Monotonicity properties for joint and generalized spectral radius and their essential versions of weighted geometric symmetrizations

Abstract We prove new monotonicity properties for joint and generalized spectral radius and their essential versions of weighted geometric symmetrizations of bounded sets of positive kernel operators on $$L^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> . To our knowledge, several proved properties are new even in the finite dimensional case.