Once More on Positive Commutators

Let A and B be bounded operators on a Banach lattice E such that the commutator C=AB-BA and the product BA are positive operators. If the product AB is a power-compact operator, then C is a quasi-nilpotent operator having a triangularizing chain of closed ideals of E. This theorem answers an open question posed in a paper by Bra\v{c}i\v{c}, Drnov\v{s}ek, Farforovskaya, Rabkin and Zem\'{a}nek.Comment: The paper will appear in Studia Mathematic

On order automorphisms of the effect algebra

We give short proofs of two \v{S}emrl's descriptions of order automorphisms of the effect algebra. This sheds new light on both formulas that look quite complicated. Our proofs rely on Moln\'{a}r's characterization of order automorphisms of the cone of all positive operators.Comment: 7 page

When powers of a matrix coincide with its Hadamard powers

We characterize matrices whose powers coincide with their Hadamard powers

An infinite-dimensional generalization of Zenger's lemma

We prove an infinite-dimensional generalization of Zenger's lemma that was used in the proof of the fact that the convex hull of the point spectrum of a linear operator is contained in its numerical range. Two relevant examples are given, and possible application in the Arrow-Debreu model is also discussed.Comment: The paper will appear in J. Math. Anal. App