In
mathematics, especially
operator theory
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 ...
, a paranormal operator is a generalization of a
normal operator
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*'' = ''N*N''.
Normal opera ...
. More precisely, a
bounded linear operator ''T'' on a complex
Hilbert space ''H'' is said to be paranormal if:
:
for every unit vector ''x'' in ''H''.
The class of paranormal operators was introduced by V. Istratescu in 1960s, though the term "paranormal" is probably due to Furuta.
Every
hyponormal operator (in particular, a
subnormal operator, a
quasinormal operator and a normal operator) is paranormal. If ''T'' is a paranormal, then ''T''
''n'' is paranormal.
[Furuta, Takayuki. ]
On the Class of Paranormal Operators
' On the other hand,
Halmos gave an example of a hyponormal operator ''T'' such that ''T''
2 isn't hyponormal. Consequently, not every paranormal operator is hyponormal.
A
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
paranormal operator is normal.
[Furuta, Takayuki]
Certain Convexoid Operators
/ref>
References
Operator theory
Linear operators
{{mathanalysis-stub