Soviet Foreign Minister Andrei Gromyko expressed the hope today that the appointment of George P. Shultz as secretary of state would lead to an improvement in Soviet-American relations.

In a congratulatory telegram to Shultz, Gromyko said, "I would like to hope that your activity in this responsible post will improve Soviet-American relations in the interest of the peoples of our countries, the interest of peace."

The wording of the telegram, published by the official news agency Tass, appeared to signal Soviet readiness for a serious dialogue with Washington and cautious hopes that Shultz may be a moderating influence on what is seen here as an implacably hostile Reagan administration.

Another signal to this effect came from Georgy Arbatov, the Kremlin's senior expert on the United States, who expressed the hope that unspecified factors "would revert American policy not only to the understanding of existing differences but also to the very serious, vitally important common interests" that require "a general improvement" in bilateral relations.

The Soviets have been perplexed by the resignation of Alexander M. Haig Jr. and they privately insist that they have not reached an assessment of what led to it.

Shultz is known here, having had dealings with senior Soviet officials as treasury secretary in the Nixon and Ford administrations. Despite his appointment, the Soviets privately express concern about the overall foreign policy course of the Reagan administration thus far.

Arbatov offered a sharply negative analysis of U.S. policies that paralleled points articulated by the defense minister, Marshal Dmitri Ustinov, earlier this week.

Arbatov blamed Reagan's policy of "confrontation" for creating an "international situation that favors the outbreak and the deepening of conflicts."

Arbatov suggested that an improvement in bilateral relations could lead to new arms control agreements in Geneva and also help reduce the prospects of regional conflicts. A resumption of a serious dialogue, he said, could create a better atmosphere for the solution of the Middle East conflict.