On the inverse of the generator of a bounded C 0-semigroup

Abstract

Let A be the generator of a uniformly bounded C 0 -semigroup in a Banach space B , and let A have a densely defined inverse A −1 . We present sufficient conditions on the resolvent (A−λI) −1 , Re λ > 0, under which A −1 is also the generator of a uniformly bounded C 0 -semigroup.Key words: uniformly bounded C 0 -semigroup, inverse of the generator, Banach space, Carleson embedding theorem. 1.Let B be a Banach space with norm · , and let E = E(B) be the set of densely defined closed linear operators in B. We denote the domain of an operator A ∈ E by D(A), the spectrum of A by σ(A), the identity operator by I , and the resolvent ofthe set of operators whose spectra lie in the half-plane Re λ 0. If x ∈ B and y ∈ B * , where B * is the dual Banach space of B with norm · * , then (x, y) stands for the value of the functional y on x.Let us recall some facts of operator semigroup theory. (See [1, Chap. 9] and [2,In what follows, an operator C 0 -semigroup is called simply a semigroup for brevity. The generator A ∈ E of T (t) is defined by the formulaand we write T (t) = e tA , t 0. A semigroup T (t) is said to be uniformly bounded if there exists a constant M 1 such that T (t) M , t 0. By G = G (B) we denote the set of generators of uniformly bounded semigroups in B.Let H (θ) [1, Chap. 9, §1] be the set of generators of semigroups holomorphic (analytic) in the sector Σ θ = {z = re iφ : 0 < r < ∞, |φ| < θ}, θ ∈ (0, π/2], and uniformly bounded in an arbitrary sector Σ θ 0 , θ 0 ∈ (0, θ). Next, let H 0 be the set of operators of the form A = A 0 + βI , where β 0 and A 0 ∈ H (θ) for some θ ∈ (0, π/2]. The set H 0 consists of generators of semigroups holomorphic in some sector Σ θ and, for β > 0, quasibounded in the terminology in [1, Chap. 9, §1.4].If A ∈ H (θ) has an inverse A −1 ∈ E (i.e., ker A = {0} and the range of A is dense in B), then A −1 also belongs to H (θ) [3,4]; in particular, A −1 ∈ G . Indeed [1, 2], A ∈ H (θ) if and only if the resolvent R(A, λ) exists for λ ∈ Σ π/2+θ and satisfies the estimateTherefore, if A ∈ H (θ) and A −1 ∈ E , thenfor λ ∈ Σ π/2+θ , and relations (1) and (2) readily imply the estimate R(A −1 , λ) (1 + c φ )|λ| −1 , λ ∈ Σ π/2+φ , for all φ ∈ (0, θ). We see that A −1 ∈ H (θ).In this paper, we extend the implication

Optimal estimates for the semigroup generated by the classical Volterra operator on L p -spaces

“…So, the Theorem provides the example of a nilpotent C 0 -semigroup It can also be shown that in this case (see [8,9]) the semigroup (e tA −1 ) t≥0 has the following integral representation:…”

“…On the other hand, it was shown in [8,9,11,12] that there exists a Banach space X and an injective linear operator on X with dense range generating a uniformly bounded C 0 -semigroup whose inverse does not generate a C 0 -semigroup. In [9] this was proved for X = l p , p ∈ (1, 2) ∪ (2, ∞).…”

The growth of a C 0-semigroup characterised by its cogenerator

“…Recently, Piskarev, Zwart [16] proved that this result (and even the estimate V n (V − I) ≤ c(1 + √ n)) is optimal. On the other side, Gomilko, Zwart, Tomilov [9] showed that on every l p -space, 1 < p < ∞, p = 2 there exists a contraction V with 1 / ∈ P σ (V) which is not the cogenerator of a C 0 -semigroup. An analogous example on the space c 0 follows from Komatsu [13, pp.…”

“…Note that although the proofs in this section are easy, the presented method seems to be promising. We also discuss the connection with the inverse of a generator and growth of the corresponding semigroup (see Zwart [21,22] Gomilko, Zwart [10], Gomilko, Zwart, Tomilov [9], de Laubenfels [14] for this aspect).…”

Is A-1an infinitesimal generator?

“…Furthermore, for every p ∈ [1, 3 ) ∪ (4, ∞) he constructed a bounded generator of a contraction semigroup on ℓ p , such that the inverse does not generate a C 0 -semigroup, see [8], thus proving that the question by deLaubenfels should be answered negatively on reflexive Banach spaces and for bounded generators. Furthermore, in [7] he gives some sufficient conditions under which A −1 is also a infinitesimal generator of a bounded C 0 -semigroup.…”