The real estate investment trusts that own the malls and shopping centers where many Sears stores are anchor tenants have waited years for the retailer's demise to renovate the sites and boost rent, although redevelopment costs may strain some plans.

Most large U.S. malls are controlled by REITs. In recent years, the REITs have cut their exposure to Sears Holdings, which filed for Chapter 11 bankruptcy on Monday. Sears said it plans to close 142 stores.

Sears now accounts for less than 1 percent of many REITs' base rental income. Gaining control of vacated sites will be costly, while rebuilding the stores can run about $10 million to $12 million for each site.

But most REIT shares gained on Monday as investors focused on the potential benefits of a Sears bankruptcy.

The REITs have looked forward to a bankruptcy to remove the eyesore of many Sears stores. They also will be able to raise the rent in contracts sometimes signed more than 20 years ago with extension options that kept leases very low.

Shares of mall REITs have fallen in recent years as investors feared online shopping would eliminate the need for many brick-and-mortar stores and as rising interest rates add to funding costs. Besides Sears, retail bankruptcies such as Toys R Us and Bon-Ton Stores created vacancies.