SELECT
  p.StartDate,
  p.EndDate,
  h.PersonID
FROM Periods p
  LEFT JOIN History h
    ON h.[From] BETWEEN p.StartDate AND p.EndDate OR
       p.StartDate BETWEEN h.[From] AND ISNULL(h.[To], '30000101')

It would affect performance, but I think it is worth just trying the odd looking between:

select x
from table1 t1
  inner join table2 t2
    on t2.date between t1.startdate and t1.enddate

Whether it works or not will depend on whether this is to be production, or just a one time thing, and how many records are involved. It may be way too slow.

Can you please try:

SELECT P.StartDate, P.EndDate, H.PersonID
FROM Period P INNER JOIN History H ON P.StartDate <= H.Fromand (P.EndDate >= H.To OR H.To IS NULL)

I have edited the SQL after reading the spec more clearly

I have edited the SQL again. I'm using INNER JOIN now.

You need to do a left join in order to show all the periods available even if there are no history entries associated with that period. The criteria would be if the History date was between the period. You would also need to check if the To date was null and include it into your results

SELECT  p.StartDate, p.EndDate, h.PersonId
FROM    Period p
        LEFT JOIN History h
            ON p.StartDate >= h.[From] AND 
            (h.[To] IS NULL OR p.EndDate <= h.[To])

at table history , set NULL to 9999-12-31

SELECT *
FROM periods a
INNER JOIN history b
ON a.FROM < b.TO AND a.TO > b.FROM