Numerical Range and Convex Sets*

1974 ◽  
Vol 17 (2) ◽  
pp. 295-296 ◽  
Author(s):  
Fredric M. Pollack
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Convex Subset ◽  
Bounded Linear Operator ◽  
Convex Sets ◽  
Numerical Range ◽  
Bounded Subset ◽  
Bounded Linear ◽  
Empty Convex ◽  
Image Position

The numerical range W(T) of a bounded linear operator T on a Hilbert space H is defined byW(T) is always a convex subset of the plane [1] and clearly W(T) is bounded since it is contained in the ball of radius ‖T‖ about the origin. Which non-empty convex bounded subsets of the plane are the numerical range of an operator? The theorem we prove below shows that every non-empty convex bounded subset of the plane is W(T) for some T.

scholarly journals A result on hermitian operators

1989 ◽  
Vol 31 (1) ◽  
pp. 71-72
Author(s):  
J. E. Jamison ◽  
Pei-Kee Lin
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Numerical Range ◽  
Complex Banach Space ◽  
Bounded Linear ◽  
Image Position

Let X be a complex Banach space. For any bounded linear operator T on X, the (spatial) numerical range of T is denned as the setIf V(T) ⊆ R, then T is called hermitian. Vidav and Palmer (see Theorem 6 of [3, p. 78] proved that if the set {H + iK:H and K are hermitian} contains all operators, then X is a Hilbert space. It is natural to ask the following question.

Relations Between Generalized Growth Conditions and Several Classes of Convexoid Operators

1977 ◽  
Vol 29 (5) ◽  
pp. 1010-1030 ◽  
Author(s):  
Takayuki Furuta
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Numerical Range ◽  
Growth Conditions ◽  
Complex Hilbert Space ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Image Position

In this paper we shall discuss some classes of bounded linear operators on a complex Hilbert space. If T is a bounded linear operator T acting on the complex Hilbert space H, then the following two inequalities always hold:where σ(T) indicates the spectrum of T, W(T) denotes the numerical range of T defined by W(T) = {(Tx, x) : ||x|| = 1 and x ∊ H} and means the closure of W(T) respectively.

scholarly journals Some convexity theorems for matrices

1971 ◽  
Vol 12 (2) ◽  
pp. 110-117 ◽  
Author(s):  
P. A. Fillmore ◽  
J. P. Williams
Keyword(s):  
Linear Operator ◽  
Operator Theory ◽  
Bounded Linear Operator ◽  
Numerical Range ◽  
Two Dimensional ◽  
Bounded Linear ◽  
Image Position

The numerical range of a bounded linear operator A on a complex Hilbertspace H is the set W(A) = {(Af, f): ║f║ = 1}. Because it is convex andits closure contains the spectrum of A, the numerical range is often a useful toolin operator theory. However, even when H is two-dimensional, the numerical range of an operator can be large relative to its spectrum, so that knowledge of W(A) generally permits only crude information about A. P. R. Halmos [2] has suggested a refinement of the notion of numerical range by introducing the k-numerical rangesfor k = 1, 2, 3, …. It is clear that W1(A) = W(A). C. A. Berger [2] has shown that Wk(A) is convex.

The power inequality on Banach spaces

1971 ◽  
Vol 69 (3) ◽  
pp. 411-415 ◽  
Author(s):  
Béla Bollobás
Keyword(s):  
Linear Operator ◽  
Banach Spaces ◽  
Normed Space ◽  
Bounded Linear Operator ◽  
Dual Space ◽  
Numerical Range ◽  
Numerical Radius ◽  
Bounded Linear ◽  
Power Inequality ◽  
Image Position

Let X be a complex normed space with dual space X′ and let T be a bounded linear operator on X. The numerical range of T is defined asand the numerical radius is v(T) = sup {|ν: νε V(T)}. Most known results and problems concerning numerical range can be found in the notes by Bonsall and Duncan (5).

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.

