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 oper ...
, 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 op ...
. More precisely, a
bounded linear operator
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \to W between two vect ...
''T'' on a complex
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natu ...
''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 mathematics, especially operator theory, a hyponormal operator is a generalization of a normal operator. In general, a bounded linear operator ''T'' on a complex Hilbert space ''H'' is said to be ''p''-hyponormal (0 < p \le 1) if:
: (in particular, a
subnormal operator 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 ...
, a
quasinormal operator In operator theory, quasinormal operators is a class of bounded operators defined by weakening the requirements of a normal operator.
Every quasinormal operator is a subnormal operator. Every quasinormal operator on a finite-dimensional Hilbert ...
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
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