* May: BA should provide service "people expect"
* Flight cancellations left 75,000 passengers stranded
* Analysts see costs of up to 80 million euros
* BA now running full schedule at Heathrow and Gatwick (Adds quotes from PM May, energy companies, Ryanair)
LONDON, May 30 (Reuters) - British Airways flights were back in the skies on Tuesday but the company faced increasing pressure over its response to the huge IT failure that left 75,000 passengers stranded over a holiday weekend and dealt a major blow to its reputation.
BA, which once marketed itself as "the world's favourite airline" suffered a public relations disaster after it had to cancel all flights from London's Heathrow and Gatwick airports on Saturday. It blamed a power surge that knocked out its computer system, disrupting flight operations, call centres and its website.
Although BA said it expected to run a full schedule from Heathrow and Gatwick on Tuesday, it was left with work to do in the longer term to restore its reputation after a long weekend of chaos and frustration for passengers.
British Prime Minister Theresa May weighed in on the issue on the campaign trial ahead of the June 8 national election.
"It is up to them to sort their IT out and to ensure that they're able to provide the services that people expect them to provide as British Airways," May said.
London-listed shares in BA's parent company IAG fell when the stock market reopened. BA said it was launching a thorough investigation to understand what happened and make sure there was no repeat.
BA had already come under fire for charging extra for food and baggage and the sight of stranded passengers trying to sleep on the floor of its gleaming Heathrow Terminal 5 building is likely to tarnish its image.
"This will certainly damage their reputation," said Angharad Griffiths, a travel agent who was at Heathrow picking up a tour group from Lisbon. "I've never had a good experience with them, even before this."
Like other European full-service airlines, BA is facing increased competition from budget rivals Ryanair and easyJet.
"British Airways' IT failure over the weekend is clearly a PR nightmare and will take a real focus in terms of handling customers' complaints and compensations claims in order to rebuild trust and confidence with the public," said Mark Simpson, analyst at Goodbody. He estimated the cost of the outage at 82 million euros ($91.6 million).
BA Chief Executive Alex Cruz had said on Monday that the power surge was so strong that it also rendered the back-up systems ineffective. The firm said a supply issue at a data centre near Heathrow sparked the surge.
Scottish and Southern Electricity Networks, which manages the electricity distribution network in Harmondsworth just north of Heathrow airport, said its network in the area was running as normal on Saturday morning.
"The power surge that BA is referring to could have taken place at the customer side of the meter," a spokesman said.
A spokeswoman for Heathrow Airport said that there were no issues with the site's private electricity network on Saturday.
Cruz, who moved to BA from low cost sister airline Vueling, denied that the outage was linked to a decision to cut staff numbers and outsource work to India.
Irish rival Ryanair, which reported record annual profits on Tuesday, said it had systems in place to avoid a similar fiasco.
"We have three IT locations in different countries across Europe," Kenny Jacobs, chief marketing officer, told reporters in London.
"If there's a power surge at one, the second kicks in, and the third one would kick in. That's what most businesses would do. That's our approach and we've never had a major outage."
Ryanair Chief Executive Michael O'Leary said his company saw very strong bookings over the weekend, but added it was unclear if this was related to the BA problems.
In the wake of the outage, Ryanair had taken to social media to poke fun at BA.
Shares in BA's parent company IAG, which also owns carriers Iberia, Aer Lingus and Vueling, fell as much as 4.5 percent on Tuesday, the first day of London trading this week. They were trading 1.8 percent lower at 1400 GMT. ($1 = 0.8956 euros)
(Additional reporting by Paul Sandle, Jon Coffey and Karolin Schaps; editing by Keith Weir)