Linear preservers of polynomial numerical hulls of matrices

The final publication is available at Elsevier via https://doi.org/10.1016/j.laa.2019.03.029. © 2019. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/Let Mn be the algebra of all n × n complex matrices, 1 ≤ k ≤ n − 1 be an integer, and φ : Mn −→ Mn be a linear operator. In this paper, it is shown that φ preserves the polynomial numerical hull of order k if and only if there exists a unitary matrix U ∈ Mn such that either φ(A)=U∗AU forallA∈Mn,orφ(A)=U∗AtU forallA∈Mn.Research supported in part by Shahid Bahonar University of Kerman, Kerman, Iran. Research supported in part by NSERC (Canada)

Reducibility of operator semigroups and values of vector states

This is a post-peer-review, pre-copyedit version of an article published in Semigroup Forum. The final authenticated version is available online at: https://doi.org/10.1007/s00233-017-9872-7Let S be a multiplicative semigroup of bounded linear operators on a complex Hilbert space H, and let Ω be the range of a vector state on S so that Ω = {⟨Sξ, ξ⟩ : S ∈ S} for some fixed unit vector ξ ∈ H. We study the structure of sets Ω of cardinality two coming from irreducible semigroups S. This leads us to sufficient conditions for reducibility and, in some cases, for the existence of common fixed points for S. This is made possible by a thorough investigation of the structure of maximal families F of unit vectors in H with the property that there exists a fixed constant ρ ∈ C for which ⟨x, y⟩ = ρ for all distinct pairs x and y in F.Research supported in part by NSERC (Canada)

A spatial version of Wedderburn’s Principal Theorem

This is an Accepted Manuscript of an article published by Taylor & Francis in 'Linear and Multilinear Algebra' on 7/2014, available online: http://www.tandfonline.com/10.1080/03081087.2014.925452.In this article we verify that ‘Wedderburn’s Principal Theorem’ has a particularly pleasant spatial implementation in the case of cleft subalgebras of the algebra of all linear transformations on a finite-dimensional vector space. Once such a subalgebra A is represented by block upper triangular matrices with respect to a maximal chain of its invariant subspaces, after an application of a block upper triangular similarity, the resulting algebra is a linear direct sum of an algebra of block-diagonal matrices and an algebra of strictly block upper triangular matrices (i.e. the radical), while the block-diagonal matrices involved have a very nice structure. We apply this result to demonstrate that, when the underlying field is algebraically closed, and (Rad(A))μ(A)−1 ≠ {0} the algebra is unicellular, i.e. the lattice of all invariant subspaces of A is totally ordered by inclusion. The quantity μ(A) stands for the length of (every) maximal chain of non-zero invariant subspaces of A.The first author was supported by the Colby College Natural Science Division Grant. The second, third and fourth authors acknowledge the support of NSERC Canada

Ranges of vector states on irreducible operator semigroups

This is a post-peer-review, pre-copyedit version of an article published in Semigroup Forum. The final authenticated version is available online at: https://doi.org/10.1007/s00233-015-9772-7Let be a linear functional of rank one acting on an irreducible semigroup S of operators on a finite- or infinite-dimensional Hilbert space. It is a well-known and simple fact that the range of cannot be a singleton. We start a study of possible finite ranges for such functionals. In particular, we prove that in certain cases, the existence of a single such functional with a two-element range yields valuable information on the structure of S.Natural Sciences and Engineering Research Counci

Mean ergodicity and spectrum of the Cesàro operator on weighted c0 spaces

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