Inequalities on the spectral radius and the operator norm of Hadamard products of positive operators on sequence spaces

Drnovšek, Roman; Peperko, Aljoša

doi:10.1215/17358787-3649524

Cited by 16 publications

(39 citation statements)

References 17 publications

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“…In [39], X. Zhan conjectured that, for non-negative n × n matrices A and B, These inequalities were established via a trace description of the spectral radius. Soon after, inequality (1.1) was reproved, generalized and refined in different ways by several authors ( [18], [19], [32], [33], [29], [7], [13], [30], [31]). Using the fact that the Hadamard product is a principal submatrix of the Kronecker product, R.A. Horn and F. Zhang proved in [18] the inequalities…”

Section: Introductionmentioning

confidence: 99%

Inequalities on the joint and generalized spectral and essential spectral radius of the Hadamard geometric mean of bounded sets of positive kernel operators

Peperko

2018

Linear and Multilinear Algebra

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Let Ψ and Σ be bounded sets of positive kernel operators on a Banach function space L. We prove several refinements of the known inequalitiesfor the generalized spectral radius ρ and the joint spectral radiusρ, where Ψ ( 1 2 ) • Σ ( 1 2 ) denotes the Hadamard (Schur) geometric mean of the sets Ψ and Σ. Furthermore, we prove that analogous inequalities hold also for the generalized essential spectral radius and the joint essential spectral radius in the case when L and its Banach dual L * have order continuous norms.Applying the techniques of [18], Z. Huang proved thatfor n×n non-negative matrices A 1 , A 2 , · · · , A m (see [19]). A.R. Schep was the first one to observe that the results from [11] and [27] are applicable in this context (see 2010 Mathematics Subject Classification. 15A42, 15A60, 47B65, 47B34, 47A10, 15B48.Key words and phrases. Hadamard-Schur geometric mean; Hadamard-Schur product; joint and generalized spectral radius; essential spectral radius; measure of noncompactness; positive kernel operators; non-negative matrices; bounded sets of operators.

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Section: Introductionmentioning

confidence: 99%

Inequalities on the joint and generalized spectral and essential spectral radius of the Hadamard geometric mean of bounded sets of positive kernel operators

Peperko

2018

Linear and Multilinear Algebra

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“…Similarly to [14], we can prove the following max-plus version of [14, Corollary 3.5, Theorem 3.9, Corollary 3.10]. In the proof it is useful to switch to the isomorphic max-times algebra setting by using the equality ν(A) = log µ(B), where B denotes a non-negative n × n matrix B = [e a ij ] and µ(B) denotes the largest max-times eigenvalue of B.…”

Section: 2mentioning

confidence: 99%

“…The matrix A admits several max-row partitions (max-column partitions of A T ), including the partition {(1), (3), (2, 4))}, which is also a max-column partition of B. The chosen maximal elements in the rows of A (in ascending order) are a 24 , a 33 , a 41 , a 13 , and the chosen maximal elements in the columns of B are b 43 , b 11 , b 22 , b 14 :…”

Section: 4mentioning

confidence: 99%

Polynomial convolutions in max-plus algebra

Rosenmann

Lehner

Peperko

2019

Linear Algebra and its Applications

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Recently, in a work that grew out of their exploration of interlacing polynomials, Marcus, Spielman and Srivastava [21] and Marcus [20] studied certain combinatorial polynomial convolutions. These convolutions preserve real-rootedness and capture expectations of characteristic polynomials of unitarily invariant random matrices, thus providing a link to free probability. We explore analogues of these types of convolutions in the setting of max-plus algebra. In this setting the max-permanent replaces the determinant, the maximum is the analogue of the expected value and real-rootedness is replaced by full canonical form. Our results resemble those of Marcus et al., however, in contrast to the classical setting we obtain an exact and simple description of all roots of the convolution of p(x) and q(x) in terms of the roots of p(x) and q(x).

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“…Note also that a positive cone Z = Y + of each Banach lattice Y or, in particular, of each Banach function space (see e.g. [1], [3], [37], [32], [13], [33] and the references cited there) satisfies these properties. For the theory of cones, wedges, linear and non-linear operators on cones and wedges, Banach ordered spaces, Banach function spaces, vector and Banach lattices and applications e.g.…”

Section: Introductionmentioning

confidence: 99%