Product Description
The intention of this book is to introduce students to active areas of research in mathematical physics in a rather direct way minimizing the use of abstract mathematics. The main features are geometric methods in spectral analysis, exponential decay of eigenfunctions, semi-classical analysis of bound state problems, and semi-classical analysis of resonance. A new geometric point of view along with new techniques are brought out in this book which have both been discovered within the past decade. This book is designed to be used as a textbook, unlike the competitors which are either too fundamental in their approach or are too abstract in nature to be considered as texts. The authors' text fills a gap in the marketplace. ... Read more

Product Description

Operator theory, system theory, scattering theory, and the theory of analytic functions of one complex variable are deeply related topics, and the relationships between these theories are well understood. When one leaves the setting of one operator and considers several operators, the situation is much more involved. There is no longer a single underlying theory, but rather different theories, some of them loosely connected and some not connected at all. These various theories, which one could call "multidimensional operator theory", are topics of active and intensive research.
The present volume contains a selection of papers in multidimensional operator theory. Topics considered include the non-commutative case, function theory in the polydisk, hyponormal operators, hyperanalytic functions, and holomorphic deformations of linear differential equations.
The volume will be of interest to a wide audience of pure and applied mathematicians, electrical engineers and theoretical physicists.

Product Description
This book presents the basic tools of modern analysis within the context of the fundamental problem of operator theory: to calculate spectra of specific operators on infinite dimensional spaces, especially operators on Hilbert spaces. The tools are diverse, and they provide the basis for more refined methods that allow one to approach problems that go well beyond the computation of spectra: the mathematical foundations of quantum physics, noncommutative K-theory, and the classification of simple C*-algebras being three areas of current research activity which require mastery of the material presented here. ... Read more

Customer Reviews (2)

Great author!
You can find the same material in a bunch of books... but the exposition by Arveson is really superb. He actually EXPLAINS things, not just a list of theorems and proofs with no apparent reason.

The book is 130p only, so it might be ideal for self study. Not everything is in the text but if you want to try the exercises (you might need further reading to do some of them) and you do the extra reading from other sources, you will end up getting a good understanding.

The chapters are: 1. Spectral theory and Banach algebras, 2. Operators on Hilbert space, 3. Asymptotics: Compact perturbations and Fredholm theory and 4. Methods and applications

approximatelly 170 exercises.

The most inspiring book on the subject I have seen, up to now.

A book to learn from.
The word "spectrum" comes from quantum theory, as does the
subject of operators in Hilbert space. Since the early days, the
subject has grown, and branched out in a variety of directions.
This fact has perhaps made it difficult for an instructor to
chose a book from which to teach the fundamental ideas;--
those ideas in a subject that stay constant, even when the
fashions change. The topics covered in Arveson's book are relevant now, and are likely to
be useful decades from now.They are beautifully presented.--
The material and the format have been tested in courses;-- and this book represents a formula that works! ... Read more

Product Description
This book constitutes a first- or second-year graduate course in operator theory. It is a field that has great importance for other areas of mathematics and physics, such as algebraic topology, differential geometry, and quantum mechanics. It assumes a basic knowledge in functional analysis but no prior acquaintance with operator theory is required. ... Read more

Customer Reviews (1)

Excellent, modern introduction to C*-algebras
The use of C*-algebras in operator theory is known as a "soft" technique, in contrast to the "hard" techniques that use deep results from analysis. The blending of algebra, topology, measure theory, and analysis to study operators has resulting in breathtaking advances, and this trend continues. Applications to physics, especially quantum field theory, has also resulted in fruitful collaborations between mathematicians and physicists. The author has given an introduction to C*-algebras in this book from the purely mathematical standpoint, but those interested in applications can gain much from the reading of this book.

