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All Journal CAUCHY: Jurnal Matematika Murni dan Aplikasi
Albania, Imam Nugraha
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Mathematics

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Function-Theoretic Operator Norm Inequalities: A Kosaki-type Generalization to Symmetric Probability Weights Albania, Imam Nugraha; Rosjanuardi, Rizky
CAUCHY: Jurnal Matematika Murni dan Aplikasi Vol 10, No 2 (2025): CAUCHY: JURNAL MATEMATIKA MURNI DAN APLIKASI
Publisher : Mathematics Department, Universitas Islam Negeri Maulana Malik Ibrahim Malang

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.18860/cauchy.v10i2.36905

Abstract

Operator norm comparisons play a fundamental role in matrix analysis, yet existing proofs often depend on operator means or interpolation techniques. This study provides a function-theoretic approach to operator norm inequalities. It also extends the classical two-term Heinz comparison to multi-term averages with arbitrary symmetric probability weights. Our approach translates each operator norm comparison into a scalar condition. The condition is derived from functional calculus for the left and right multiplication operators. We examine positive-definiteness and infinite divisibility through Fourier-measure representations. We also use elementary closure properties. For positive operators and any unitarily invariant norm, the two-term Heinz symmetrization is dominated by the binomial average when the exponent differs from one-half by at most one divided by twice the number of terms. For general symmetric probability weights, domination occurs exactly when the exponent lies within a specific threshold. This threshold equals the smallest positive distance from the midpoint to any index carrying nonzero weight. The proposed function-theoretic framework yields necessary and sufficient thresholds to unify the binomial and general symmetric cases.
Order Ideals on Lexicographic Direct Sum of Three Totally Ordered Abelian Groups Latifah, Dian; Gozali, Sumanang Muhtar; Rosjanuardi, Rizky; Albania, Imam Nugraha
CAUCHY: Jurnal Matematika Murni dan Aplikasi Vol 9, No 1 (2024): CAUCHY: JURNAL MATEMATIKA MURNI DAN APLIKASI
Publisher : Mathematics Department, Universitas Islam Negeri Maulana Malik Ibrahim Malang

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.18860/ca.v9i1.24116

Abstract

Order ideals play an important role in the study of abstract algebra, especially in the study of ordered groups. In this paper, we focus on the study of order ideals in lexicographic direct sums of totally ordered Abelian groups. We begin by examining the order ideals in the group of integers , and the group of real numbers  It is shown that there are no non-trivial order ideals in both groups. Next, we revisit the order ideals in the lexicographic direct sum of two totally ordered Abelian groups,  The only non-trivial order ideal of  is  Furthermore, our study extends to the lexicographic direct sum of three totally ordered Abelian groups:  and  We investigate the non-trivial order ideals in these structures. It is stated that the non-trivial order ideals of  are only  and  Furthermore, the non-trivial order ideals of  are only  and .Keywords: order ideal; lexicographic order; direct sum.
Co-Authors DIAN LATIFAH, DIAN Rizky Rosjanuardi Sumanang Muhtar Gozali