ON THE FAVARD CLASSES OF SEMIGROUPS ASSOCIATED WITH PSEUDO-RESOLVENTS

WOJCIECH CHOJNACKI AND JAN KISYŃSKI

Abstract. A pseudo-resolvent on a Banach space, indexed by positive numbers and tempered at infinity, gives rise to a bounded strongly continuous one-parameter semigroup S on a closed subspace of the ambient Banach space. We prove that the range space of the pseudo-resolvent contains the domain of the generator of S, and is contained in the Favard class of S, which consists of all uniformly Lipschitz vectors for S. We explore when some or all of these three spaces coincide.

1. Introduction

Let A be a Banach algebra over the field ℂ of complex numbers. A family {r_{λ}}_{λ∈U} of elements of A, indexed by a subset U ⊂ ℂ, is called a pseudo-resolvent in A if the following Hilbert equation is satisfied:

(1)  r_{λ} − r_{μ} = (μ − λ)r_{λ}r_{μ}  (λ, μ ∈ U).

Denote by ℝ the set of all real numbers, by ℝ_{+} the set of all nonnegative numbers, and by ℝ_{+}^{•} the set of all positive numbers. Given a pseudo-resolvent r = {r_{λ}}_{λ∈ℝ_{+}^{•}} in A, let

c_{r} = sup{λ^{n}∥r_{λ}^{n}∥ | n ∈ ℕ, λ ∈ ℝ_{+}^{•}}.

A pseudo-resolvent r indexed by ℝ_{+}^{•} and such that c_{r} is finite will be said to be tempered at infinity. Note, immediately, that if r is tempered at infinity, then lim_{μ→∞} r_{μ} = 0, and so, in view of (1),

(2)  r_{λ} = lim_{μ→∞} μr_{μ}r_{λ}

for each λ ∈ ℝ_{+}^{•}.

Hereafter we assume that all vector spaces and algebras considered are complex.

Let E be a Banach space, and let L(E) be the Banach algebra of all continuous linear operators in E equipped with the usual norm derived from that of E. Let A be a closed linear operator on E with domain D_{A} and range R_{A}. The resolvent set ρ_{A} of A is the set of λ ∈ ℂ for which λ − A is one-to-one with range equal to E. For each λ ∈ ρ_{A}, setting

R_{λ}(A) = (λ − A)^{−1}

defines the resolvent operator of A corresponding to λ; it is bounded by the closed graph theorem. The resolvent set ρ_{A} is open, and the family R(A) = {R_{λ}(A)}_{λ∈ρ_{A}} is a special instance of pseudo-resolvent, called the resolvent of A.

Let R = {R_{λ}}_{λ∈U} with U ⊂ ℂ be a pseudo-resolvent in L(E). It immediately follows from the Hilbert equation (1) that all the R_{λ} have a common null space and a common range. These spaces are called the null space and range space of R, and are denoted N_{R} and R_{R}, respectively. If N_{R} is zero, then R is a resolvent of a closed operator whose domain coincides with R_{R}.

1991 Mathematics Subject Classification. Primary 47D03; Secondary 47A10, 46J25.

Key words and phrases. Favard class, one-parameter semigroup, pseudo-resolvent, uniformly Lipschitz vector.

Let S = {S_{t}}_{t∈ℝ_{+}} be a bounded strongly continuous one-parameter semigroup of operators in L(E). The boundedness of S means, of course, that

m_{S} := sup_{t∈ℝ_{+}} ∥S_{t}∥ < +∞,

and strong continuity is synonymous with the continuity in the strong operator topology on L(E). Let A be the (infinitesimal) generator of S. The operator A is closed, has dense domain, and its resolvent set contains ℝ_{+}^{•}. The resolvent R(A) = {R_{λ}(A)}_{λ∈ℝ_{+}^{•}} is tempered at infinity and c_{R(A)} = m_{S}. For each λ ∈ ℝ_{+}^{•}, R_{λ}(A) can explicitly be expressed in terms of S as follows:

(3)  R_{λ}(A)x = ∫_{ℝ_{+}} e^{−λt}S_{t}x dt  (x ∈ E).

