series of Distributions. 2.3 Fourier Series. 2.4 Fourier Transforms and Integrals. 2.5 Differential Equations in Distributions. 2.6 Weak Derivatives and Sobolev Spaces. 3 One-Dimensional Boundary Value Problems. 3.1 Review. 3.2 Boundary Value Problems for Second-Order Equations. 3.3 Boundary Value Problems for Equations of Order p. 3.4 Alternative Theorems. 3.5 Modified Green's Functions. 4 Hilbert and Banach Spaces. 4.1 Functions and Transformations. 4.2 Linear Spaces. 4.3 Metric Spaces, Normed Linear Spaces, and Banach Spaces. 4.4 Contractions and the Banach Fixed-Point Theorem. 4.5 Hilbert Spaces and the Projection Theorem. 4.6 Separable Hilbert Spaces and Orthonormal Bases. 4.7 Linear Functionals and the Riesz Representation Theorem. 4.8 The Hahn-Banach Theorem and Reflexive Banach Spaces. 5 Operator Theory. 5.1 Basic Ideas and Examples. 5.2 Closed Operators. 5.3 Invertibility: The State of an Operator. 5.4 Adjoint Operators. 5.5 Solvability Conditions. 5.6 The Spectrum of an Operator. 5.7 Compact Operators. 5.8 Extremal Properties of Operators. 5.9 The Banach-Schauder and Banach-Steinhaus Theorems. 6 Integral Equations. 6.1 Introduction. 6.2 Fredholm Integral Equations. 6.3 The Spectrum of a Self-Adjoint Compact Operator. 6.4 The Inhomogeneous Equation. 6.5 Variational Principles and Related Approximation Methods. 7 Spectral Theory of Second-Order Differential Operators. 7.1 Introduction The Regular Problem. 7.2 Weyl s Classification of Singular Problems. 7.3 Spectral Problems with a Continuous Spectrum. 8 Partial Differential Equations. 8.1 Classification of Partial Differential Equations. 8.2 Well-Posed Problems for Hyperbolic and Parabolic Equations. 8.3 Elliptic Equations. 8.4 Variational Principles for Inhomogeneous Problems. 8.5 The Lax-Milgram Theorem. 9 Nonlinear Problems. 9.1 Introduction and Basic Fixed-Point Techniques. 9.2 Branching Theory. 9.3 Perturbation Theory for Linear Problems. 9.4 Techniques for Nonlinear Problems. 9.5 The Stability of the Steady State. 10 Approximation Theory and Methods. 10.1 Nonlinear Analysis Tools for Banach Spaces. 10.2 Best and Near-Best Approximation in Banach Spaces. 10.3 Overview of Sobolev and Besov Spaces. 10.4 Applications to Nonlinear Elliptic Equations. 10.5 Finite Element and Related Discretization Methods. 10.6 Iterative Methods for Discretized Linear Equations. 10.7 Methods for Nonlinear Equations. Index.

Abstract: 1. Gaussian Hilbert spaces 2. Wiener chaos 3. Wick products 4. Tensor products and Fock spaces 5. Hypercontractivity 6. Distributions of variables with finite chaos expansions 7. Stochastic integration 8. Gaussian stochastic processes 9. Conditioning 10. Limit theorems for generalized U-statistics 11. Applications to operator theory 12. Some operators from quantum physics 13. The Cameron-Martin shift 14. Malliavin calculus 15. Transforms Appendices.

