A study of generalized numerical ranges

by Tsing, Nam-kiu

i

Abstract of thesis entitled

'A STUDY OF GENERALIZED NUMERICAL RANGES'

submitted by Nam-Kiu Tsing for the degree of

Doctor of Philosophy

at the University of Hong Kong in April, 1983.

This thesis consists of five chapters. Chapter I and III contain

the joint work of the author and his supervisor Dr. Y.H. Au-Yeung.

In Chapter I, we consider the set

n

{( L

i=l

... ,

n

L c.x.A x~): i=l l l P l

c.x.A x*, l l 1 i

is an orthonormal set in Fn}

where F is the real field

ffi or the complex field ~,

n IR

p

Hermitian (symmetric)

c ) E: n

and

... ,

any

matrices with elements in F. We shall give a unified proof of the

convexity of

A2) (with n > 2 if F m). With a similar cr

Wc(Al, A2' A3) for n > 2 is also proved.

method, convexity of

Equivalent statements of these results in terms of definiteness and

inclusion relation are considered and some applications are given .

.?

In Chapter II, the relation between the sets

weAl

{xAx*: X E: ~n, xx* = I}

and

n {xAy* + yAx*: x, y E: ~ , xx*

yy*

1, xy*

a} ,

-------------------------------------------

~

ii

where A is any n x n complex matrix, is studied. We shall also give

a characterization of the diameter and the minimal width of W(A)

In Chapter III, we shall show that if A is an n x n normal

matrix with eigenvalues AI' ... , An which are not collinear, then

the set

n

{I A.x.Ax~: {xl' ... , xn} is an orthonormal set in ~n} l l l

i=l

is not convex which answers a conjecture posed by Marcus.

In Chapter IV, let A be any complex matrix of order n ,

(Yl, ... , Yk) E ~k where 1 < k ~ n. We shall prove that the set

lS star-shaped. Consequently, the set

k

{ I y.X.AX~:{xl' ... ,~} is an orthonormal set in ~n} i=l l l l

is star-shaped, which answers a conjecture posed by Straus.

In Chapter V, we shall prove the convexity of the set

{xAy*: x, y E ~n, xx*

yy*

1, xy* EK}

where A is any n x n complex matrix and K a convex set in IT. As

a consequence, the set

{tr(CUAU*): U unitary}

is convex if C is any n x n complex matrix of rank one.