Extreme points of numerical ranges of quasihyponormal operators

It is shown that a quasihypnormal operator on a Hilbert space having 0 as a boundary point of its numerical range is hyponormal. A necessary and sufficient condition is given for the extreme points of the numerical range of a quasihyponormal operator to be eigenvalues. It is also established that if T is bounded and there is IIxll = 1 such that IITxll = IITII and that < Tx, x > is a boundary point of the numerical range of T. then T has eigenvalues. Finally, an example is included of a paranormal operator which is not convexoid and such that T -∝ I is not paranormal for certain values of ∝.