The Favard class of S, Fav(S), is the subspace of E defined by

Fav(S) = {x ∈ E | lim sup_{t→+0} t^{−1}∥S_{t}x − x∥ < +∞}

(see [3, p. 84]). In this definition, the upper limit can safely be replaced by the lower limit (cf. [3, proof of Theorem 2.1.2(a)]). As a consequence of S being bounded, it is easily seen that Fav(S) coincides with the space of all uniformly Lipschitz vectors for S; that is

Fav(S) = {x ∈ E | sup_{t∈ℝ_{+}^{•}} t^{−1}∥S_{t}x − x∥ < +∞}

(see [5, Def. 3.17]). A basic, simple result is that

(4)  D_{A} ⊂ Fav(S)

(cf. [3, Thm. 2.1.2], [5, p. 67]).

Suppose now that R = {R_{λ}}_{λ∈ℝ_{+}^{•}}, a pseudo-resolvent in L(E), is tempered at infinity. It is an immediate consequence of (1) that, for every λ ∈ ℝ_{+}^{•}, R_{R} is invariant for R_{λ}. Clearly, the closure \overline{R_{R}} of R_{R} is also invariant for all the R_{λ}. For each λ ∈ ℝ_{+}^{•}, define

r(R)_{λ} = R_{λ} ↾ \overline{R_{R}},

where the symbol S ↾ D denotes the restriction of the operator S to D. Obviously, r(R) = \{r(R)_{λ}\}_{λ∈ℝ_{+}^{•}} is a pseudo-resolvent in L(\overline{R_{R}}). Observe that, on account of (2),

x = lim_{μ→∞} μR_{μ}x

for each x ∈ R_{R}. As \{μ∥R_{μ}∥\}_{μ∈ℝ_{+}^{•}} is bounded, this equality extends over all x ∈ \overline{R_{R}}. Hence we immediately find that N_{r(R)} is zero and that R_{r(R)} is dense in \overline{R_{R}}. Consequently, r(R) is a resolvent of a closed densely defined operator A on \overline{R_{R}}. Of course, D_{A} = R_{r(R)}. Taking into account that c_{r(R)} ≤ c_{R} and applying the Hille–Yosida Theorem (cf. [3, Thm. 1.3.6], [8, Thm. 2.21], [12, p. 358]), we see that A is the generator of a bounded strongly continuous semigroup S = \{S_{t}\}_{t∈ℝ_{+}} on \overline{R_{R}}, satisfying m_{S} = c_{r(R)}. This semigroup will be termed the semigroup associated with R. Semigroups associated with pseudo-resolvents tempered at infinity arise naturally in Markov processes theory (cf. [9, Chap. XII, §5, p. 290–302], [15]).

For a semigroup S associated with a pseudo-resolvent R, inclusion (4) can be restated as

R_{r(R)} ⊂ Fav(S).

It emerges that, in view of the special nature of S, the above relation can be refined. In this note, we prove the following:

Theorem 1. Let E be a Banach space, let R = {R_{λ}}_{λ∈ℝ_{+}^{•}} be a pseudo-resolvent in L(E), tempered at infinity, and let S be the semigroup associated with R. Then

(5)  R_{r(R)} ⊂ R_{R} ⊂ Fav(S).

Observe that, of the two inclusions above, the one on the left-hand side is trivial, so only the inclusion on the right-hand side needs to be verified.

In addition, we tackle the problem of determining when some or both of the inclusions in (5)can be replaced by equalities. We prove that for dual pseudo-resolvents (on dual Banach spaces) the right-hand side inclusion reduces to equality. As a complementary result, a class of dual pseudo-resolvents is revealed for which the left-hand side inclusion is strict. It is also shown that, for pseudo-resolvents on reflexive Banach spaces, both inclusions become equalities. Finally, two examples are given to demonstrate that, in general, both inclusions in (5) can be simultaneously strict.

Interestingly, A. Bobrowski has recently given a necessary and sufficient condition for the left-hand side inclusion in (5) to become equality (cf. [2, Prop. 2.2]).

2. Proof of the main result

This section is devoted to the proof of Theorem 1. We start by presenting a preliminary material.

Denote by λ_{ℝ_{+}} the Lebesgue measure on ℝ_{+}. As is customary, abbreviate dλ_{ℝ_{+}}(t) to dt. Let L^{1}(ℝ_{+}) be the space of equivalence classes (under equality λ_{ℝ_{+}}-almost everywhere) of complex-valued Lebesgue integrable functions f on ℝ_{+}. With the addition and scalar multiplication derived from the pointwise addition and scalar multiplication of the functions, and with the norm given by

∥f∥_{1} = ∫_{ℝ_{+}} |f(t)| dt

(where the same symbol f is used to denote both a function and its equivalence class), L^{1}(ℝ_{+}) is a complex Banach space. With convolution

(f ∗ g)(t) = ∫_{0}^{t} f(t − s)g(s) ds (λ_{ℝ_{+}}-a. e. t ∈ ℝ_{+})

as the product, it becomes a complex Banach algebra. For each λ ∈ ℝ, denote by ϵ_{λ} the function

ϵ_{λ}(t) = e^{λt} (t ∈ ℝ_{+}).

