Abstract Harmonic Analysis

Approximation of unbounded operators by bounded operators and related extremal problems

“…The properties of amenable semigroups and invariant means on them can be found, for example, in [120], [121]; in particular, it is known that a commutative semigroup is amenable. We shall say that problem (6.6) is invariant, or more precisely, (Θ, T)-invariant, if, for any (x, y) G W and s G 5, there exists an element y' £Y with the properties y = r(s)y', (9(s) and in problem (6.6) there exists an extremal operator that is (Τ, Θ)-invariant.…”

Arithmetic of semigroups of series in multiplicative systems

“…If two or more terms of this sort were present, then the decomposition of f would contain at least two terms from W k by virtue of step 2, which contradicts the condition. 7. The decomposition of f 1 contains only one term from W k < .…”

On left thickness of subsets in semigroups

“…If S is only left cancellative, then (4) implies (1) provided F -B(S) (see [8,Theorem 17.15 (iii) implies (i)]). Theorem 3.6.…”

“…Using Theorem 3.1 and the arguments used in Granirer [6, Lemma 3, p. 99] we obtain the implications (1) => (2) => (3) => (4). For (3) => (1), we use Theorem 3.1 again and the arguments in Hewitt and Ross [8,Theorem 17.15,p. 235].…”