Normal Operator

Information about Normal Operator

In mathematics, especially functional analysis, a normal operator on a Hilbert space $\text{[math]}$ (or more generally in a C* algebra) is a continuous linear operator

$\text{[math]}$

that commutes with its hermitian adjoint ''N*:

$\text{[math]}$

Normal operators are characterized by the spectral theorem.

A bounded operator T is normal if and only if ||Tx|| = ||T*x|| for all x. ^[1] If N is a normal operator, then N and N* have the same kernel and range. Consequently, the range of N is dense if and only if N is injective.

Examples of normal operators:

unitary operators ( $\text{[math]}$ )
Hermitian operators ( $\text{[math]}$ )
positive operators ( $\text{[math]}$ )
orthogonal projection operators ( $\text{[math]}$ )
normal matrices can be seen as normal operators if one takes the Hilbert space to be $\text{[math]}$ .

Notes

1. ^ We have $\text{[math]}$ , $\text{[math]}$ and the fact that $\text{[math]}$ for all $\text{[math]}$ implies that $\text{[math]}$

Mathematics (colloquially, maths or math) is the body of knowledge centered on such concepts as quantity, structure, space, and change, and also the academic discipline that studies them. Benjamin Peirce called it "the science that draws necessary conclusions".
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Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of vector spaces and operators acting upon them. It has its historical roots in the study of functional spaces, in particular transformations of functions, such as the Fourier
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Hilbert space, named after the David Hilbert, generalizes the notion of Euclidean space in a way that extends methods of vector algebra from the two-dimensional plane and three-dimensional space to infinite-dimensional spaces.
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C*-algebras are an important area of research in functional analysis, a branch of mathematics. The prototypical example of a C*-algebra is a complex algebra, A, of linear operators on a complex Hilbert space with two additional properties:

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In topology and related areas of mathematics a continuous function is a morphism between topological spaces. Intuitively, this is a function f where a set of points near f(x) always contain the image of a set of points near x.
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In mathematics, a linear map (also called a linear transformation or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication.
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commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.
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adjoint operator. Adjoints of operators generalize conjugate transposes of square matrices to (possibly) infinite-dimensional situations. If one thinks of operators on a Hilbert space as "generalized complex numbers", then the adjoint of an operator plays the role of the complex
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In mathematics, particularly linear algebra and functional analysis, the spectral theorem is any of a number of results about linear operators or about matrices. In broad terms the spectral theorem provides conditions under which an operator or a matrix can be diagonalized (that
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In functional analysis (a branch of mathematics), a bounded linear operator is a linear transformation L between normed vector spaces X and Y for which the ratio of the norm of L(v) to that of v
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unitary operator is a bounded linear operator U : H → H on a Hilbert space H satisfying

where U^∗ is the adjoint of U, and I :
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In mathematics, on a finite-dimensional inner product space, a self-adjoint operator is one that is its own adjoint, or, equivalently, one whose matrix is Hermitian, where a Hermitian matrix is one which is equal to its own conjugate transpose.
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positive element if its spectrum consists of positive real numbers. Equivalently, A has a hermitian square root, that is an element B of the C*-algebra satisfying B*=B and B²=A.
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projection is a linear transformation P from a vector space to itself such that P² = P. Projections map the whole vector space to a subspace and leave the points in that subspace unchanged.
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normal matrix if

A*A=AA*

where A* is the conjugate transpose of A. (If A is a real matrix, A*=A^T and so it is normal if A^TA = AA^T.
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In operator theory, quasinormal operators is a class of bounded operators defined by weakening the requirements of a normal operator.

Definition and some properties

Definition

Let A be a bounded operator on a Hilbert space H, then A
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In mathematics, especially operator theory, subnormal operators are bounded operators on a Hilbert space defined by weakening the requirements for normal operators. Some examples of subnormal operators are isometries and Toeplitz operators with analytic symbols.
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