Positivity preservers forbidden to operate on diagonal blocks. (English) Zbl 1527.15037
The author considers entrywise positivity preservers, i.e., functions that operate entrywise on matrices and preserve the class of positive semidefinite matrices in all dimensions. The classical Schur product theorem in [I. Schur, J. Reine Angew. Math. 140, 1–28 (1911; JFM 42.0367.01)] implies that absolutely monotonic functions (i.e., power series with nonnegative coefficients) preserve positivity on matrices of all dimensions.
Celebrated results in [I. J. Schoenberg, Duke Math. J. 9, 96–108 (1942; Zbl 0063.06808); W. Rudin, Duke Math. J. 26, 617–622 (1959; Zbl 0092.28302)] show the converse, i.e., there are no other such functions. D. Guillot and B. Rajaratnam [Trans. Am. Math. Soc. 367, No. 1, 627–649 (2015; Zbl 1305.15075)] classified the entrywise positivity preservers in all dimensions which act only on the off-diagonal entries. These two results are at opposite ends and in both cases the preservers have to be absolutely monotonic.
In this paper, it is explained that the above mentioned results are two extreme cases among other possibilities. This is essentially done by forbidding the entrywise functions \(f\) from operating on diagonal/principal blocks. This idea makes it possible to unify the two different looking results into one and the same framework. In this process of unification, the author provides dimension-free, nonabsolutely monotonic positivity preservers when \(f\) is forbidden from acting on certain diagonal/principal blocks. This is the first time that nonabsolutely monotonic functions are found to preserve positivity in all dimensions when acting in a certain way.
Three results are presented and carefully proved in the paper according to the nature of \(f\). Concretely:
(1) When \(f\) is forbidden only from \(1\times 1\) diagonal blocks (or diagonal entries);
(2) When \(f\) is forbidden from larger diagonal/principal blocks;
(3) When \(f\) is forbidden from at least two overlapping \(2\times 2\) diagonal/principal blocks.
Celebrated results in [I. J. Schoenberg, Duke Math. J. 9, 96–108 (1942; Zbl 0063.06808); W. Rudin, Duke Math. J. 26, 617–622 (1959; Zbl 0092.28302)] show the converse, i.e., there are no other such functions. D. Guillot and B. Rajaratnam [Trans. Am. Math. Soc. 367, No. 1, 627–649 (2015; Zbl 1305.15075)] classified the entrywise positivity preservers in all dimensions which act only on the off-diagonal entries. These two results are at opposite ends and in both cases the preservers have to be absolutely monotonic.
In this paper, it is explained that the above mentioned results are two extreme cases among other possibilities. This is essentially done by forbidding the entrywise functions \(f\) from operating on diagonal/principal blocks. This idea makes it possible to unify the two different looking results into one and the same framework. In this process of unification, the author provides dimension-free, nonabsolutely monotonic positivity preservers when \(f\) is forbidden from acting on certain diagonal/principal blocks. This is the first time that nonabsolutely monotonic functions are found to preserve positivity in all dimensions when acting in a certain way.
Three results are presented and carefully proved in the paper according to the nature of \(f\). Concretely:
(1) When \(f\) is forbidden only from \(1\times 1\) diagonal blocks (or diagonal entries);
(2) When \(f\) is forbidden from larger diagonal/principal blocks;
(3) When \(f\) is forbidden from at least two overlapping \(2\times 2\) diagonal/principal blocks.
Reviewer: Ctirad Matonoha (Praha)
MSC:
| 15B48 | Positive matrices and their generalizations; cones of matrices |
| 15A86 | Linear preserver problems |
| 26A21 | Classification of real functions; Baire classification of sets and functions |
| 15A24 | Matrix equations and identities |
| 15A39 | Linear inequalities of matrices |
| 15A45 | Miscellaneous inequalities involving matrices |
| 30B10 | Power series (including lacunary series) in one complex variable |
Keywords:
entrywise positivity preservers; absolutely monotonic functions; forbidden blocks; Herz functionReferences:
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