It is directly verified that the family ϵ = {ϵ_{−λ}}_{λ∈ℝ_{+}^{•}} is a pseudo-resolvent in L^{1}(ℝ_{+}) satisfying c_{ϵ} = 1. This special pseudo-resolvent enters critically the following result (cf. [13]; see also [1, 4]):

Theorem 2. Let A be a Banach algebra and let {r_{λ}}_{λ∈ℝ_{+}^{•}} be a pseudo-resolvent in A. Then r is tempered at infinity if and only if there exists a continuous Banach algebra homomorphism H: L^{1}(ℝ_{+}) → A such that H(ϵ_{−λ}) = r_{λ} for each λ ∈ ℝ_{+}^{•}. Furthermore, if a continuous homomorphism H: L^{1}(ℝ_{+}) → A satisfying H(ϵ_{−λ}) = r_{λ} for each λ ∈ ℝ_{+}^{•} exists, then it is unique and ∥H∥ = c_{r}.

The above theorem is indicative of the role played by operator-valued homomorphisms of L^{1}(ℝ_{+}) in studying pseudo-resolvents tempered at infinity. This role is greatly reinforced by the algebraic version of the Hille–Yosida Theorem, which we present next (see [13] for a proof).

For a Banach algebra homomorphism H: L^{1}(ℝ_{+}) → L(E), where E is a Banach space, denote by R_{H} the subspace of E defined as

R_{H} = {x ∈ E | x = H(f)y for some f ∈ L^{1}(ℝ_{+}) and y ∈ E}.

For each t ∈ ℝ_{+}, let T_{t}^{+} : L^{1}(ℝ_{+}) → L^{1}(ℝ_{+}) be the forward shift by t given by

(T_{t}^{+}f)(x) = 1_{[t,+∞)}(x)f(x − t) (f ∈ L^{1}(ℝ_{+}), λ_{ℝ_{+}}-a. e. x ∈ ℝ_{+}),

where, as usual, the symbol 1_{A} denotes the characteristic function of the set A. Clearly, the family T^{+} = {T_{t}^{+}}_{t∈ℝ_{+}} is a strongly continuous semigroup of linear contractions on L^{1}(ℝ_{+}).

Theorem 3. Let E be a Banach space. If H: L^{1}(ℝ_{+}) → L(E) is a Banach algebra homomorphism, then there exists a unique strongly continuous semigroup S = {S_{t}}_{t∈ℝ_{+}} on R_{H} such that, for each t ∈ ℝ_{+}, each f ∈ L^{1}(ℝ_{+}), and each x ∈ E,

(6)  S_{t}H(f)x = H(T_{t}^{+}f)x.

The semigroup S is bounded and m_{S} = ∥H∥.

We now draw out a link between Theorems 2 and 3. Let E be a Banach space and let R = {R_{λ}}_{λ∈ℝ_{+}^{•}} be a pseudo-resolvent in L(E), tempered at infinity. By Theorem 2, there is a unique Banach algebra homomorphism H: L^{1}(ℝ_{+}) → L(E) such that R_{λ} = H(ϵ_{−λ}) for each λ ∈ ℝ_{+}^{•}. It is clear that R_{R} ⊂ R_{H}. A consequence of the linear span of \{ϵ_{−λ} | λ ∈ ℝ_{+}^{•}\} being dense in L^{1}(ℝ_{+}) is the inclusion R_{H} ⊂ \overline{R_{R}}. It turns out that R_{H} is closed, so in fact

(7)  R_{H} = \overline{R_{R}}.

The closedness of R_{H} is an immediate consequence of two results:

• the simple observation that L^{1}(ℝ_{+}) possesses a bounded approximate identity; one such approximate identity is the family {λϵ_{−λ}}_{λ∈ℝ_{+}^{•}};

• a deep theorem found independently by E. Hewitt [11], P. C. Curtis and A. Figá-Talamanca [7], and S. L. Gulick, T. S. Liu and A. C. M. van Rooij [10], which generalises the so-called factorisation theorem of P. J. Cohen [6]; it states that R_{H} is closed for any continuous homomorphism H: B → L(E), where B is a Banach algebra with a bounded approximate identity, and E is a Banach space.

