Hyponormal Operator

	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: : $(T^T)^p \ge (TT^)^p$ (That is to say, $(T^T)^p - (TT^)^p$ is a positive operator.) If $p = 1$ , then ''T'' is called a hyponormal operator. If $p = 1/2$ , then ''T'' is called a semi-hyponormal operator. Moreover, ''T'' is said to be log-hyponormal if it is invertible and : $\log (T^T) \ge \log (TT^).$ 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 Al ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	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 or closed operators, and consideration may be given to nonlinear operators. The study, which depends heavily on the topology of function spaces, is a branch of functional analysis. If a collection of operators forms an algebra over a field, then it is an operator algebra. The description of operator algebras is part of operator theory. Single operator theory Single operator theory deals with the properties and classification of operators, considered one at a time. For example, the classification of normal operators in terms of their spectra falls into this category. Spectrum of operators The spectral theorem is any of a number of results about linear operators or about matrices. In broad terms the spectral theorem provides cond ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	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'' = ''NN''. Normal operators are important because the spectral theorem holds for them. The class of normal operators is well understood. Examples of normal operators are unitary operators: ''N'' = ''N−1'' Hermitian operators (i.e., self-adjoint operators): ''N'' = ''N'' Skew-Hermitian operators: ''N'' = −''N'' positive operators: ''N'' = ''MM'' for some ''M'' (so ''N'' is self-adjoint). A normal matrix is the matrix expression of a normal operator on the Hilbert space C''n''. Properties Normal operators are characterized by the spectral theorem. A compact normal operator (in particular, a normal operator on a finite-dimensional linear space) is unitarily diagonalizable. Let T be a bounded operator. The following are equivalent. T is normal. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	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 preserves the operations of vector addition and scalar multiplication. The same names and the same definition are also used for the more general case of modules over a ring; see Module homomorphism. If a linear map is a bijection then it is called a . In the case where V = W, a linear map is called a (linear) ''endomorphism''. Sometimes the term refers to this case, but the term "linear operator" can have different meanings for different conventions: for example, it can be used to emphasize that V and W are real vector spaces (not necessarily with V = W), or it can be used to emphasize that V is a function space, which is a common convention in functional analysis. Sometimes the term ''linear function'' has the same meaning as ''linear map' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	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 naturally and frequently in mathematics and physics, typically as function spaces. Formally, a Hilbert space is a vector space equipped with an inner product that defines a distance function for which the space is a complete metric space. The earliest Hilbert spaces were studied from this point of view in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz. They are indispensable tools in the theories of partial differential equations, quantum mechanics, Fourier analysis (which includes applications to signal processing and heat transfer), and ergodic theory (which forms the mathematical underpinning of thermodynamics). John von Neumann coined the term ''Hilbert space'' for the abstract concept that under ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	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. A bounded operator ''A'' on ''H'' is said to be subnormal if ''A'' has a normal extension. In other words, ''A'' is subnormal if there exists a Hilbert space ''K'' such that ''H'' can be embedded in ''K'' and there exists a normal operator ''N'' of the form :N = \begin A & B\\ 0 & C\end for some bounded operators :B : H^ \rightarrow H, \quad \mbox \quad C : H^ \rightarrow H^. Normality, quasinormality, and subnormality Normal operators Every normal operator is subnormal by definition, but the converse is not true in general. A simple class of examples can be obtained by weakening the properties of unitary operators. A unitary operator is an isometry with dense range. Consider n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	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: : \, 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 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 ancien ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	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. An example of such an operator is a normal operator (or some of its generalization). A closely related operator is a spectraloid operator: an operator whose spectral radius coincides with its numerical radius. In fact, an operator ''T'' is convexoid if and only if T - \lambda is spectraloid for every complex number \lambda. See also Aluthge transform In mathematics and more precisely in functional analysis, the Aluthge transformation is an operation defined on the set of bounded operators of a Hilbert space. It was introduced by Ariyadasa Aluthge to study p-hyponormal linear operators. Def ... References T. Furuta. Certain convexoid operators' Operator theory {{mathanalysis-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	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 or closed operators, and consideration may be given to nonlinear operators. The study, which depends heavily on the topology of function spaces, is a branch of functional analysis. If a collection of operators forms an algebra over a field, then it is an operator algebra. The description of operator algebras is part of operator theory. Single operator theory Single operator theory deals with the properties and classification of operators, considered one at a time. For example, the classification of normal operators in terms of their spectra falls into this category. Spectrum of operators The spectral theorem is any of a number of results about linear operators or about matrices. In broad terms the spectral theorem provides cond ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]