Abstract: 0. Introduction.- 1. Classical Time-Decaying Forces.- 2. Classical 2-Body Hamiltonians.- 3. Quantum Time-Decaying Hamiltonians.- 4. Quantum 2-Body Hamiltonians.- 5. Classical N-Body Hamiltonians.- 6. Quantum N-Body Hamiltonians.- A. Miscellaneous Results in Real Analysis.- A.1 Some Inequalities.- A.2 The Fixed Point Theorem.- A.3 The Hamilton-Jacobi Equation.- A.4 Construction of Some Cutoff Functions.- A.5 Propagation Estimates.- A.6 Comparison of Two Dynamics.- A.7 Schwartz's Global Inversion Theorem.- B. Operators on Hilbert Spaces.- B.1 Self-adjoint Operators.- B.2 Convergence of Self-adjoint Operators.- B.3 Time-Dependent Hamiltonians.- B.4 Propagation Estimates.- B.5 Limits of Unitary Operators.- B.6 Schur's Lemma.- C. Estimates on Functions of Operators.- C.1 Basic Estimates of Commutators.- C.2 Almost-Analytic Extensions.- C.3 Commutator Expansions I.- C.4 Commutator Expansions II.- D. Pseudo-differential and Fourier Integral Operators.- D.0 Introduction.- D.1 Symbols of Operators.- D.2 Phase-Space Correlation Functions.- D.3 Symbols Associated with a Uniform Metric.- D.4 Pseudo-differential Operators Associated with a Uniform Metric.- D.5 Symbols and Operators Depending on a Parameter.- D.6 Weighted Spaces.- D.7 Symbols Associated with Some Non-uniform Metrics.- D.8 Pseudo-differential Operators Associated with the Metric 91.- D.9 Essential Support of Pseudo-differential Operators.- D.10 Ellipticity.- D.12 Non-stationary Phase Method.- D.13 FIO's Associated with a Uniform Metric.- D.14 FIO's Depending on a Parameter.- References.

Abstract: Stochastic Differential Equations.- Representations of Solutions.- Regularity of Solutions.- One-dimensional Diffusions.- Nondivergence form Operators.- Martingale Problems.- Divergence Form Operators.- The Malliavin Calculus.

Abstract: The entities A, B, X, Y in the title are operators, by which we mean either linear transformations on a finite-dimensional vector space (matrices) or bounded (= continuous) linear transformations on a Banach space. (All scalars will be complex numbers.) The definitions and statements below are valid in both the finite-dimensional and the infinite-dimensional cases, unless the contrary is stated. 1991 Mathematics Subject Classification 15A24, 47A10, 47A62, 47B47, 47B49, 65F15, 65F30.

Abstract: Preface. Notations. 1. G-Convergence of Abstract Operators. 2. Strong G-Convergence of Nonlinear Elliptic Operators. 3. Homogenization of Elliptic Operators. 4. Nonlinear Parabolic Operators. A: Homogenization of Nonlinear Difference Schemes. B: Open Problems. References. Index.

Abstract: This article is concerned with the operators from harmonic analysis which are naturally associated to a multiple parameter family of dilations. We are especially interested here in dealing with questions from the theory of such operators whose answers cannot be obtained by a reduction to the case of product operators. We also introduce a new tool in order to carry out this multiparameter analysis which comes from the classical theory. This is the concept of "sharp weighted inequalities," where, for operators such as the Hilbert transform or classical square function, one asks for Hilbert space inequalities which imply the correct estimates of the norm on LP of these operators as P becomes large. 0. Introduction. In this article, our main goal is to study classes of oper ators associated with multiparameter groups of dilations on Rn. In some sense the consideration of these operators is a natural "next step" or simplest case after those of the classical Calderon-Zygmund theory, and the product space theory. It turns out that in order to obtain estimates for the multiparameter operators we shall be led to the following type of extremely classical question: How does one obtain L2 estimates on R1 for the Hilbert transform (or other singular integrals) which yield sharp estimates on the LP(Rl) operator norm of H as p approaches 1 (or oc)? These "sharp weighted inequalities" are of some independent interest, and their study will be seen to be essentially equivalent to the deep work of Chang, Wilson, and Wolff (3) on the exponential square integrability of functions whose area integral is bounded. They also have other applications; for example, they provide an immediate extension of the Chang-Wilson-Wolff estimates to product spaces as given by Pipher in (12). At this point, let us be more specific about the exact nature of the multipa rameter operators which we shall consider. Consider, in R3, the dilations given by ps,t(x, y, z) = (sx, ty9 stz), for s91 > 0. Then there is a maximal operator (first considered by Zygmund) Ml and singular integrals Tl (introduced by Ricci and Stein (13)) naturally associated to these dilations. The maximal operator Af3 is defined by

