Concept

# Multiplication operator

Summary
In operator theory, a multiplication operator is an operator Tf defined on some vector space of functions and whose value at a function φ is given by multiplication by a fixed function f. That is, for all φ in the domain of Tf, and all x in the domain of φ (which is the same as the domain of f). This type of operator is often contrasted with composition operators. Multiplication operators generalize the notion of operator given by a diagonal matrix. More precisely, one of the results of operator theory is a spectral theorem that states that every self-adjoint operator on a Hilbert space is unitarily equivalent to a multiplication operator on an L2 space. Consider the Hilbert space X = L2[−1, 3] of complex-valued square integrable functions on the interval . With f(x) = x2, define the operator for any function φ in X. This will be a self-adjoint bounded linear operator, with domain all of X = L2[−1, 3] and with norm 9. Its spectrum will be the interval (the range of the function x→ x2 defined on ). Indeed, for any complex number λ, the operator Tf − λ is given by It is invertible if and only if λ is not in , and then its inverse is which is another multiplication operator. This can be easily generalized to characterizing the norm and spectrum of a multiplication operator on any Lp space.
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.
Related concepts (15)
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow the methods of linear algebra and calculus to be generalized from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as function spaces. Formally, a Hilbert space is a vector space equipped with an inner product that induces a distance function for which the space is a complete metric space.
Shift operator
In mathematics, and in particular functional analysis, the shift operator, also known as the translation operator, is an operator that takes a function x ↦ f(x) to its translation x ↦ f(x + a). In time series analysis, the shift operator is called the lag operator. Shift operators are examples of linear operators, important for their simplicity and natural occurrence. The shift operator action on functions of a real variable plays an important role in harmonic analysis, for example, it appears in the definitions of almost periodic functions, positive-definite functions, derivatives, and convolution.
Functional calculus
In mathematics, a functional calculus is a theory allowing one to apply mathematical functions to mathematical operators. It is now a branch (more accurately, several related areas) of the field of functional analysis, connected with spectral theory. (Historically, the term was also used synonymously with calculus of variations; this usage is obsolete, except for functional derivative. Sometimes it is used in relation to types of functional equations, or in logic for systems of predicate calculus.
Related courses (2)
PHYS-331: Functional analysis (for PH)
Ce cours ambitionne de présenter les mathématiques de la mécanique quantique, et plus généralement de la physique quantique. Il s'adresse essentiellement aux physiciens, ou a des mathématiciens intére
COM-309: Quantum information processing
Information is processed in physical devices. In the quantum regime the concept of classical bit is replaced by the quantum bit. We introduce quantum principles, and then quantum communications, key d
Related lectures (5)
Spectral Decomposition of Bounded Self-Adjoint Operators
Explores the spectral decomposition of self-adjoint operators on Hilbert spaces.
Essential Adjoints: Spectral Decomposition and Symmetric Operators
Explores spectral decomposition, essential self-adjointness, and symmetric operators in Hilbert spaces.
Harmonic Oscillator
Explores the harmonic oscillator, covering energy functions, equations of motion, and operator constructions.