SOME ASPECTS OF NUMERICAL RANGES OF BOUNDED LINEAR OPERATORS IN A COMPLEX HILBERT SPACE

Abstract

With mathematics analyst’ interest shifting from finite-dimensional inner product spaces to infinite-dimensional Hilbert spaces and with consequent shift of matrices to linear operation, their focus of attention changed from quadratic forms to numerical ranges of linear operators. In case of bounded linear operator, the closure of the numerical aspect that makes the study of numerical range more appealing and worthy of the increasing attention currently directed towards it. First, we give an alternative proof to the most important property of numerical range that for any bounded any linear operator, the numerical range is a convex set. Secondly, we show that for a hyponormal operator, the convex hull of the spectrum is the closure of numerical range. We also show that the same holds for subnormal operator. Lastly, we prove that if numerical range is closed, then every point arc in the boundary of the numerical range at which the boundary is not a differentiable arc is an eigenvalue for T.