scholarly journals SPECTRUM, NUMERICAL RANGE AND DAVIS-WIELANDT SHELL OF A NORMAL OPERATOR

2009 ◽  
Vol 51 (1) ◽  
pp. 91-100 ◽  
Author(s):  
CHI-KWONG LI ◽  
YIU-TUNG POON
Keyword(s):  
Hilbert Space ◽  
Separable Hilbert Space ◽  
Convex Set ◽  
Numerical Range ◽  
Normal Operator ◽  
Additional Information ◽  
Commuting Operators ◽  
Boundary Points ◽  
Reducing Subspace ◽  
Joint Numerical Range

AbstractWe denote the numerical range of the normal operator T by W(T). A characterization is given to the points in W(T) that lie on the boundary. The collection of such boundary points together with the interior of the the convex hull of the spectrum of T will then be the set W(T). Moreover, it is shown that such boundary points reveal a lot of information about the normal operator. For instance, such a boundary point always associates with an invariant (reducing) subspace of the normal operator. It follows that a normal operator acting on a separable Hilbert space cannot have a closed strictly convex set as its numerical range. Similar results are obtained for the Davis-Wielandt shell of a normal operator. One can deduce additional information of the normal operator by studying the boundary of its Davis-Wielandt shell. Further extension of the result to the joint numerical range of commuting operators is discussed.

scholarly journals The boundary of the numerical range

1981 ◽  
Vol 22 (1) ◽  
pp. 69-72 ◽  
Author(s):  
G. de Barra
Keyword(s):  
Hilbert Space ◽  
Convex Hull ◽  
Similar Condition ◽  
Point Spectrum ◽  
Numerical Range ◽  
Normal Operator ◽  
Density Condition ◽  
Compact Normal Operator

In [1] it was shown that for a compact normal operator on a Hilbert space the numerical range was the convex hull of the point spectrum. Here it is shown that the same holds for a semi-normal operator whose point spectrum satisfies a density condition (Theorem 1). In Theorem 2 a similar condition is shown to imply that the numerical range of a semi-normal operator is closed. Some examples are given to indicate that the condition in Theorem 1 cannot be relaxed too much.

Extreme Points of the Convex set of Stochastic Maps on A C*-Algebra

1998 ◽  
Vol 01 (04) ◽  
pp. 599-609 ◽  
Author(s):  
K. R. Parthasarathy
Keyword(s):  
Hilbert Space ◽  
Separable Hilbert Space ◽  
Convex Set ◽  
Convex Combination ◽  
Extreme Points ◽  
Bounded Operators ◽  
Completely Positive ◽  
C Algebra ◽  
Permutation Matrices ◽  
Normal Maps

Let [Formula: see text] be a unital C*-subalgebra of the C*-algebra ℬ(ℋ) of all bounded operators on a complex separable Hilbert space ℋ. Let [Formula: see text] denote the convex set of all unital, linear, completely positive and normal maps of [Formula: see text] into itself. Using Stinespring's theorem, we present a criterion for an element [Formula: see text] to be extremal. When [Formula: see text], this criterion leads to an explicit description of the set of all extreme points of [Formula: see text]. We also obtain a quantum probabilistic analogue of the classical Birkhoff's theorem2 that every bistochastic matrix can be expressed as a convex combination of permutation matrices.

Boundary Points of the Numerical Range of an Operator

1975 ◽  
Vol 17 (5) ◽  
pp. 689-692 ◽  
Author(s):  
C.-S. Lin
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Numerical Range ◽  
Bounded Linear ◽  
Boundary Points

The purpose of this note is to investigate boundary points of the numerical range of an operator in terms of inner and outer center points. Some applications on commutators are given.Throughout this note, an operator will always mean a bounded linear operator on a Hilbert space X.

scholarly journals Spectral properties of n-normal operators

Filomat ◽  
2018 ◽  
Vol 32 (14) ◽  
pp. 5063-5069 ◽  
Author(s):  
Muneo Chō ◽  
Biljana Nacevska
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Spectral Properties ◽  
Normal Operator ◽  
Complex Hilbert Space ◽  
Bounded Linear ◽  
Reducing Subspace ◽  
Normal Operators ◽  
Isolated Point ◽  
Zero Number

