$\sigma$-Biderivations and $\sigma$-commuting maps of triangular algebras

Let $\text{A}$ be an algebra and $\sigma$ an automorphism of $\text{A}$. A linear map $d$ of $\text{A}$ is called a $\sigma$-derivation of $\text{A}$ if $d(xy) = d(x)y + \sigma(x)d(y)$,forall$x, y \in \A$\mathrm{.}Abilinearmap$D: \A \times \A \to \A$issaidtobea$\sigma$-biderivationof$\A$ifitisa$\sigma$-derivation in each component. An additive map $\mathrm{\Theta }$ of $\text{A}$ is $\sigma$-commuting if it satisfies $\mathrm{\Theta }(x)x-\sigma (x)\mathrm{\Theta }(x)=0$, for all $x\in \text{A}$. In this paper, we introduce the notions of inner and extremal $\sigma$-biderivations and of proper $\sigma$-commuting maps. One of our main results states that, under certain assumptions, every $\sigma$-biderivation of a triangular algebras is the sum of an extremal $\sigma$-biderivation and an inner $\sigma$-biderivation. Sufficient conditions are provided on a triangular algebra for all of its $\sigma$-biderivations (respectively, $\sigma$-commuting maps) to be inner (respectively, proper). A precise description of $\sigma$-commuting maps of triangular algebras is also given. A new class of automorphisms of triangular algebras is introduced and precisely described. We provide many classes of triangular algebras whose automorphisms can be precisely described.