Can the Numerical Range of a Nilpotent Operator be a Disc?

Akio Arimoto

Department of Mathematics Musashi Institute of Technology

T

0nT 1n

cos1

w Tn

W T w T

: nilpotent linear operator with norm 1 , i.e.

Assumption 1:

is a disc with the radius

and the center at origin.

Assumption 2:

Conclusion：

for some

Assertion in this talk

Notation

1:W T T B

： Numerical Radius

: Numerical range

1B : a unit ball in a Hilbert 　space

H

1sup :w T T B

Known results

• For a 2x2 matrix with eigenvalues 1 2, T

W T is an elliptical disc with as foci 1 2,

minor axis 1

2 2* 21 22c tr T T

major axis

22 1 c

Chi-Kwong Li, Proceeding of AMS, 1996, vol 124, no.7, 1985-1986

Toeplitz- Hausdorff ‘s Theorem

W T is a convex set in the Gauss plane.

O.Toeplitz, Das algebraische Analogon zu einem Satz von Fejer, Math.Z.2(1918),187-197

F.Hausdorff, Der Wertvorat einer Bilinearform,

Math.Z.(1919),314-316

Some Examples

1 0

0 0T

Ex.1.

0 0

1 0T

Ex.2.

Ex.3. 0 0

1 1T

Ex.4. 0 0 1

1 0 0

0 1 0

T

Ex.5. 0 0 0

1 0 0

0 0 1

T

Ex.6.

0 0 0

1 0 0

1 1 0

T

My undergraduate student Aono found the following example.

Counter example for Karaev’s paper(2004,Proceedings of AMS)

Ex.6. shows that nilpotency

is not a sufficient condition for

to be a disc. W T

3 0T Indeed

This is my motivation to start this study.

Haargerup and de la Harpe [HH] shown

that for a nilpotent

cos1

w T Tn

T

This is a consequence of a Fejer theorem :

1 0 cos1

f fn

1

1

0n

ikk

k n

f f e

Suppose

1T and that there exists a unit vector

1B with cos1

Tn

Let Vbe the linear span of 2 1, , , , nT T T

T

Theorem A.[HH p.375]

1 0,nT 0nT satisfies

Then V is an n- dimensional subspace of H

and the restriction of T to Vis unitarily equivalent to the n-dimensional shift on

0 0 0 0 0

1 0 0 0 0

0 1 0 0 0

0 0 0 0 0

0 0 0 1 0

S

We can restrict our problem to a finite matrix case even for the infinite dimensional space!

nC

Lemma

is a disc with the radius cos1n

and the center at zero.0 0 0 0 0

1 0 0 0 0

0 1 0 0 0

0 0 0 0 0

0 0 0 1 0

S

See example 2

: 1, nW S S C

Proof of the Lemma

2

1

0 0 0 0

0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

i

i

n i

ni

e

e

D

e

e

iDS e SD

2

23

0 00 00 0

0 01 0 00 0

00 0

00 0 1 00 0

0 0 0

ii

ii

nini

ee

eDS e

ee

2

( 1)

0 0

0 0

0

0

0 0 0

i

i

n i

e

SD e

e

•

1D

iDS e SD

*D D I

is unitary.D

•

1

i ie S e DS D

SD D W S

ie DS D

iW S e W S for

where

S nC 1

W S must be a disc so

because of Hausdorff-Toeplitz theorem.

• If we take a unit vector

0 0 cos ,1

Sn

0 1

The Haagerup - de la Harpe’s inequality

must be the equality Q.E.D.

( ) cos1

w Sn

1

2

0

1

2sin

1 1n

k n

kC

n n

we have

Theorem B.[HH,p.374]

Let 1 1,

i ji j T T , 1i j

1 1, 0 0

i ji j S S

If 1,2 1,2

then

1

2

0

1

2sin

1 1n

k n

kC

n n

Theorem 1. (by Arimoto)

1 0, 0n nT T

cos1

w Tn

1T

W T is a disc.

Proof of Theorem

2,1 0 0 2,1cos1

ie T Sn

1( 1) 1 1 1 10 0

i ji k k j k je T e T S S

, 1,2 ,k j n

For some

kj kj ( from Theorem B)

, 0,1, , 1kfor some c k n

1

0 00

nk

kk

D c S

2 10 0 0 0, , , nS S S

are linearly independent, so

1

0

nik k

kk

c e T

we now define

1 12

0 0 00 0

1n n

k jk j

k j

D c c S S

1 1

0 0

n nik k ij j

k jk j

c c e T e T

by using the same kc

2

1 11

0 0

n nik k ij j

k jk j

T c c e T e T

1 1

10 0

0 0

n ni k j

k jk j

e c c S S

0 0 0 0

i i ie SD D e S

ie T

0 0 2,1cos1

ie T Sn

where we used

ie T W T for any θ

Apply again the Toeplitz-Hausdorff theorem,

0 0 2,1cos1

ie T Sn

is a disc with the radius W T

cos1

w Tn

References

• [HH] Uffe Haagerup and Pierre de la Harpe, The Numerical Radius of a Nilpotent Operator on a Hilbert Space, Proceedings of Amer.Math.Soc. 115,(1992)

• [K] Mubariz T. Karaev, The Numerical Range of a Nilpotent Operator on a Hilbert Space, Proc. Amer.Math.Soc. ,2004

• [Wu]Pey-Yuan Wu( 呉培元） Polygons and Numerical ranges,Mathematical Monthly,107(2000)pp.528-540

• [Wu-Gau]P-Y.Wu and Hwa-Long Gau( 高） Numerical Range of S(Φ),Linear and Multilinear Algebra 45(1998),pp.49-73

Poncelet’s theorem

1 2,C C Algebraic curves of order 2 (examples: ellipes)

Poncelet’s theorem

0 1 1 0, , , ,n nP P P P P

0 1 1 0, , , ,n nL L L L L

If for some

Then starting from any other 0P on 1C

0 1 1 0, , , ,n nP P P P P

0 1 1 0, , , ,n nL L L L L

A nxn matrix

1/ 2* 1rank I A A

1A

then 1C being unit circle center 0

and 2C W A

2C W A has Poncelet ‘s property

Starting from any point on 1C0P

We have an n+1-gon

0 1 1 0, , , ,n nP P P P P

Also see • Hwa-Long Gau and Pei Yuan Wu

Numerical range and Poncelet propertyTaiwanese J.Math, vol.7,no2.173-193(2003)