For a bounded linear operator T on a complex Hilbert space and n ? N, T is said to be n-normal if T*Tn = TnT*. In this paper we show that if T is a 2-normal operator and satisfies ?(T) ? (-?(T)) ? {0}, then T is isoloid and ?(T) = ?a(T). Under the same assumption, we show that if z and w are distinct eigenvalues of T, then ker(T-z)? ker(T-w). And if non-zero number z ? C is an isolated point of ?(T), then we show that ker(T-z) is a reducing subspace for T. We show that if T is a 2-normal operator satisfying ?(T) ?(-?(T)) = 0, then Weyl?s theorem holds for T. Similarly, we show spectral properties of n-normal operators under similar assumption. Finally, we introduce (n,m)-normal operators and show some properties of this kind of operators.

scholarly journals An inequality for similarity condition numbers of unbounded operators with Schatten - von Neumann Hermitian components

Filomat ◽  
2016 ◽  
Vol 30 (13) ◽  
pp. 3415-3425 ◽  
Author(s):  
Michael Gil’
Keyword(s):  
Hilbert Space ◽  
Unbounded Operator ◽  
Differential Operators ◽  
Separable Hilbert Space ◽  
Normal Operator ◽  
Condition Numbers ◽  
Unbounded Operators ◽  
Similarity Condition ◽  
Von Neumann ◽  
Operator Functions

Let H be a linear unbounded operator in a separable Hilbert space. It is assumed the resolvent of H is a compact operator and H ? H* is a Schatten - von Neumann operator. Various integro-differential operators satisfy these conditions. Under certain assumptions it is shown that H is similar to a normal operator and a sharp bound for the condition number is suggested. We also discuss applications of that bound to spectrum perturbations and operator functions.

scholarly journals Centers of mass for operator-families

1992 ◽  
Vol 34 (1) ◽  
pp. 123-126
Author(s):  
Makoto Takaguchi
Keyword(s):  
Hilbert Space ◽  
Numerical Range ◽  
Complex Hilbert Space ◽  
Bounded Operators ◽  
Centers Of Mass ◽  
Image Position ◽  
Joint Numerical Range

Let Hbe a complex Hilbert space and let B(H) be the algebra of (bounded) operators on H. Let A =(A,…,An) be an n-tuple of operators on H. The joint numerical range of A is the subset W(A) of ℂn such that

scholarly journals Absorption in Invariant Domains for Semigroups of Quantum Channels

2021 ◽  
Author(s):  
Raffaella Carbone ◽  
Federico Girotti
Keyword(s):  
Hilbert Space ◽  
Ergodic Theory ◽  
Markov Processes ◽  
Separable Hilbert Space ◽  
Quantum Channels ◽  
Quantum Markov Processes ◽  
Positive Recurrent ◽  
Markov Evolution ◽  
Absorption Probabilities ◽  
Invariant Domains

AbstractWe introduce a notion of absorption operators in the context of quantum Markov processes. The absorption problem in invariant domains (enclosures) is treated for a quantum Markov evolution on a separable Hilbert space, both in discrete and continuous times: We define a well-behaving set of positive operators which can correspond to classical absorption probabilities, and we study their basic properties, in general, and with respect to accessibility structure of channels, transience and recurrence. In particular, we can prove that no accessibility is allowed between the null and positive recurrent subspaces. In the case, when the positive recurrent subspace is attractive, ergodic theory will allow us to get additional results, in particular about the description of fixed points.

Formally Normal Operators Having no Normal Extensions

1965 ◽  
Vol 17 ◽  
pp. 1030-1040 ◽  
Author(s):  
Earl A. Coddington
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Null Space ◽  
Normal Operator ◽  
Closed Linear Operator ◽  
Normal Operators ◽  
The Given ◽  
Densely Defined ◽  
Made In

The domain and null space of an operator A in a Hilbert space will be denoted by and , respectively. A formally normal operatorN in is a densely defined closed (linear) operator such that , and for all A normal operator in is a formally normal operator N satisfying 35 . A study of the possibility of extending a formally normal operator N to a normal operator in the given , or in a larger Hilbert space, was made in (1).

scholarly journals Quasi-posinormal operators

2010 ◽  
Vol 7 (3) ◽  
pp. 1282-1287
Author(s):  
Baghdad Science Journal
Keyword(s):  
Hilbert Space ◽  
Normal Operator ◽  
Basic Properties ◽  
Hyponormal Operators ◽  
The Relationship

In this paper, we introduce a class of operators on a Hilbert space namely quasi-posinormal operators that contain properly the classes of normal operator, hyponormal operators, M–hyponormal operators, dominant operators and posinormal operators . We study some basic properties of these operators .Also we are looking at the relationship between invertibility operator and quasi-posinormal operator .

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