How can I check the LiteSpeed backup file name used in a restore performed a few days ago ? (4320333)

I need to establish the filename used by a LiteSpeed restore which was performed a few days ago, how can I list the file names?

N/A

If LiteSpeed Verbose logging was enabled the log file will record the filename used - as long as you know the date/time the restore took place you can find the appropriate log file and therefore the filename.  
If no LiteSpeed logging was enabled, you can check the information recorded in the LiteSpeedLocal database for the server in question.  Check the LiteSpeedActivity table for the database you are interested in. You will be able to retrieve the rows based on StartTime and in the ActivityDetail column you can view the filename, amongst other details, used on that specific restore job. The ActivityTypeID will equal the value 3 for a Full restore or 4 for a Log Restore.

A similar query may also be executed:

use msdb
SELECT DISTINCT TOP 20
s1.type,
s1.backup_start_date,
s1.backup_finish_date,s2.physical_device_name
 from msdb..backupset s1 inner join msdb..backupmediafamily s2
 on s1.media_set_id = s2.media_set_id
 where s1.database_name ='Northwind' 
and s1.backup_start_date >= '5/25/2008 06:41:25 AM'
 --  You will need to change to the database you are querying
 and s1.type in('D','L','I')  -- sl.type in ('D','L') means full or Transaction Log backups  (if 'I', Differential backups) or ('F' for File Group, 'G' for File Group Differential)
 --and s1.backup_start_date >= (select max(backup_start_date) from msdb..backupset where database_name ='Northwind' and type ='D')  
 order by s1.backup_start_date desc