Normal element - Wikipedia

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

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. (1999). Topological Vector Spaces. GTM. Vol. 3. New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. {{cite book}}: Invalid |ref=harv (help)

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v t e 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
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Spectral Theorem	Borel functional calculus Min-max theorem Positive operator-valued measure Projection-valued measure Riesz projector Rigged Hilbert space Spectral theorem Spectral theory of compact operators Spectral theory of normal C*-algebras
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Miscellaneous	Abstract index group Banach algebra cohomology Cohen–Hewitt factorization theorem Extensions of symmetric operators Fredholm theory Limiting absorption principle Schröder–Bernstein theorems for operator algebras Sherman–Takeda theorem Unbounded operator
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References