Let B be a unital commutative semi-simple Banach algebra. We study endomorphisms of B which are simultaneously Riesz operators. Clearly compact and power compact endomorphisms are Riesz. Several general theorems about Riesz endomorphisms are proved, and these results are then applied to the question of when Riesz endomorphisms of certain algebras are necessarily power compact.

We extend the notion of the canonical extension of automorphisms of type III factors to the case of endomorphisms with finite statistical dimensions. Following the automorphism case, we introduce two notions for endomorphisms of type III factors: modular endomorphisms and Connes-Takesaki modules. Several applications to compact groups of automorphisms and subfactors of type III factors are give...

We give an algebro-geometric approach towards the dynamics of automorphisms/endomorphisms of projective varieties or compact Kähler manifolds, try to determine the building blocks of automorphisms /endomorphisms, and show the relation between the dynamics of automorphisms/endomorphisms and the geometry of the underlying manifolds.

We present a formulation of the SRB (Sinai-Ruelle-Bowen) property for invariant measures of C2 endomorphisms (maybe non-invertible and with singularities) of a compact manifold via their inverse limit spaces, and prove that this property is necessary and sufficient for Pesin’s entropy formula. This result is a non-invertible endomorphisms version of a result of Ledrappier, Strelcyn and Young.

Let D be the open unit disc and, as usual, let H(D) be the algebra of bounded analytic functions on D with ‖f‖ = supz∈D |f(z)|. With pointwise addition and multiplication, H(D) is a well known uniform algebra. In this note we characterize the compact endomorphisms of H(D) and determine their spectra. We show that although not every endomorphism T of H(D) has the form T (f)(z) = f(φ(z)) for some...

For a compact space X we consider extending endomorphisms of the algebra C(X) to be endomorphisms of Arens-Hoffman and Cole extensions of C(X). Given a non-linear, monic polynomial p ∈ C(X)[t], with C(X)[t]/pC(X)[t] semi-simple, we show that if an endomorphism of C(X) extends to the Arens-Hoffman extension with respect to p then it also extends to the simple Cole extension with respect to p. We...

We first show that a bounded linear operator $ T $ on a real Banach space $ E $ is quasicompact (Riesz, respectively) if and only if $T': E_{mathbb{C}}longrightarrow E_{mathbb{C}}$ is quasicompact  (Riesz, respectively), where the complex Banach space $E_{mathbb{C}}$ is a suitable complexification of $E$ and $T'$ is the complex linear operator on $E_{mathbb{C}}$ associated with $T$. Next, we pr...