When powers of a matrix coincide with its Hadamard powers
We characterize matrices whose powers coincide with their Hadamard powers.
An irreducible semigroup of idempotents
We construct a semigroup of bounded idempotents with no nontrivial invariant closed subspace. This answers a question which was open for some time.
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 answers an open question posed by Bračič et al. [Positivity 14 (2010)], where the study of positive commutators of positive operators was initiated.
Bounds for the spectral radius of positive operators
Let be a non-zero positive vector of a Banach lattice , and let be a positive linear operator on with the spectral radius . We find some groups of assumptions on , and under which the inequalities hold. An application of our results gives simple upper and lower bounds for the spectral radius of a product of positive operators in terms of positive eigenvectors corresponding to the spectral radii of given operators. We thus extend the matrix result obtained by Johnson and Bru which...
Volterra kernel operators on Banach function spaces
On operator bands
A multiplicative semigroup of idempotent operators is called an operator band. We prove that for each K>1 there exists an irreducible operator band on the Hilbert space which is norm-bounded by K. This implies that there exists an irreducible operator band on a Banach space such that each member has operator norm equal to 1. Given a positive integer r, we introduce a notion of weak r-transitivity of a set of bounded operators on a Banach space. We construct an operator band on that is weakly...
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