Abstract: For Belavin's elliptic quantum R-matrix, we construct an L-operator as a set of difference operators acting on functions on the type A weight space. According to the fundamental relation RLL=LLR, taking the trace of the L-operator gives a set of commuting difference operators. We show that for the above mentioned L-operator this approach gives Macdonald type operators with elliptic theta function coefficient, actually equivalent to Ruijsenaars' operators. The relationship between the difference L-operator and Krichever's Lax matrix is given, and an explicit formula for elliptic commuting differential operators is derived. We also study the invariant subspace for the system which is spanned by symmetric theta functions on the weight space.

Abstract: These lectures will focus on those properties of maximal monotone operators which are valid in arbitrary real Banach spaces.

Abstract: An introduction to linear PDE - generalized functions and wave front sets, pseudodifferential operators and Fourier integral operators linear PDE with multiple characteristics - the symplectic case, the involutive case, perturbations of powers of operators of principal type general theory for nonlinear PDE - local solvability for nonlinear equations with multiple characteristics.

Abstract: In this paper we characterize the composition operators on Orlicz spaces and study some of their properties.

Abstract: In this paper, by applying (p,k)-epi mapping theory, we introduce a new definition of spectrum for nonlinear operators which contains all eigenvalues, as in the linear case. Properties of this spectrum are given and comparison is made with the other definitions of spectra. We also give applications of the new theory.

Abstract: Floquet theory for ordinary differential operators with periodic coefficients is well known and fundamental to study the stability of solutions and spectral properties of the operators. Generalization of Floquet theory to differential operators with almost periodic or more random coefficients have been done by several authors, by applying C*-algebra theory, dynamical theory and probability theory. The purpose of this paper is to give a general point of view of Floquet theory using the Weyl's functions systematically, which was initiated by Johnson-Moser. 0 1997 Published by Elsevier Science Ltd

Abstract: This paper addresses the question of understanding quantum algorithms in terms of unitary operators. Many quantum algorithms can be expressed as applications of operators formed by conjugating so-called classical operators. The operators that are used for conjugation are determined by the problem and any additional structure possessed by the Hilbert space that is acted upon. We prove many new commutative laws between these different operators, and we use those to phrase and analyze old and new problems and algorithms. As an example, we review the Abelian subgroup problem. We then introduce the problem of determining a group homomorphism, and we give classical and quantum algorithms for it. We also generalize Deutsch''s problem and improve the previous best algorithms for earlier generalizations of it.

