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

Can the Numerical Range of a Nilpotent Operator be a Disc? Akio Arimoto Department of Mathematics Musashi Institute of Technology

Assertion in this talk Assumption 1: : nilpotent linear operator with norm 1 , i.e. for some Assumption 2: Conclusion： is a disc with the radius and the center at origin.

Notation ： Numerical Radius : Numerical range : a unit ball in a Hilbertspace

Known results • For a 2x2 matrix with eigenvalues is an ellipticaldisc with as foci minor axis major axis Chi-Kwong Li, Proceeding of AMS, 1996, vol 124, no.7, 1985-1986

Toeplitz- Hausdorff ‘s Theorem 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 Ex.1.

Ex.2.

Ex.3.

Ex.4.

Ex.5.

Ex.6. 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. Indeed This is my motivation to start this study.

Haargerup and de la Harpe [HH] shown that for a nilpotent This is a consequence of a Fejer theorem :

Theorem A.[HH p.375] Suppose satisfies and that there exists a unit vector with Let be the linear span of

Then is an n-dimensional subspace of and the restriction of to is unitarily equivalent to the n-dimensional shift on We can restrict our problem to a finite matrix case even for the infinite dimensional space!

Lemma is a disc with the radius and the center at zero. See example 2

Proof of the Lemma

is unitary.

for where so must be a disc because of Hausdorff-Toeplitz theorem.

If we take a unit vector we have The Haagerup - de la Harpe’s inequality must be the equality Q.E.D.

Theorem B.[HH,p.374] Let If then

Theorem 1. (by Arimoto) is a disc.

Proof of Theorem For some ( from Theorem B)

are linearly independent, so

we now define by using the same

where we used

foranyθ Apply again the Toeplitz-Hausdorff theorem, is a disc with the radius

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 Algebraic curves of order 2 (examples: ellipes)

Poncelet’s theorem If for some Then starting from any other on

nxn matrix then being unit circle center 0 and has Poncelet ‘s property

Starting from any point on We have an n+1-gon Also see • Hwa-Long Gau and Pei Yuan Wu Numerical range and Poncelet propertyTaiwanese J.Math, vol.7,no2.173-193(2003)

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