Local Spectral Theory for Unbounded Product of Nonnegative Selfadjoint Operators

Abstract An unbounded product $$T=AB$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>T</mml:mi> <mml:mo>=</mml:mo> <mml:mi>A</mml:mi> <mml:mi>B</mml:mi> </mml:mrow> </mml:math> of two nonnegative selfadjoint operators A and B , where $$\rho (T) \ne \emptyset $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ρ</mml:mi> <mml:mo>(</mml:mo> <mml:mi>T</mml:mi> <mml:mo>)</mml:mo> <mml:mo>≠</mml:mo> <mml:mi>∅</mml:mi> </mml:mrow> </mml:math> and A is bounded, is proved to have the single valued extension property (SVEP) and, more significantly, to be a $$\mathbb {C}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>C</mml:mi> </mml:math> -generalized scalar operator. This property plays a central role as, it not only implies the Dunford’s property for the product AB , but it also allows any operator T that is quasi-affine to a nonnegative selfadjoint operator S to satisfy $$\sigma (T)=\sigma (S)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>σ</mml:mi> <mml:mo>(</mml:mo> <mml:mi>T</mml:mi> <mml:mo>)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>σ</mml:mi> <mml:mo>(</mml:mo> <mml:mi>S</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . These results have required a deeper study of the spectral connection between operators S having SVEP and Dunford properties and operators T that are quasi-affine to S .