Lecture

Non-Bounded Operators: Spectral Theory

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Description

This lecture delves into the spectral theory of non-bounded operators, defining the essential spectrum, the resolvent set, and the spectrum. It explains the concept of the resolvent operator and why the focus is on closed operators to achieve a spectral decomposition similar to the bounded case.

Official source

Ontological neighbourhood

Mathematics

Analysis: Functional analysis

Related concepts (18)

Spectrum (functional analysis)

In mathematics, particularly in functional analysis, the spectrum of a bounded linear operator (or, more generally, an unbounded linear operator) is a generalisation of the set of eigenvalues of a matrix. Specifically, a complex number is said to be in the spectrum of a bounded linear operator if either has no set-theoretic inverse; or the set-theoretic inverse is either unbounded or defined on a non-dense subset. Here, is the identity operator. By the closed graph theorem, is in the spectrum if and only if the bounded operator is non-bijective on .

Resolvent formalism

In mathematics, the resolvent formalism is a technique for applying concepts from complex analysis to the study of the spectrum of operators on Banach spaces and more general spaces. Formal justification for the manipulations can be found in the framework of holomorphic functional calculus. The resolvent captures the spectral properties of an operator in the analytic structure of the functional.

Unbounded operator

In mathematics, more specifically functional analysis and operator theory, the notion of unbounded operator provides an abstract framework for dealing with differential operators, unbounded observables in quantum mechanics, and other cases. The term "unbounded operator" can be misleading, since "unbounded" should sometimes be understood as "not necessarily bounded"; "operator" should be understood as "linear operator" (as in the case of "bounded operator"); the domain of the operator is a linear subspace, not necessarily the whole space; this linear subspace is not necessarily closed; often (but not always) it is assumed to be dense; in the special case of a bounded operator, still, the domain is usually assumed to be the whole space.

Resolvent set

In linear algebra and operator theory, the resolvent set of a linear operator is a set of complex numbers for which the operator is in some sense "well-behaved". The resolvent set plays an important role in the resolvent formalism. Let X be a Banach space and let be a linear operator with domain . Let id denote the identity operator on X. For any , let A complex number is said to be a regular value if the following three statements are true: is injective, that is, the corestriction of to its image has an inverse ; is a bounded linear operator; is defined on a dense subspace of X, that is, has dense range.

Spectral theory

In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operators in a variety of mathematical spaces. It is a result of studies of linear algebra and the solutions of systems of linear equations and their generalizations. The theory is connected to that of analytic functions because the spectral properties of an operator are related to analytic functions of the spectral parameter.

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