mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, 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 operato ...

, a paranormal operator is a generalization of a

normal operator In mathematics, especially functional analysis, a normal operator on a complex number, complex Hilbert space H is a continuous function (topology), continuous linear operator N\colon H\rightarrow H that commutator, commutes with its Hermitian adjo ...

. 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 mapping V \to W between two vector spaces that pr ...

''T'' on a complex

Hilbert space In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...

''H'' is said to be paranormal if: :

\, T^2x\,  \ge \, Tx\, ^2

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 isometry, isometries and Toeplitz operator ...

, 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 spa ...

and a normal operator) is paranormal. If ''T'' is a paranormal, then ''T''^''n'' is paranormal. 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, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...

paranormal operator is normal.

References

Operator theory Linear operators {{mathanalysis-stub