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It is a consequence of the Morse-Bott Lemma on Banach spaces that a smoothMorse-Bott function on an open neighborhood of a critical point in a Banachspace obeys a Lojasiewicz gradient inequality with the optimal exponent onehalf. In this article we prove converses for analytic functions on Banachspaces: If the Lojasiewicz exponent of an analytic function is equal to onehalf at a critical point, then the function is Morse-Bott and thus its criticalset nearby is an analytic submanifold. The main ingredients in our proofs arethe Lojasiewicz gradient inequality for an analytic function on afinite-dimensional vector space and the Morse Lemma for functions on Banachspaces with degenerate critical points that generalize previous versions in theliterature, and which we also use to give streamlined proofs of theLojasiewicz-Simon gradient inequalities for analytic functions on Banachspaces.

On the Morse-Bott property of analytic functions on Banach spaces with Lojasiewicz exponent one half