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The primarily objective of the book is to serve as a primer on the theory of bounded linear operators on separable Hilbert space. The book presents the spectral theorem as a statement on the existence of a unique continuous and measurable functional calculus. It discusses a proof without digressing into a course on the Gelfand theory of commutative Banach algebras. The book also introduces the reader to the basic facts concerning the various von Neumann–Schatten ideals, the compact operators, the trace-class operators and all bounded operators. .

Operator theory --- Functional analysis --- Mathematics --- analyse (wiskunde) --- functies (wiskunde) --- wiskunde

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Operator theory --- Analytical spaces --- 517.983 --- Linear operators. Linear operator equations --- Integral operators. --- Lp spaces. --- 517.983 Linear operators. Linear operator equations --- Opérateurs intégraux

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Operator algebras --- Algebras, Operator --- Operator theory --- Topological algebras --- Algebraic topology

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This book provides readers with a concise introduction to current studies on operator-algebras and their generalizations, operator spaces and operator systems, with a special focus on their application in quantum information science. This basic framework for the mathematical formulation of quantum information can be traced back to the mathematical work of John von Neumann, one of the pioneers of operator algebras, which forms the underpinning of most current mathematical treatments of the quantum theory, besides being one of the most dynamic areas of twentieth century functional analysis. Today, von Neumann’s foresight finds expression in the rapidly growing field of quantum information theory. These notes gather the content of lectures given by a very distinguished group of mathematicians and quantum information theorists, held at the IMSc in Chennai some years ago, and great care has been taken to present the material as a primer on the subject matter. Starting from the basic definitions of operator spaces and operator systems, this text proceeds to discuss several important theorems including Stinespring’s dilation theorem for completely positive maps and Kirchberg’s theorem on tensor products of C*-algebras. It also takes a closer look at the abstract characterization of operator systems and, motivated by the requirements of different tensor products in quantum information theory, the theory of tensor products in operator systems is discussed in detail. On the quantum information side, the book offers a rigorous treatment of quantifying entanglement in bipartite quantum systems, and moves on to review four different areas in which ideas from the theory of operator systems and operator algebras play a natural role: the issue of zero-error communication over quantum channels, the strong subadditivity property of quantum entropy, the different norms on quantum states and the corresponding induced norms on quantum channels, and, lastly, the applications of matrix-valued random variables in the quantum information setting.

Physics. --- Quantum Physics. --- Quantum Computing. --- Mathematical Methods in Physics. --- Functional Analysis. --- Mathematical Physics. --- Functional analysis. --- Quantum theory. --- Mathematical physics. --- Physique --- Analyse fonctionnelle --- Théorie quantique --- Physique mathématique --- Physics --- Physical Sciences & Mathematics --- Atomic Physics --- Quantum computers. --- Quantum physics. --- Quantum dynamics --- Quantum mechanics --- Quantum physics --- Mechanics --- Thermodynamics --- Physical mathematics --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Mathematics --- Quantum theory --- Mathematics. --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics --- Computers