Circular numerical ranges of partial isometries

Abstract

Let be an -by- partial isometry whose numerical range is a circular disc with centre and radius . In this paper, we are concerned with the possible values of and . We show that must be if is at most and conjecture that the same is true for the general . As for the radius, we show that if , then the set of all possible values of is . Indeed, it is shown more precisely that for , , the possible values of are those in the interval . In the proof process, we also characterize -by- partial isometries which are (unitarily) irreducible. The paper is concluded with a question on the rotational invariance of nilpotent partial isometries with circular numerical ranges centred at the origin.

Keywords:

• numerical range
• partial isometry
• irreducible matrix
• nilpotent matrix
• rotationally invariant matrix

AMS Subject Classifications:

• 15A60
• 47A12

Acknowledgements

The contents of this paper was reported by the third author in the 12th Workshop on Numerical Ranges and Numerical Radii (WORNA) at Sanya, Hainan, China on July 30, 2014. Thanks are due to the organizer C.-K. Li and the staff, especially Yanyu Fang, for all the work they have done for the workshop. We thank the (anonymous) referee for his incisive comments, which lead to improvements of our Example 4.5. The two paragraphs after it are essentially due to him.

Additional information

Funding

The authors acknowledge supports from the Ministry of Science and Technology of the Republic of China under projects MOST 103-2115-M-008-006, MOST 103-2115-M-009-001 and NSC 102-2115-M-009-007, respectively.