Secretary of State James A. Baker III was back in his seventh floor, State Department suite yesterday, having cut short a Wyoming vacation that caused speculation about whether he was, as the Financial Times of London remarked, the "Mr. Invisible" of the Persian Gulf crisis.

Baker went to a Wyoming ranch Aug. 16 after attending President Bush's meeting with Jordan's King Hussein in Kennebunkport, Maine. And, just as questions have been asked about the propriety of the president remaining in the ocean breezes of his summer home while thousands of Americans are bogged down in the sands of the Arabian peninsula, Baker's absence has prompted gossip about whether he has been too far from the center of the action.

That idea is dismissed as "invalid to the point of being ludicrous" by Margaret Tutwiler, Baker's spokesman and executive assistant. She said, "He really didn't get a vacation at all because he spent more than half of every single day he was in Wyoming on the phone keeping in touch with the department, with the president and with his foreign minister counterparts in the Middle East and Europe."

In the State Department, mid-level officials generally seem to take the attitude that the boss, like everyone else, is entitled to some occasional time off. In Baker's absence, they say, the department has run quite well under the direction of Deputy Secretary Lawrence S. Eagleburger and Robert M. Kimmitt, undersecretary for political affairs.

Still, many of these same officials add, in the federal bureaucracy where appearances often are as important as substance, Baker may have made a fairly big tactical mistake by absenting himself from Washington at this particular time. For one thing, it has caused speculation, particularly in the European news media, that he is at odds with the White House over Bush's emphasis on making the buildup of military forces in Saudi Arabia the main thrust of the U.S. confrontation with Iraq.

State Department officials dismiss that idea as totally uninformed. Instead, they say, the problem created by Baker's absence is that his biggest asset -- his influence as Bush's most trusted foreign policy adviser -- was not immediately available to State at a time when events were pushing the Defense Department into the role of lead agency in managing U.S. policy in the gulf.

"It's no secret that Jim Baker is not the most popular secretary of recent times among the troops because of his tendency to exclude the career bureaucracy from much of his decision making and rely on a relatively closed circle of insiders," one senior official said.

"In the present situation, people expected that he would hold up the department's side in internal administration debates and give it more of a policy input than has been the case," the official continued. "That didn't happen. The State Department hasn't exactly been pushed aside, but there have been lots of little, day-to-day matters where its influence could have been greater, where Eagleburger and Kimmitt don't have the personal status with the president to tip the balance of an argument in State's favor."

As one example, he and other officials say that the State Department has had problems asserting itself as the principal source of political intelligence and analysis about events in Iraq since the Aug. 2 invasion of Kuwait.

April Glaspie, the U.S. ambassador to Baghdad who currently is in Washington, and John Kelly, the assistant secretary for Middle East affairs, have worked hard to fill that role, the officials said, but without Baker's presence literally behind them, their views frequently have played second fiddle to those of experts from the National Security Council, the Pentagon and the Central Intelligence Agency.

Tutwiler scoffs at the idea that Baker has to be standing by Bush's side to make his influence felt. She said:

"Jim Baker doesn't play that game. He's not in a contest over visibility. That cheapens the seriousness of the situation. What he is involved in is the daily substance of U.S. policy. Every day! He did not leave it in Washington. He took it with him to Wyoming."

She pointed out that between the outbreak of the crisis and the start of his vacation, Baker had made whirlwind trips to Moscow to join with Soviet Foreign Minister Eduard Shevardnadze in an unprecedented U.S.-Soviet declaration denouncing Iraq's action, to Ankara to firm up Turkey's willingness to cut off the flow of Iraqi oil through its territory and to Brussels to consult with America's North Atlantic Treaty Organization allies about how best to deal with the crisis.

While in Wyoming, she added, he was on the phone to Eagleburger and Kimmitt "five or six times a day" and "spent much of his remaining time on calls to foreign capitals."

One of those calls -- to Shevardnadze last Thursday -- is known to have been a key factor in bringing Washington and Moscow into agreement on the resolution adopted by the U.N. Security Council on Saturday authorizing the world's navies to use military force against attempts to evade U.N. sanctions on Iraq.

In any case, the arguments about the wisdom of Baker's 10-day absence are likely to become moot quickly.

On his first day back, he quickly returned to a routine that yesterday saw him join Bush in briefing congressional leaders and then discussing the Persian Gulf situation with Omani Foreign Minister Yusef Bin Alawi and Australian Foreign Minister Gareth Evans.

As the senior State Department official said, "If there was any damage to the department's standing during his absence, the results clearly are not going to be very long lasting."