Relative positions of half-sided modular inclusions

Abstract Let $$K_1 \subset H$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>K</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>⊂</mml:mo> <mml:mi>H</mml:mi> </mml:mrow> </mml:math> and $$K_2 \subset H$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>K</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>⊂</mml:mo> <mml:mi>H</mml:mi> </mml:mrow> </mml:math> be half-sided modular inclusions in a common standard subspace H . We prove that the inclusion $$K_1 \subset K_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>K</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>⊂</mml:mo> <mml:msub> <mml:mi>K</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> </mml:math> holds if and only if we have an inclusion of spectral subspaces of the generators of the positive one-parameter groups associated to the half-sided modular inclusions $$K_1 \subset H$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>K</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>⊂</mml:mo> <mml:mi>H</mml:mi> </mml:mrow> </mml:math> and $$K_2 \subset H$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>K</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>⊂</mml:mo> <mml:mi>H</mml:mi> </mml:mrow> </mml:math> . From this we give a characterization of this situation in terms of (operator-valued) symmetric inner functions. We illustrate these characterizations with some examples of non-trivial phenomena occurring in this setting.