On a Family of Generalized Numerical Ranges

1974 ◽  
Vol 26 (3) ◽  
pp. 678-685
Author(s):  
C.-S. Lin
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Banach Algebra ◽  
Orthogonal Projection ◽  
Bounded Linear Operator ◽  
Unitary Operator ◽  
Linear Operators ◽  
Minkowski Distance ◽  
Bounded Linear ◽  
Image Position

Throughout this note, an operator will always mean a bounded linear operator acting on a Hilbert space X into itself, unless otherwise stated. The class Cρ (0 < ρ < ∞ ) of operators, considered by Sz.-Nagy and Foiaş [5], is defined as follows: An operator T is in Cρ if Tnx = pPUnx for all x ∊ X, n = 1, 2, . . . , where U is a unitary operator on some Hilbert space Y containing X as a subspace, and P is the orthogonal projection of Y onto X. In [2] Holbrook defined the operator radii wρ(·) (0 < ρ ≦ ∞ ) as the generalized Minkowski distance functionals on the Banach algebra of bounded linear operators on X, i.e.,and w∞(T) = r(T), the spectral radius of T [2, Theorem 5.1].

Steepest Descent and Least Squares Solvability

1974 ◽  
Vol 17 (2) ◽  
pp. 275-276 ◽  
Author(s):  
C. W. Groetsch
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Least Squares ◽  
Bounded Linear Operator ◽  
Steepest Descent ◽  
Normal Equation ◽  
Bounded Linear ◽  
Least Squares Solution ◽  
Image Position

Let T be a bounded linear operator defined on a Hilbert space H. An element z∈H is called a least squares solution of the equationif . It is easily shown that z is a least squares solution of (1) if and only if z satisfies the normal equation

A Spectral Theory for Duality Systems of Operators on a Banach Space

1970 ◽  
Vol 22 (5) ◽  
pp. 994-996 ◽  
Author(s):  
J. G. Stampfli
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Linear Operator ◽  
Banach Spaces ◽  
Spectral Theory ◽  
Bounded Linear Operator ◽  
Bounded Linear ◽  
Banach Theorem ◽  
Duality Map ◽  
Image Position

This note is an addendum to my earlier paper [8]. The class of adjoint abelian operators discussed there was small because the compatibility relation between the operator and the duality map was too restrictive. (In effect, the relation is appropriate for Hilbert space, but ill-suited for other Banach spaces where the unit ball is not round.) However, the techniques introduced in [8] permit us to readily obtain a spectral theory (of the Dunford type) for a wider class of operators on Banach spaces, as we shall show.A duality system for the operator T is an ordered sextuple(i) T is a bounded linear operator mapping the Banach space B into B,(ii) ϕ is a duality map from B to B*. Thus, for x ∊ B, ϕ(x) = x* ∊ B*, where ‖x‖ = ‖x*‖ and x*(x) = ‖x‖2. The existence of ϕ follows easily from the Hahn-Banach Theorem.

scholarly journals Bounded point evaluations for cyclic Hilbert space operators

2003 ◽  
Vol 4 (2) ◽  
pp. 301
Author(s):  
A. Bourhim
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Bounded Linear ◽  
Hilbert Space Operators ◽  
Point Evaluations ◽  
Analytic Bounded Point Evaluations

<p>In this talk, to be given at a conference at Seconda Università degli Studi di Napoli in September 2001, we shall describe the set of analytic bounded point evaluations for an arbitrary cyclic bounded linear operator T on a Hilbert space H and shall answer some questions due to L. R. Williams.</p>

scholarly journals Exponential Function of a bounded Linear Operator on a Hilbert Space.

2014 ◽  
Vol 11 (3) ◽  
pp. 1267-1273
Author(s):  
Baghdad Science Journal
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Exponential Function ◽  
Bounded Linear Operator ◽  
Bounded Linear ◽  
Exponential Operator

In this paper, we introduce an exponential of an operator defined on a Hilbert space H, and we study its properties and find some of properties of T inherited to exponential operator, so we study the spectrum of exponential operator e^T according to the operator T.

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