Submultiplicative

Submultiplicative describes a property in mathematics where a product-like operation does not grow faster than the product of its factors. More precisely, a function f defined on a semigroup with operation ⋅ is called submultiplicative if f(x ⋅ y) ≤ f(x) f(y) for all x and y in the domain. The concept appears in several contexts, including sequences, norms, and arithmetic functions.

In analysis and linear algebra, a common instance is a submultiplicative norm. A norm ||·|| on an algebra or on the space of linear operators is submultiplicative if ||xy|| ≤ ||x|| · ||y|| for all x and y. This holds for standard matrix norms and, in particular, for the operator (spectral) norm and many induced norms. Submultiplicativity yields important bounds on powers, since ||T^n|| ≤ ||T||^n for any linear operator T.

For sequences, a sequence (a_n) with a_0 = 1 is called submultiplicative if a_{m+n} ≤ a_m a_n for all nonnegative integers m, n. A key consequence, known as Fekete’s lemma, is that the limit lim_{n→∞} (a_n)^{1/n} exists and equals inf_{n} (a_n)^{1/n}. This limit often serves as a growth rate or radius of convergence indicator.

Applications include bounding the growth of powers of operators, establishing spectral radius formulas, and providing a framework for submultiplicative arithmetic functions, where f(mn) ≤ f(m) f(n). Common examples include exponential sequences a_n = c^n, which are multiplicative (hence submultiplicative) with equality, and norms that satisfy ||AB|| ≤ ||A|| ||B||.