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Mathematics - Analysis | Operator Theory

Operator Theory

Alpay, Daniel (Ed.)

2015, 1550 p.

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  • First reference work on operator theory with its long history and wide range of topics
  • Describes the multiple applications to, and connections with, other fields of mathematics, physics and engineering
  • Comprises ten sections which are interconnected -with notions such as positivity and reproducing kernel percolating to most of them
A one-sentence definition of operator theory could be: The study of (linear) continuous operations between topological vector spaces, these being in general (but not exclusively) Fréchet, Banach, or Hilbert spaces (or their duals). Operator theory is thus a very wide field, with numerous facets, both applied and theoretical. There are deep connections with complex analysis, functional analysis, mathematical physics, and electrical engineering, to name a few. Fascinating new applications and directions regularly appear, such as operator spaces, free probability, and applications to Clifford analysis. In our choice of the sections, we tried to reflect this diversity. This is a dynamic ongoing project, and more sections are planned, to complete the picture. We hope you enjoy the reading, and profit from this endeavor.

Content Level » Research

Keywords » Schur analysis - de Branges spaces - indefinite inner product spaces - infinite dimensional analysis - noncommutative analysis - reproducing kernel Hilbert spaces

Related subjects » Analysis - Applications

Table of contents 

Indefinite Hamiltonian Systems.- Locally Definitizable Operators: The Local Structure of Spectrum.- Multi-valued Operators/Linear Relations Between Krein Spaces.- Reproducing Kernel Krein Spaces.- Schur Analysis in an Indefinite Setting.- Symmetric and Isometric Relations.- The Algebraic Ricatti Equation and Its Role in Indefinite Inner Product Spaces.- The Critical Point Infinity Associated with Indefinite Sturm-Liouville Problems.- A Von Neumann Algebra over the Adele Ring and the Euler Totient Function.- Arithmetic Functions in Harmonic Analysis and Operator Theory.- Unbounded Operators, Lie Algebras, and Local Representations.- Linear Transforms in Signal and Optical Systems.- Realization of Herglotz-Nevanlinna Functions by Conservative Systems.- Robust Stabilization of Linear Control Systems Using A Frequency Domain Approach.- Semi-Separable Systems: Representations, Inversion, and Limiting Behavior.- Synchronization Problems for Spatially Invariant Infinite Dimensional Linear Systems.- Commutative Dilation Theory.- Operator Theory and Function Theory in Drury-Arveson Space and Its Quotients.- Taylor Functional Calculus.- Sampling Theory and Reproducing Kernel Hilbert Spaces.- Quaternionic Analysis: Application to Boundary Value Problems.- An Introduction to Hilbert Module Approach to Multivariable Operator Theory.- Applications of Hilbert Module Approach to Multivariable Operator Theory.- Perturbations of Unbounded Fredholm Linear Operators in Banach Spaces.

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