NUMERICAL RANGE AND SUB-SELF-ADJOINT OPERATORS

NUMERICAL RANGE AND SUB-SELF-ADJOINT OPERATORS

R. Chettouh, S. Bouzenada

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Abstract

In this paper, we show that the numerical range of a bounded linear operator T on a complex Hilbert space is a line segment if and only if there are scalars and such that T = T + I, and we determine the equation of the straight support of this numerical range in terms of  and : An operator T is called sub-self-adjoint if their numerical range is a line segment. The class of sub-self-adjoint operators contains every self-adjoint operator and contained in the class of normal operators. We show that this class is uniformly closed, invariant under unitary equivalence and invariant under ane transformation. Some properties of the sub-self-adjoint operators and their numerical ranges are investigated.

Keywords

Numerical range, self-adjoint operator, normal operator.