What is the agency relationship between directors of public ltd company and the shareholders?

A corporation's board of directors are not agents of a corporation while corporate officers are. Although individual directors resemble agents in the sense that they owe a fiduciary duty of loyalty to the entity they serve, the distinguishing difference is that they are generally not subject to another's control, are elected by stockholders for set terms and are entitled to use their own business judgment in managing the corporation's affairs. See Restatement (Second) of Agency § 14C.

As noted in Restatement (Third) of Agency § 1.01 cmt. f(2), directors' powers originate as the legal consequence of their election and are not conferred or delegated by shareholders.