Spectral theory is introduced in chapter 1, with a definition of Banach algebras immediately given on the first page. There are many advantages is presenting spectral theory in this general context, and the author illustrates these advantages throughout the chapter. For Banach algebras with a unit, Gelfand's theorem, giving the non-emptiness of the spectrum, is proven. The author also discusses the Gelfand representation, that says essentially that abelian Banach algebras act like continuous functions. He then restricts his attention to compact and Fredholm operators, and discusses their index theory. It is here that the reader can see the origin of the idea of a compact operator acting as a "zero element" in algebras of operators. This is readily apparent in the definition of the Calkin algebra, which is the quotient algebra modulo the compact operators.
By defining an involution on a Banach algebra, the author introduces C*-algebras in chapters 2 and 3, and proves the famous Gelfand-Naimark theorem. The latter allows one to view any C*-algebra as a collection of bounded operators on a separable Hilbert space. In addition, the spectral theorem for normal operators is proven. These two theorems give one confidence in the power of the theory of C*-algebras to study operators, but their nonconstructive nature sometimes is not of much use for calculating explicitly the spectra or to find the Hilbert space that serves as a representative example in the Gelfand-Naimark result.The author also introduces the concept of a hereditary C*-subalgebras, which is a generalization of the concept of an idea, and, like the algebra of compact operators, are always simple if they are subalgebras of simple C*-algebras. Also, the author introduces the important class of Toeplitz operators. These are intimately connected with "hard analysis" and have important applications in physics and cryptography. The author also uses them, via the consideration of the Toeplitz algebra, to introduce concepts in K-theory.

The theory of von Neumann algebras, or W*-algebras as they are sometimes called, is discussed in chapter 4. His viewpoint of them is characteristically modern, as essentially a noncommutative measure theory. This viewpoint meshes will with current research in the field of noncommutative geometry. Proofs of the double commutant theorem and the Kaplansky density theorem are given. The presence of the weak operator topology makes these objects of primary interest to applications in quantum physics, as it is this topology which is physically relevant. The famous "type" characterization of projections in von Neumann algebras is given in an addendum to the chapter.

The representation theory of C*-algebras is considered in chapter 5. The author shows that topological irreducibility is equivalent to algebraic irreducibility for C*-algebras. The reader can see the role that ideals, especially the "primitive" ideals, play in the representation theory. The author also discusses CCR algebras, but he calls them "liminal" algebras. The theory of liminal algebras is of upmost importance in applications to quantum physics (physicists still call them CCR algebras).

Chapter 6 is an introduction to the construction of C*-algebras using direct limits and tensor products of given C*-algebras. And here again, the physicist reader will find a useful class of algebras, namely the AF-algebras, which are used heavily in mathematical statistical mechanics. Nuclear C*-algebras, which are the most well-behaved class under the operation of tensor product, are discussed briefly.

The last chapter of the book is the most interesting, for it deals with the K-theory of C*-algebras. The Brown-Douglas-Fillmore theory was briefly mentioned in an addendum to chapter 2. This theory could be considered a precursor to latter work on K-theory of operator algebras. The author explains the origin of the K-groups K0(A) and K1(A) assigned to a C*-algebra, and how they can be used to study some properties of A. The K-groups are constructed by first forming the set P(A) consisting of the union of all projections in the collection of n by n matrices over an arbitrary *-algebra A. Then for a unital *-algebra A, a notion of stable equivalance is defined for elements of P(A). The reader familiar with the K-theory of vector bundles will see the similarity in this definition. Here the two projections are stably equivalent if they are equivalent under the direct sum of the n x n identity operator. The equivalance classes under stability are then enveloped via the Grothendieck group, giving the K-group K0(A). Then the K0-group shown to be a complete isomorphism invariant for unital AF-algebras, i.e. two unital AF-algebras are *-isomorphic iff there is a unital order isomorphism between their K0-groups. Three fundamental results in K-theory are then discussed: "weak exactness", which gives an exact sequence of K0-groups given an exact sequence of C*-algebras; "homotopy invariance", which shows that mappings between the K0-groups of two C*-algebras are equal if the mappings between the C*-algebras are homotopic; "continuity": which gives a notion of continuity for the K0-functor. Then after defining a notion of stability for the K0-functor, the author proves an analog of Bott periodicity. This involves of course the construction of the K1-group, which is done in terms of the suspension of the C*-algebra, in complete analogy with the vector bundle case. ... Read more

