The Ninth Workshop on Numerical Ranges and Numerical Radii

In this talk I will discuss some instances in quantum computing where numerical range techniques arise. I will also try to formulate some open problems. Elliptical range theorems for generalized numerical ranges of quadratic operators Speaker Chi-Kwong Li, William and Mary, [email protected] Co-authors Yiu-Tung Poon, Iowa State University, [email protected]; Nung-Sing Sze, University of Connecticut, [email protected] Abstract The classical numerical range of a quadratic operator is an elliptical disk. This result is extended to different kinds of generalized numerical ranges. In particular, it is shown that for a given quadratic operator, the rank-k numerical range, the essential numerical range, and the q-numerical range are elliptical disks; the c-numerical range is a sum of elliptical disks, and the Davis-Wielandt shell is an ellipsoid with or without interior. Preservers of the joint higher rank numerical rangeThe classical numerical range of a quadratic operator is an elliptical disk. This result is extended to different kinds of generalized numerical ranges. In particular, it is shown that for a given quadratic operator, the rank-k numerical range, the essential numerical range, and the q-numerical range are elliptical disks; the c-numerical range is a sum of elliptical disks, and the Davis-Wielandt shell is an ellipsoid with or without interior. Preservers of the joint higher rank numerical range Speaker Jennifer Mahle, College of William and Mary, [email protected] Co-authors Sean Clark and Chi-Kwong Li, College of William and Mary. Abstract It is shown that linear preservers of the joint higher rank numerical range of mtuples of square matrices have the formIt is shown that linear preservers of the joint higher rank numerical range of mtuples of square matrices have the form (A1, . . . , Am) 7→ (UA1U, . . . , UAmU) or (A1, . . . , Am) 7→ (UA1U, . . . , UAmU) for some unitary matrix U . Moreover, it is shown that that the linearity assumption can be replaced by additivity and surjectivity. To achieve this, we show that additive maps mapping the cone of positive semi-definite matrices onto itself must be linear.