Remarks on submultiplicative functions
2019
artykuł naukowy
angielski
- submultiplicativitive function
- seminorm
- Orlicz function
- square property
EN The real functions satisfying the inequality Ф (uv) ≤ KФ (u) Ф (v) for some positive K which occur among others in [5], [3], [4], and referred there as submultiplicative, are discussed. A simplifying remark that Ф satisfies this inequality iff KФ is submultiplicative in the standard sense, is done. It is shown that, under general conditions, the standard submultiplicativity of Ф and the inequality Ф (u) Ф (1/u) ≤ 1 imply that Ф must be multiplicative. Applying a result of Bhatt [1], we observe that if p is a nontrivial seminorm on a Banach algebra X such that the set { p(x2)/[p(x)]2 : ∈ X, p (x) ≠ 0} is a singleton {λ}, then s = λp is a submultiplicative seminorm on X.
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