Abstract: In this paper we generalize the following consequence of a wellknown result of Nagy: if T and T−1 are power bounded operators, then T is a polynomially bounded operator. Let H be a separable, infinite dimensional, complex Hilbert space, and let L(H) denote the algebra of all bounded linear operators on H. Recall that an operator T ∈ L(H) is called power bounded (notation: T ∈ PW(H)) if there exists a constant M(≥ 1) such that ‖T n‖ ≤M, n ∈ N, (1) and T is called polynomially bounded (notation: T ∈ PB(H)) if there exists a constant M(≥ 1) such that ‖p(T )‖ ≤M‖p‖∞ (2) for every polynomial p, where ‖p‖∞ =sup{|p(z)| : z ∈ C , |z| ≤ 1}. The smallest number M satisfying (1) (resp., (2)) is called the power bound ( resp., the polynomial bound ) of T and will be denoted by Mw(T ) (resp., Mp(T )), or simply Mw (resp., Mp) when no confusion is possible. One knows (cf. [1, 2, 4]) that PW(H) strictly contains the class PB(H), but there is a theorem of Nagy [3] which says that every T ∈ PW(H) such that T−1 exists and belongs to PW(H) is similar to a unitary operator, and therefore is polynomially bounded. The purpose of this note is to establish the following two stronger results than the above-mentioned consequence of Nagy’s theorem. Theorem 1.1. Suppose T ∈ PW(H)(with Mw(T ) > 1) and the following inequality holds for some positive number α and a strictly increasing sequence {nk} ⊂ N : 1/nk nk ∑ j=0 T ∗jT j ≥ α(I − Pker(T )), (3) where Pker(T ) is the (orthogonal) projection on the kernel of T . Then T ∈ PB(H) and the polynomial bound Mp of T satisfies Mp ≤Mw(T ) ( Mw(T ) 2 − 1 αlnMw(T ) )1/2 + 1. (4) Received by the editors June 19, 1995 and, in revised form, November 20, 1995. 1991 Mathematics Subject Classification. Primary 47B99. c ©1997 American Mathematical Society 1435 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 1436 EUGEN J. IONASCU Theorem 1.2. Suppose T ∈ PW(H) and the following inequality holds for some positive number α and a strictly increasing sequence {tk}k∈N of real numbers converging to 1 : (1− tk) ∞ ∑ j=0 tjkT ∗jT j ≥ α(I − Pker(T )). (5) Then T ∈ PB(H), and the polynomial bound Mp of T satisfies Mp ≤ ( 14 α )1/2 M w. (6) As mentioned above, the following is an immediate consequence of either Theorem 1.1 or Theorem 1.2 . Corollary 1.3 (Nagy [4]). If T ∈ PW(H) is invertible and T−1 ∈ PW(H), then T is polynomially bounded . In order to prove Theorem 1.1, we use the following lemma, which is well known [5]. Lemma 1.4. Suppose S ∈ L(H) is such that S is a contraction for some integer m ≥ 2. Then S is similar to a contraction, and, in particular, A = (I + S∗S + ... + S∗(m−1)S(m−1))1/2 (7) is an invertible operator that satisfies ‖ASA−1‖ ≤ 1. (8) Proof. Clearly A is an invertible selfadjoint operator. To establish (8) it is enough to check that ‖ASA−1h‖ ≤ ‖h‖ , h ∈ H. (9) For a given h, define g = A−1h, and hence (9) becomes equivalent to 〈A2Sg, Sg〉 ≤ 〈A2g, g〉. (10) Using (7), we see that (10) is equivalent to m ∑

Abstract: In this paper we investigate several interesting properties of the fractional calculus operators, the Carlson-Shaffer linear operator L(a,c) and a certain integral operator , associated with various subclasses of analytic and univalent functions. We also investigate the relationships of some of these classes with the Hardy space H ∞ (of bounded analytic functions in the open unit disk μ and with some families of multiplier transformations.

Abstract: This paper investigates two constraints for the connected operator class. For binary images, connected operators are those that treat grains and pores of the input in an all or nothing way, and therefore they do not introduce discontinuities. The first constraint, called connected-component (c.c.) locality, constrains the part of the input that can be used for computing the output of each grain and pore. The second, called adjacency stability, establishes an adjacency constraint between connected components of the input set and those of the output set. Among increasing operators, usual morphological filters can satisfy both requirements. On the other hand, some (non-idempotent) morphological operators such as the median cannot have the adjacency stability property. When these two requirements are applied to connected and idempotent morphological operators, we are lead to a new approach to the class of filters by reconstruction. The important case of translation invariant operators and the relationships between translation invariance and connectivity are studied in detail. Concepts are developed within the binary (or set) framework; however, conclusions apply as well to flat non-binary (gray-level) operators.

Abstract: Classical properties, associated with a large phase space domain with a smooth boundary as compared to the Planck constant, are shown to be expressed quantum-mechanically by a family of quantum projection operators with the help of known theorems in microlocal analysis. Under the conditions of Egorov’s theorem, the conservation of this correspondence under classical/quantum dynamics is asserted.

Abstract: In this paper, several sufficient conditions for boundedness of the Hilbert transform between two weighted Lp-spaces are obtained. Invariant A8 weights are obtained. Several characterizations of invariant A8 weights are given. We also obtain some sufficient conditions for products of two Toeplitz operators of Hankel operators to be bounded on the Hardy space of the unit circle using Orlicz spaces and Lorentz spaces.

Abstract: We study Toeplitz operators on the harmonic Bergman spacebp(B), whereB is the open unit ball inRn(n≥2), for 1