Normal element

In mathematics, an element x of a *-algebra is normal if it satisfies $xx^{*}=x^{*}x.$

This definition stems from the definition of a normal linear operator in functional analysis, where a linear operator A from a Hilbert space into itself is called unitary if $AA^{*}=A^{*}A,$ where the adjoint of A is A^∗ and the domain of A is the same as that of A^∗. See normal operator for a detailed discussion. If the Hilbert space is finite-dimensional and an orthonormal basis has been chosen, then the operator A is normal if and only if the matrix describing A with respect to this basis is a normal matrix.

References

Reed, M.; Simon, B. (1972). Methods of Mathematical Physics. Vol 2. Academic Press.
Teschl, G. (2009). Mathematical Methods in Quantum Mechanics; With Applications to Schrödinger Operators. Providence: American Mathematical Society.
Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.

Functional analysis (topics – glossary)

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Banach Besov Fréchet Hilbert Hölder Nuclear Orlicz Schwartz Sobolev topological vector
Properties	barrelled complete dual (algebraic/topological) locally convex reflexive separable

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Advanced topics

Spectral theory and ^*-algebras
Basic concepts	Involution/-algebra Banach algebra B-algebra C-algebra Noncommutative topology Projection-valued measure Spectrum Spectrum of a C-algebra Spectral radius Operator space
Main results	Gelfand–Mazur theorem Gelfand–Naimark theorem Gelfand representation Polar decomposition Singular value decomposition Spectral theorem Spectral theory of normal C*-algebras
Special Elements/Operators	Isospectral Normal operator Hermitian/Self-adjoint operator Unitary operator Unit
Spectrum	Krein–Rutman theorem Normal eigenvalue Spectrum of a C*-algebra Spectral radius Spectral asymmetry Spectral gap
Decomposition of a spectrum	(Continuous Point Residual) Approximate point Compression Discrete Spectral abscissa
Spectral Theorem	Borel functional calculus Min-max theorem Projection-valued measure Riesz projector Rigged Hilbert space Spectral theorem Spectral theory of compact operators Spectral theory of normal C*-algebras
Special algebras	Amenable Banach algebra With an Approximate identity Banach function algebra Disk algebra Uniform algebra
Finite-Dimensional	Alon–Boppana bound Bauer–Fike theorem Numerical range Schur–Horn theorem
Generalizations	Dirac spectrum Essential spectrum Pseudospectrum Structure space (Shilov boundary)
Miscellaneous	Abstract index group Banach algebra cohomology Cohen–Hewitt factorization theorem Extensions of symmetric operators Limiting absorption principle Unbounded operator
Examples	Wiener algebra
Applications	Almost Mathieu operator Corona theorem Hearing the shape of a drum (Dirichlet eigenvalue) Heat kernel Kuznetsov trace formula Lax pair Proto-value function Ramanujan graph Rayleigh–Faber–Krahn inequality Spectral geometry Spectral method Spectral theory of ordinary differential equations Sturm–Liouville theory Superstrong approximation Transfer operator Transform theory Weyl law

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Normal element

See also

References