scholarly journals On extreme points of the joint numerical range of commuting normal operators

1976 ◽  
Vol 67 (2) ◽  
pp. 473-476 ◽  
Author(s):  
Pushpa Juneja
Keyword(s):  
Numerical Range ◽  
Extreme Points ◽  
Normal Operators ◽  
Joint Numerical Range

scholarly journals Joint numerical ranges for unbounded normal operators

1985 ◽  
Vol 28 (2) ◽  
pp. 225-232 ◽  
Author(s):  
Huang Danrun
Keyword(s):  
Unbounded Operator ◽  
Operator Algebras ◽  
Numerical Range ◽  
Bounded Operators ◽  
Unbounded Operators ◽  
Joint Spectrum ◽  
Normal Operators ◽  
Intimate Relations ◽  
Joint Numerical Range

For bounded operators, the theory of the joint numerical range has been developed by various authors [1,2,3,4,5]. Especially, the properties of commuting normal n-tuples are discussed in detail. Our purpose here is to show that many results in the above references still hold in the case of unbounded normal operators (see Theorem 2.3, Corrollary 3.5, Theorem 4.1, Theorem 4.2). Besides, the operator algebras are closely related to the theory of joint spectrum and joint numerical ranges in the boundedcase (cf. [1,3]). How about unbounded operators? It seems that one must consider unbounded operator algebras. Some work has been done in this direction for the joint spectrum of unbounded normal operators [9]. In the last section of this paper, we provide some intimate relations between the joint numerical range and the unbounded operator algebras for unbounded normal operators.

scholarly journals ON EXTREME POINTS OF THE JOINT NUMERICAL RANGE

1990 ◽  
Vol 21 (4) ◽  
pp. 399-403
Author(s):  
YOUNGOH YANG
Keyword(s):  
Numerical Range ◽  
Extreme Points ◽  
The Relationship ◽  
Joint Numerical Range

Our purpose is to study the relationship between the joint numerical range and joint essential numerical range. We give an example of an operator such that the set of all extreme points of the closure of its essential numerical range is not a subset of the set of all exti:eme points of its numerical range. We shall investigate the extreme points of a convex joint essential numerical range.

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 Convexity of the joint numerical range: topological and differential geometric viewpoints

2004 ◽  
Vol 376 ◽  
pp. 143-171 ◽  
Author(s):  
Eugene Gutkin ◽  
Edmond A. Jonckheere ◽  
Michael Karow
Keyword(s):  
Numerical Range ◽  
Joint Numerical Range

scholarly journals Pre-images of extreme points of the numerical range, and applications

2016 ◽  
pp. 1043-1058 ◽  
Author(s):  
Ilya M. Spitkovsky ◽  
Stephan Weis
Keyword(s):  
Numerical Range ◽  
Extreme Points

A note on the convexity of the indefinite joint numerical range

2014 ◽  
Vol 27 ◽  
Author(s):  
Hiroshi Nakazato ◽  
Natalia Bebiano ◽  
Joao Da Providencia
Keyword(s):  
Numerical Range ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Hermitian Matrices ◽  
Krein Spaces ◽  
Necessary And Sufficient ◽  
Joint Numerical Range

This note investigates the convexity of the indefinite joint numerical range of a tuple of Hermitian matrices in the setting of Krein spaces. Its main result is a necessary and sufficient condition for convexity of this set. A new notion of “quasi-convexity” is introduced as a refinement of pseudo-convexity.

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