Yes. To be specific (pre BR):
West Kirby and New Brighton to Liverpool Central (electrified) = LMS lines.
Rock Ferry to Liverpool Central = Mersey Railway.
Operationally, however, the MR also ran the weekday New Brighton services and LMS ran the Rock Ferry services on Sundays.
West Kirby to Hooton,
Birkenhead to Chester/Helsby and beyond = GWR/LMS joint.
Seacombe- Wrexham/Chester = LNER (though the Seacombe branch itself was owned by the LMS).
West Kirby and New Brighton to Liverpool Central (electrified) = LMS lines.
Rock Ferry to Liverpool Central = Mersey Railway.
Operationally, however, the MR also ran the weekday New Brighton services and LMS ran the Rock Ferry services on Sundays.
West Kirby to Hooton,
Birkenhead to Chester/Helsby and beyond = GWR/LMS joint.
Seacombe- Wrexham/Chester = LNER (though the Seacombe branch itself was owned by the LMS).