Product Description
This invaluable book presents reviews of some recent topics in the theory of Schrödinger operators. It includes a short introduction to the subject, a survey of the theory of the Schrödinger equation when the potential depends on the time periodically, an introduction to the so-called FBI transformation (also known as coherent state expansion) with application to the semi-classical limit of the S-matrix, an overview of inverse spectral and scattering problems, and a study of the ground state of the Pauli–Fierz model with the use of the functional integral. The material is accessible to graduate students and non-expert researchers. ... Read more

Product Description
This volume is the first in the literature on the increasinglypopular Ordered Weighted Averaging (OWA) operators. These OWAoperators make it possible to change the form of aggregation from the`pessimistic' minimum-type aggregation through all intermediate typesincluding the conventional arithmetic mean and nonconventionalaggregations, to the `optimistic' maximum-type aggregations. Includedare contributions from a number of fields where these operators havebeen applied. These fields are decision analysis under uncertainty,learning and classification, multi-person decision-making andconsensus formation, and flexible database querying and informationretrieval. ... Read more

Product Description

Product Description
Deals with similarity and unitary equivalence and unilateral and bilateral shifts. Covers contractions, quasisimilarity, decompositions, models, applications and similarity. DLC: Hilbert space. ... Read more

Customer Reviews (1)

Nice and well organized introduction to a difficult subject
The book is a very nice introduction to the subject of Models and Decompositions of operators. In reality, it is a post-introductory book. Anyone that has already studied operator Theory in Hilbert spaces and wantsa deeper understanding of this subject, should have a look here, mainlybecause the book coherently organizes this subject that would otherwiserequire a large bibliographical research.Models and Decompositons arelooked as a sub-product of the search for the "Holy Graal" ofoperator theory: The Invariant Subspace Problem. Although it is not a bookabout that subject, it is certainly a relevant book to anyone entering thatfield, because it mentions relatively recent results that were previouslyconjectures in that area.Particularly interesting, in my opinion, arechapters 7 and 8. Chapter 7, called "Applications", in which thetheory developed in the previous chapters is applied and many interestingrelated problems, some of them still open, relative to strong and weakstability of operators are mentioned. Chapter 8 discusses the problem ofsimilarity to contractions, and leaves the reader with two related openquestions. ... Read more

Product Description
Together with "Theory of Operator Algebras I, III" (EMS 124 and 127), this book, written by one of the most prominent researchers in the field of operator algebras, summarises the scientific work of the author focusing on von Neumann algebras and non-commutative integration.The book provides essential and comprehensive information to graduate students and researchers in mathematics and mathematical physics. ... Read more

Product Description
This book provides an introduction to functional analysis with an emphasis on the theory of linear operators and its application to differential equations, integral equations, infinite systems of linear equations, approximation theory, and numerical analysis. As textbook designed for senior undergraduate and graduate students, it begins with the geometry of Hilbert spaces and proceeds to the theory of linear operators on these spaces including Banach spaces. Presented as a natural continuation of linear algebra, the book provides a firm foundation in operator theory which is an essential part of mathematical training for students of mathematics, engineering, and other technical sciences. Enriched new version of the book "Basic Operator Theory" by I. Gohberg and S. Goldberg (0-8176-4262-5). ... Read more

Product Description
This book is based on the CBMS Regional Conference held at the University of California, Irvine, in July 1985. Jordan algebras have found interesting applications in seemingly unrelated areas of mathematics such as operator theory, the foundations of quantum mechanics, complex analysis in finite and infinite dimensions, and harmonic analysis on homogeneous spaces. The author describes some relevant results and puts them in a general framework, based on the concepts of $JB$-algebra and $JB^\ast$-triple. ... Read more

Product Description
This monograph presents a complete and self-contained solution to the long-standing problem of giving a geometric description of state spaces of C*-algebras and von Neumann algebras, and of their Jordan algebraic analogs (JB-algebras and JBW-algebras). The material, which previously has appeared only in research papers and required substantial prerequisites for a reader's understanding, is made accessible here to a broad mathematical audience.

Key features include:

* The properties used to describe state spaces are primarily of a geometric nature, but many can also be interpreted in terms of physics. There are numerous remarks discussing these connections

* A quick introduction to Jordan algebras is given; no previous knowledge is assumed and all necessary background on the subject is given

* A discussion of dynamical correspondences, which tie together Lie and Jordan structures, and relate the observables and the generators of time evolution in physics

* The connection with Connes' notions of orientation and homogeneity in cones is explained

* Chapters conclude with notes placing the material in historical context

* Prerequisites are standard graduate courses in real and complex variables, measure theory, and functional analysis

* Excellent bibliography and index

In the authors' previous book, "State Spaces of Operator Algebras: Basic Theory, Orientations and C*-products" (ISBN 0-8176-3890-3), the role of orientations was examined and all the prerequisites on C*-algebras and von Neumann algebras, needed for this work, were provided in detail. These requisites, as well as all relevant definitions and results with reference back to "State Spaces," are summarized in an appendix, further emphasizing the self-contained nature of this work.

"Geometry of State Spaces of Operator Algebras" is intended for specialists in operator algebras, as well as graduate students and mathematicians in other areas. ... Read more

Product Description

This course-tested book explains in detail the theory of linear Hilbert-space operators and their use in quantum physics. The central mathematical tool of the book is the spectral theory of self-adjoint operators; in order to make the exposition self-contained, selected topics of functional analysis are included. An introduction to the theory of operator sets and algebras is also presented. This mathematical material is then used for a systematic analysis of the operator structure of quantum theory. Logical building of the theory is discussed as well as its practical aspects, such as spectral properties of quantum mechanical Hamiltonians, scattering theory, and more.

The second edition was extended by two new chapters devoted to properties of quantum waveguides and quantum graphs. The bibliography was amended by about 130 new items.

Product Description
The present selection of refereed papers is dedicated to Moshe Livic on the occasion of his eightieth anniversary. It covers many areas of operator theory and its applications, reflecting the breadth and the profound impact of his work. In particular, some of his most recent ideas on 2D-systems are presented. Other contributions cover important avenues of modern operator theory in such fields as interpolation theory (also in the so-called nonstationary setting), direct and inverse problems for the string equation and for nonselfadjoint differential operators, operator models and function theory.The volume appeals to a wide audience of pure and applied mathematicians, electrical engineers and theoretical physicists. ... Read more

Product Description
The main aim of this book is to introduce the reader to the concept of comparison algebra, defined as a type of C*-algebra of singular integral operators. The first part of the book develops the necessary elements of the spectral theory of differential operators as well as the basic properties of elliptic second order differential operators. The author then introduces comparison algebras and describes their theory in L2-spaces and L2-Soboler spaces, and in particular their importance in solving functional analytic problems involving differential operators. The book is based on lectures given in Sweden and the USA. ... Read more

Product Description
Jordan's classification theorem for linear transformations on a finite-dimensional vector space is a natural highlight of the deep relationship between linear algebra and the arithmetical properties of polynomial rings. Because the methods and results of finite-dimensional linear algebra seldom extend to or have analogs in infinite-dimensional operator theory, it is therefore remarkable to have a class of operators which has a classification theorem analogous to Jordan's classical result and has properties closely related to the arithmetic of the ring $H^{\infty}$ of bounded analytic functions in the unit disk. $C_0$ is such a class and is the central object of study in this book. A contraction operator belongs to $C_0$ if and only if the associated functional calculus on $H^{\infty}$ has a nontrivial kernel. $C_0$ was discovered by Bela Sz.-Nagy and Ciprian Foias in their work on canonical models for contraction operators on Hilbert space. Besides their intrinsic interest and direct applications, operators of class $C_0$ are very helpful in constructing examples and counterexamples in other branches of operator theory. In addition, $C_0$ arises in certain problems of control and realization theory. In this survey work, the author provides a unified and concise presentation of a subject that was covered in many articles. The book describes the classification theory of $C_0$ and relates this class to other subjects such as general dilation theory, stochastic realization, representations of convolution algebras, and Fredholm theory. This book should be of interest to operator theorists as well as theoretical engineers interested in the applications of operator theory. In an effort to make the book as self-contained as possible, the author gives an introduction to the theory of dilations and functional models for contraction operators. Prerequisites for this book are a course in functional analysis and an acquaintance with the theory of Hardy spaces in the unit disk. In addition, knowledge of the trace class of operators is necessary in the chapter on weak contractions. ... Read more