ON NUMERICAL RANGE OF SOME CLASSES OF OPERATORS IN HILBERT SPACES

Abstract

The numerical range of operators in Hilbert spaces has been researched by several authors. The properties of the numerical range play an important role in identifying the behaviour of operators in Hilbert spaces. It is a well-known fact in operator theory that the spectrum of an operator is contained in the closure of the numerical range. The aim of our study was to establish the condition that gives the generalization that the spectrum of an operator is contained in the numerical range. Such spectral properties and the location of the numerical range in the complex plane were significant in determining the behavior of different classes of operators. We compared and analyzed known properties of the numerical range in the complex Hilbert space and narrowed this results to compact operators, spectroloid operators and partial isometries. Our results benefits other areas of mathematics and applied sciences such as quantum computing, physics and analysis. But more notably numerical range are used in engineering as rough estimates of eigenvalues of operators.