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

: 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 ： Numerical Radius : Numerical range : a unit ball in a Hilbert space

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

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), F.Hausdorff, Der Wertvorat einer Bilinearform, Math.Z.(1919),

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 :

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

Thenis an n- dimensional subspace of and the restriction ofto 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 must be a discso because of Hausdorff-Toeplitz theorem.

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

Theorem B.[HH,p.374] Let Ifthen

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

for any θ 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 [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 property Taiwanese J.Math, vol.7,no (2003)