scholarly journals Line segments on the boundary of the numerical ranges of some tridiagonal matrices

2015 ◽  
Vol 30 ◽  
pp. 693-703 ◽  
Author(s):  
Ilya Spitkovsky ◽  
Claire Thomas
Keyword(s):  
Numerical Range ◽  
Main Diagonal ◽  
Geometric Progression ◽  
Diagonal Form ◽  
Line Segments ◽  
Tridiagonal Matrices

Tridiagonal matrices are considered for which the main diagonal consists of zeroes, the sup-diagonal of all ones, and the entries on the sub-diagonal form a geometric progression. The criterion for the numerical range of such matrices to have line segments on its boundary is established, and the number and orientation of these segments is described.

scholarly journals Numerical ranges of powers of hermitian elements

1986 ◽  
Vol 28 (1) ◽  
pp. 37-45 ◽  
Author(s):  
M. J. Crabb ◽  
C. M. McGregor
Keyword(s):  
Banach Algebra ◽  
Numerical Range ◽  
Previous Knowledge ◽  
Line Segments ◽  
Straight Line ◽  
Image Position ◽  
Unital Banach Algebra

An element k of a unital Banach algebra A is said to be Hermitian if its numerical rangeis contained in ℝ; equivalently, ∥eitk∥ = 1(t ∈ ℝ)—see Bonsall and Duncan [3] and [4]. Here we find the largest possible extent of V(kn), n ∈ ℕ, given V(k) ⊆ [−1, 1], and so ∥k∥ ≤ 1: previous knowledge is in Bollobás [2] and Crabb, Duncan and McGregor [7]. The largest possible sets all occur in a single example. Surprisingly, they all have straight line segments in their boundaries. The example is in [2] and [7], but here we give A. Browder's construction from [5], partly published in [6]. We are grateful to him for a copy of [5], and for discussions which led to the present work. We are also grateful to J. Duncan for useful discussions.

Line segments and elliptic arcs on the boundary of a numerical range

2008 ◽  
Vol 56 (1-2) ◽  
pp. 131-142 ◽  
Author(s):  
Hwa-Long Gau ◽  
Pei Yuan Wu
Keyword(s):  
Numerical Range ◽  
Line Segments

scholarly journals On some reciprocal matrices with elliptical components of their Kippenhahn curves

2021 ◽  
Vol 10 (1) ◽  
pp. 117-130
Author(s):  
Muyan Jiang ◽  
Ilya M. Spitkovsky
Keyword(s):  
Numerical Range ◽  
Main Diagonal ◽  
Higher Rank

Abstract By definition, reciprocal matrices are tridiagonal n-by-n matrices A with constant main diagonal and such that ai , i +1 ai +1, i = 1 for i = 1, . . ., n − 1. We establish some properties of the numerical range generating curves C(A) (also called Kippenhahn curves) of such matrices, in particular concerning the location of their elliptical components. For n ≤ 6, in particular, we describe completely the cases when C(A) consist entirely of ellipses. As a corollary, we also provide a complete description of higher rank numerical ranges when these criteria are met.

scholarly journals THE SPECTRAL SCALE AND THE NUMERICAL RANGE

2003 ◽  
Vol 14 (02) ◽  
pp. 171-189 ◽  
Author(s):  
CHARLES A. AKEMANN ◽  
JOEL ANDERSON
Keyword(s):  
Hilbert Space ◽  
Von Neumann Algebra ◽  
Point Spectrum ◽  
Numerical Range ◽  
Spectral Information ◽  
Von Neumann ◽  
Line Segments ◽  
Information Base ◽  
Arbitrary Operator ◽  
Spectral Scale

Suppose that c is an operator on a Hilbert Space H such that the von Neumann algebra N generated by c is finite. Let τ be a faithful normal tracial state on N and set b1 = (c + c*)/2 and b2 = (c - c*)/2i. Also write B for the spectral scale of {b1, b2} relative to τ. In previous work by the present authors, some joint with Nik Weaver, B has been shown to contain considerable spectral information about the operator c. In this paper we expand that information base by showing that the numerical range of c is encoded in B also. We begin by proving that the k-numerical range of an arbitrary operator d in B(H) coincides with the numerical range of d when the von Neumann algebra generated by d contains no finite rank operators. Thus, the k-numerical range is not useful for most operators considered here. We next show that the boundary of the numerical range of c is exactly the set of radial complex slopes on B at the origin. Further, we show that points on this boundary that lie in the numerical range are visible as line segments in the boundary of B. Also, line segments on the boundary which lie in the numerical range show up as faces of dimension two in the boundary of B. Finally, when N is abelian, we prove that the point spectrum of c appears as complex slopes of 1-dimensional faces of B.

scholarly journals The c-numerical range of tridiagonal matrices

2001 ◽  
Vol 335 (1-3) ◽  
pp. 55-61 ◽  
Author(s):  
Mao-Ting Chien ◽  
Hiroshi Nakazato
Keyword(s):  
Numerical Range ◽  
Tridiagonal Matrices

scholarly journals Linearity and weak convergence on the boundary of numerical range

1983 ◽  
Vol 35 (2) ◽  
pp. 221-226 ◽  
Author(s):  
K. C. Das ◽  
B. D. Craven
Keyword(s):  
Weak Convergence ◽  
Linear Space ◽  
Line Segment ◽  
Numerical Range ◽  
Extreme Points ◽  
Weak Limit ◽  
Alternative Proof ◽  
Line Segments ◽  
On Line ◽  
Convergent Sequences

AbstractStampfli and Embry have shown that a point of the numerical range of an operator is extreme if and only if a set of vectors corresponding to it is linear. This is generalized here to show that a point of the closure of the numerical range is extreme if and only if a corresponding set of sequences forms a linear space. A more geometric alternative proof is given for a theorem of Das and Garske concerning weak convergence to zero at the unattained extreme points of the closure of the numerical range.The result is shown to hold also for lone extreme points of the numerical range which lie on line segments on its boundary. Further, a bound is obtained on the norm of the weak limit of the weakly convergent sequences corresponding to points on a line segment on the boundary of numerical range.

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