Some extension of Anderson’s theorem in semi-Hilbert spaces

Let A be a positive bounded linear operator acting on a complex Hilbert space <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi mathvariant="script">H</m:mi> </m:math> H . Our aim in this paper is to establish useful properties related to the A -numerical range and A -spectrum of A -bounded operators. In addition, we prove Anderson’s theorem for certain classes of semi-Hilbertian operators. Illustrative examples are also given.