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

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

2. 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.

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

4. 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

5. 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

6. Some Examples Ex.1.

7. Ex.2.

8. Ex.3.

9. Ex.4.

10. Ex.5.

11. Ex.6. My undergraduate student Aono found the following example. Counter example for Karaev’s paper(2004,Proceedings of AMS)

13. Ex.6. shows that nilpotency is not a sufficient condition for to be a disc. Indeed This is my motivation to start this study.

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

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

16. 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!

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

18. Proof of the Lemma

20. is unitary.

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

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

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

25. Theorem 1. (by Arimoto) is a disc.

26. Proof of Theorem For some ( from Theorem B)

27. are linearly independent, so

28. we now define by using the same

29. where we used

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

31. 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

32. Poncelet’s theorem Algebraic curves of order 2 (examples: ellipes)

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

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

35. 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)