Unconditional basic sequences in function spaces with applications to Orlicz spaces

Abstract We find conditions on a function space $${\varvec{L}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>L</mml:mi> </mml:mrow> </mml:math> that ensure that it behaves as an $$L_p$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:math> -space in the sense that any unconditional basis of a complemented subspace of $${\varvec{L}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>L</mml:mi> </mml:mrow> </mml:math> either is equivalent to the unit vector system of $$\ell _2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ℓ</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> or has a subbasis equivalent to a disjointly supported basic sequence. This dichotomy allows us to classify the symmetric basic sequences of $${\varvec{L}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>L</mml:mi> </mml:mrow> </mml:math> . Several applications to Orlicz function spaces are provided.