Who among the following, as the proprietor of the Rose theatre, repaired it in 1592 and in 1595, and spent substantial sums on renovating and expanding his theatre?

The correct option is Philip Henslowe.

Philip Henslowe was a key figure in Elizabethan theatre, best known as an entrepreneurial impresario and the builder of the Rose theatre in 1587. The Rose was the first purpose-built playhouse in Bankside, Southwark, situated just across the Thames from the City of London.

As the proprietor of the Rose, Henslowe kept detailed financial records in his account book, today famously referred to as Henslowe's Diary. His records show that he invested substantial sums of money to renovate, repair, and expand the Rose, most notably in the years 1592 and 1595. The 1592 renovation was particularly extensive, involving structural changes to the stage and the addition of a protective roof (or "heavens") over it, which accommodated large and popular acting troupes like Lord Strange's Men.

The other individuals listed are also prominent figures of the era but had different theater associations:
- James Burbage built the Theatre (1576) and bought the Blackfriars Theatre (1596).
- Richard Burbage, James's son, was a legendary actor and major shareholder in the Globe Theatre.
- Francis Langley was the builder and owner of the Swan theatre (1595).