In mathematics, especially functional analysis, a normal operator on a complex Hilbert space H is a continuous linear operator N : H → H that commutes with its hermitian adjoint N*, that is: NN* = NN.
Normal operators are important because the spectral theorem holds for them. The class of normal operators is well understood. Examples of normal operators are
unitary operators: N = N−1
Hermitian operators (i.e., self-adjoint operators): N* = N
Skew-Hermitian operators: N* = −N
positive operators: N = MM* for some M (so N is self-adjoint).
A normal matrix is the matrix expression of a normal operator on the Hilbert space Cn.
Normal operators are characterized by the spectral theorem. A compact normal operator (in particular, a normal operator on a finite-dimensional linear space) is unitarily diagonalizable.
Let be a bounded operator. The following are equivalent.
is normal.
is normal.
for all (use ).
The self-adjoint and anti–self adjoint parts of commute. That is, if is written as with and then
If is a normal operator, then and have the same kernel and the same range. Consequently, the range of is dense if and only if is injective. Put in another way, the kernel of a normal operator is the orthogonal complement of its range. It follows that the kernel of the operator coincides with that of for any Every generalized eigenvalue of a normal operator is thus genuine. is an eigenvalue of a normal operator if and only if its complex conjugate is an eigenvalue of Eigenvectors of a normal operator corresponding to different eigenvalues are orthogonal, and a normal operator stabilizes the orthogonal complement of each of its eigenspaces. This implies the usual spectral theorem: every normal operator on a finite-dimensional space is diagonalizable by a unitary operator. There is also an infinite-dimensional version of the spectral theorem expressed in terms of projection-valued measures. The residual spectrum of a normal operator is empty.
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.
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
This course is an introduction to the spectral theory of linear operators acting in Hilbert spaces. The main goal is the spectral decomposition of unbounded selfadjoint operators. We will also give el
The aim of the course is to review mathematical concepts learned during the bachelor cycle and apply them, both conceptually and computationally, to concrete problems commonly found in engineering and
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.
In mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may be presented abstractly by their characteristics, such as bounded linear operators or closed operators, and consideration may be given to nonlinear operators. The study, which depends heavily on the topology of function spaces, is a branch of functional analysis. If a collection of operators forms an algebra over a field, then it is an operator algebra.
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 .
Explores the verification of PVM properties and the spectral theorem for unbounded operators in quantum mechanics.
Covers the concept of operator projections onto subspaces, focusing on characteristic functions.
Explores vector spaces, linear transformations, matrices, eigenvalues, inner products, and operators.
Nontrivial spectral properties of non-Hermitian systems can lead to intriguing effects with no counterparts in Hermitian systems. For instance, in a two-mode photonic system, by dynamically winding around an exceptional point (EP) a controlled asymmetric-s ...
NATURE PORTFOLIO2023
We establish shape holomorphy results for general weakly- and hyper-singular boundary integral operators arising from second-order partial differential equations in unbounded two-dimensional domains with multiple finite-length open arcs. After recasting th ...
New York2024
In diverse fields such as medical imaging, astrophysics, geophysics, or material study, a common challenge exists: reconstructing the internal volume of an object using only physical measurements taken from its exterior or surface. This scientific approach ...