Hyponormal operator
Encyclopedia
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 (
) if:
(That is to say,
is a positive operator.) If 
, then T is called a hyponormal operator. If 
, then T is called a semi-hyponormal operator. Moreoever, T is said to be log-hyponormal if it is invertible and
An invertible p-hyponormal operator is log-hyponormal. On the other hand, not every log-hyponormal is p-hyponormal.
The class of semi-hyponormal operators was introduced by Xia, and the class of p-hyponormal operators was studied by Aluthge, who used what is today called the Aluthge transformation.
Every subnormal operator
(in particular, a normal operator) is hyponormal, and every hyponormal operator is a paranormal
convexoid operator
. Not every paranormal operator is, however, hyponormal.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, especially operator theory
Operator theory
In mathematics, operator theory is the branch of functional analysis that focuses on bounded linear operators, but which includes closed operators and nonlinear operators.Operator theory also includes the study of algebras of operators....
, a hyponormal operator is a generalization of a normal operator
Normal operator
In mathematics, especially functional analysis, a normal operator on a complex Hilbert space H is a continuous linear operatorN:H\to Hthat commutes with its hermitian adjoint N*: N\,N^*=N^*N....
. In general, a bounded linear operator T on a complex Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
H is said to be p-hyponormal (
) if:(That is to say,
is a positive operator.) If , then T is called a hyponormal operator. If , then T is called a semi-hyponormal operator. Moreoever, T is said to be log-hyponormal if it is invertible andAn invertible p-hyponormal operator is log-hyponormal. On the other hand, not every log-hyponormal is p-hyponormal.
The class of semi-hyponormal operators was introduced by Xia, and the class of p-hyponormal operators was studied by Aluthge, who used what is today called the Aluthge transformation.
Every subnormal operator
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 analytic symbols.-Definition:Let H be a Hilbert space...
(in particular, a normal operator) is hyponormal, and every hyponormal operator is a paranormal
Paranormal operator
In mathematics, especially operator theory, a paranormal operator is a generalization of a normal operator. 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....
convexoid operator
Convexoid operator
In mathematics, especially operator theory, a convexoid operator is a bounded linear operator T on a complex Hilbert space H such that the closure of the numerical range coincides with the convex hull of its spectrum....
. Not every paranormal operator is, however, hyponormal.