By virtue of Theorem 3, there is a unique semigroup S on R_{H} for which (6) holds. We claim that S coincides with the semigroup associated with R. Indeed, by (7), S can viewed as a semigroup on \overline{R_{R}}. Observe that

(8)  ∫_{ℝ_{+}} e^{−λt}T_{t}^{+}f dt = ϵ_{−λ} ∗ f  (f ∈ L^{1}(ℝ_{+})).

Now, if x ∈ R_{H}, then there exist f ∈ L^{1}(ℝ_{+}) and y ∈ E such that x = H(f)y. In view of (8) and the boundedness of S, we have, for each λ ∈ ℝ_{+}^{•},

\begin{aligned}∫_{ℝ_{+}} e^{−λt}S_{t}x dt & = ∫_{ℝ_{+}} e^{−λt}H(T_{t}^{+}f)y dt = H (∫_{ℝ_{+}} e^{−λt}T_{t}^{+}f dt) y \\ & = H(ϵ_{−λ} ∗ f)y = H(ϵ_{−λ})H(f)y = R_{λ}x. \end{aligned} ; (9)

Let A be the generator of S. Comparison of (3) and (9) shows that R_{λ} = R_{λ}(A) for each λ ∈ ℝ_{+}^{•}. Since every operator with non-void resolvent set is uniquely determined by any of its resolvent operators, it is now clear that A coincides with the generator of the semigroup associated with R. This establishes the claim.

With these preliminaries in place, we are now in position to prove Theorem 1.

Proof of Theorem 1. As already noted, we only need to establish the right-hand side inclusion in (5). Let H: L^{1}(ℝ_{+}) → L(E) be a continuous homomorphism H: L^{1}(ℝ_{+}) → L(E) such that H(ϵ_{−λ}) = R_{λ} for each λ ∈ ℝ_{+}^{•}; the existence of H is guaranteed by Theorem 2. Fix λ ∈ ℝ_{+}^{•} arbitrarily. Given x ∈ R_{R}, select y ∈ E so that x = R_{λ}y. In view of the remarks above, the associated semigroup S satisfies (6). Thus, for each t ∈ ℝ_{+},

(10)  S_{t}x = S_{t}H(ϵ_{−λ})y = H(T_{t}^{+}ϵ_{−λ})y.

One verifies at once that

T_{t}^{+}ϵ_{−λ} = ϵ_{−λ} + λϵ_{−λ} ∗ 1_{(0,t]} − 1_{(0,t]}.

It follows that

∥T_{t}^{+}ϵ_{−λ} − ϵ_{−λ}∥ ≤ (λ∥ϵ_{−λ}∥ + 1) ∥1_{(0,t]}∥ = 2t,

whence, by (10),

sup_{t∈ℝ_{+}^{•}} t^{−1}∥S_{t}x − x∥ ≤ 2 ∥H∥ ∥y∥.

Thus x ∈ Fav S, which establishes the desired inclusion. □

3. The case of dual pseudo-resolvents

Let E be a Banach space and let E′ be its dual space. For an operator T ∈ L(E), let T′ denote its dual operator in L(E′). The same notation applies when T is a densely defined operator on E. Recall that if T is a densely defined operator on E, then T′ is closed but not necessarily densely defined on E′—the space D_{T′} is merely weakly^{∗} dense in E′.

Let R = {R_{λ}}_{λ∈ℝ_{+}^{•}} be a pseudo-resolvent in L(E). The family R′ = {R′_{λ}}_{λ∈ℝ_{+}^{•}} is a pseudo-resolvent in L(E′) called the dual pseudo-resolvent of R. Since c_{R′} = c_{R}, we see that if R is tempered at infinity, then so too is R′. Assume henceforth that R is tempered at infinity. Denote by S the semigroup associated with R, and by S^{+} the semigroup associated with R′. Let A and A^{+} denote the corresponding generators. In this section, we elaborate on the inclusions in (5) for the Favard class of S^{+}.

Theorem 4. Under the assumptions as above, we have R_{R′} = Fav(S^{+}).

In order to prove the theorem, we need an auxiliary result.

Proposition 1. