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This book shows how operator theory interacts with function theory in one and several variables. The authors develop the theory in detail, leading the reader to the cutting edge of contemporary research. It starts with a treatment of the theory of bounded holomorphic functions on the unit disc. Model theory and the network realization formula are used to solve Nevanlinna-Pick interpolation problems, and the same techniques are shown to work on the bidisc, the symmetrized bidisc, and other domains. The techniques are powerful enough to prove the Julia-Carathéodory theorem on the bidisc, Lempert's theorem on invariant metrics in convex domains, the Oka extension theorem, and to generalize Loewner's matrix monotonicity results to several variables. In Part II, the book gives an introduction to non-commutative function theory, and shows how model theory and the network realization formula can be used to understand functions of non-commuting matrices.

Operator theory. --- Holomorphic functions. --- Geometric function theory. --- Hilbert space.

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The normal matrix model with algebraic potential has gained a lot of attention recently, partially in virtue of its connection to several other topics as quadrature domains, inverse potential problems and the Laplacian growth. In this present paper the authors consider the normal matrix model with cubic plus linear potential. In order to regularize the model, they follow Elbau & Felder and introduce a cut-off. In the large size limit, the eigenvalues of the model accumulate uniformly within a certain domain Omega that they determine explicitly by finding the rational parametrization of its boundary. The authors also study in detail the mother body problem associated to Omega. It turns out that the mother body measure mu_* displays a novel phase transition that we call the mother body phase transition: although partial Omega evolves analytically, the mother body measure undergoes a "one-cut to three-cut" phase transition.

Matrices --- Random matrices. --- Phase transformations (Statistical physics) --- Functions, Meromorphic. --- Riemann-Hilbert problems. --- Integral transforms. --- Norms.

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This textbook introduces spectral theory for bounded linear operators by focusing on (i) the spectral theory and functional calculus for normal operators acting on Hilbert spaces; (ii) the Riesz-Dunford functional calculus for Banach-space operators; and (iii) the Fredholm theory in both Banach and Hilbert spaces. Detailed proofs of all theorems are included and presented with precision and clarity, especially for the spectral theorems, allowing students to thoroughly familiarize themselves with all the important concepts. Covering both basic and more advanced material, the five chapters and two appendices of this volume provide a modern treatment on spectral theory. Topics range from spectral results on the Banach algebra of bounded linear operators acting on Banach spaces to functional calculus for Hilbert and Banach-space operators, including Fredholm and multiplicity theories. Supplementary propositions and further notes are included as well, ensuring a wide range of topics in spectral theory are covered. Spectral Theory of Bounded Linear Operators is ideal for graduate students in mathematics, and will also appeal to a wider audience of statisticians, engineers, and physicists. Though it is mostly self-contained, a familiarity with functional analysis, especially operator theory, will be helpful.

Operator theory. --- Functional analysis. --- Operator Theory. --- Functional Analysis. --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Functional analysis --- Spectral theory (Mathematics) --- Hilbert space --- Measure theory --- Transformations (Mathematics)

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This textbook provides an introduction to representations of general ∗-algebras by unbounded operators on Hilbert space, a topic that naturally arises in quantum mechanics but has so far only been properly treated in advanced monographs aimed at researchers. The book covers both the general theory of unbounded representation theory on Hilbert space as well as representations of important special classes of ∗-algebra, such as the Weyl algebra and enveloping algebras associated to unitary representations of Lie groups. A broad scope of topics are treated in book form for the first time, including group graded ∗-algebras, the transition probability of states, Archimedean quadratic modules, noncommutative Positivstellensätze, induced representations, well-behaved representations and representations on rigged modules. Making advanced material accessible to graduate students, this book will appeal to students and researchers interested in advanced functional analysis and mathematical physics, and with many exercises it can be used for courses on the representation theory of Lie groups and its application to quantum physics. A rich selection of material and bibliographic notes also make it a valuable reference.

Operator theory. --- Mathematical physics. --- Associative rings. --- Rings (Algebra). --- Topological groups. --- Lie groups. --- Operator Theory. --- Mathematical Physics. --- Associative Rings and Algebras. --- Topological Groups, Lie Groups. --- Groups, Lie --- Lie algebras --- Symmetric spaces --- Topological groups --- Groups, Topological --- Continuous groups --- Algebraic rings --- Ring theory --- Algebraic fields --- Rings (Algebra) --- Physical mathematics --- Physics --- Functional analysis --- Mathematics --- Hilbert algebras. --- Hilbert space. --- Operator algebras. --- Science --- Mathematics.

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The first quantum revolution started in the early 20th century and gave us new rules that govern physical reality. Accordingly, many devices that changed dramatically our lifestyle, such as transistors, medical scanners and lasers, appeared in the market. This was the origin of quantum technology, which allows us to organize and control the components of a complex system governed by the laws of quantum physics. This is in sharp contrast to conventional technology, which can only be understood within the framework of classical mechanics. We are now in the middle of a second quantum revolution. Although quantum mechanics is nowadays a mature discipline, quantum engineering as a technology is now emerging in its own right. We are about to manipulate and sense individual particles, measuring and exploiting their quantum properties. This is bringing major technical advances in many different areas, including computing, sensors, simulations, cryptography and telecommunications. The present collection of selected papers is a clear demonstration of the tremendous vitality of the field. The issue is composed of contributions from world leading researchers in quantum optics and quantum information, and presents viewpoints, both theoretical and experimental, on a variety of modern problems.

entangled states --- two atoms --- two-modes --- cavity QED setup --- entanglement --- interference phenomenon --- superposition of quantum states --- quantum tomograms --- quantum optics --- nonclassicality --- quantum resource theories --- non-Gaussianity --- photon-number-resolving detectors --- multiport devices --- Fock states --- quantum tomography --- photon losses --- relativistic dynamics --- no-interaction theorem --- world line condition --- circular gauge --- Landau gauge --- arbitrary linear gauge --- stepwise variation --- center-of-orbit coordinates --- relative coordinates --- elliptic and hyperbolic solenoids --- angular momentum --- magnetic moment --- squeezing --- mutually unbiased bases --- group representations --- graphs --- quantum information --- E = mc2 from Heisenberg’s uncertainty relations --- one symmetry for quantum mechanics and special relativity --- coherent states --- harmonic oscillator --- SU(2) coherent states --- 2D coherent states --- resolution of the identity --- uncertainty principle --- isotropic harmonic oscillator --- anisotropic harmonic oscillator --- Sudarshan --- apology --- non-hermitian operators --- real spectrum --- nonlinear algebras --- nonclassical states --- bound entanglement --- entanglement witness --- Hilbert–Schmidt measure --- optimization algorithms --- probability representation --- quantizer–dequantizer --- qubit --- quantum suprematism --- n/a --- E = mc2 from Heisenberg's uncertainty relations --- Hilbert-Schmidt measure --- quantizer-dequantizer --- Research.