Biderivations of triangular algebras

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Let A be a triangular algebra. A bilinear map φ : A × A → A is called a biderivation if it is a derivation with respect to both arguments. In this paper we define the concept of an extremal biderivation, and prove that under certain conditions a biderivation of a triangular algebra A is a sum of an extremal and an inner biderivation. The main result is then applied to (block) upper triangular matrix algebras and nest algebras. We also consider the question when a derivation of a triangular algebra is an inner derivation.