Growth of semigroups in discrete and continuous time

Abstract

We show that the growth rates of solutions of the abstract differential equationsẋ(t) = Ax(t),ẋ(t) = A −1 x(t), and the difference equation x d (n + 1) = (A + I)(A − I) −1 x d (n) are closely related. Assuming that A generates an exponentially stable semigroup, we show that on a general Banach space the lowest growth rate of the semigroup (e A −1 t) t≥0 is O(4 √ t), and for ((A + I)(A − I) −1) n it is O(4 √ n). The similarity in growth holds for all Banach spaces. In particular, for Hilbert spaces the best estimates are O(log(t)) and O(log(n)), respectively. Furthermore, we give conditions on A such that the growth rate of ((A + I)(A − I) −1) n is O(1), i.e., the operator is power bounded.

The Cayley transform of the generator of a polynomially stable $$C_0$$-semigroup

“…whether the corresponding Cayley transform is power bounded for every generator of a uniformly bounded C 0 -semigroup, as mentioned in Section 5.5 of [5]. However, some sufficient conditions for Cayley transforms to be power bounded have been obtained; see, e.g., [12,14,15,17]. In particular, it is well known that A generates a C 0 -semigroup of contractions on a Hilbert space if and only if the corresponding Cayley transform is a contraction [21,Theorem III.8.1].…”

“…In the finite-dimensional case, a matrix and its Cayley transform share the same stability properties, but this does not hold in the infinite-dimensional case. In fact, in the Banach space setting, the Cayley transform of the generator of even an exponentially stable C 0 -semigroup may not be power bounded [15, Lemma 2.1]. For the case of Hilbert spaces, it is still unknown…”

The Cayley transform of the generator of a polynomially stable $C_0$-semigroup

“…In the finitedimensional case, a matrix and its Cayley transform share the same stability properties, but this does not hold in the infinite-dimensional case. In fact, in the Banach space setting, the Cayley transform of the generator of even an exponentially stable C 0 -semigroup may not be power bounded [15,Lemma 2.1]. For the case of Hilbert spaces, it is still unknown whether the corresponding Cayley transform is power bounded for every generator of a uniformly bounded C 0 -semigroup, as mentioned in Section 5.5 of [5].…”

“…For the case of Hilbert spaces, it is still unknown whether the corresponding Cayley transform is power bounded for every generator of a uniformly bounded C 0 -semigroup, as mentioned in Section 5.5 of [5]. However, some sufficient conditions for Cayley transforms to be power bounded are obtained; see, e.g., [12,14,15,17]. In particular, it is well known that A generates a C 0semigroup of contractions on a Hilbert space if and only if the corresponding Cayley transform is a contraction [21,Theorem III.8.1].…”

Stability analysis in continuous and discrete time

“…Furthermore, if A and A −1 generate a bounded semigroup, the powers of the cogenerator is bounded as well. Chapter 7 is based on [19].…”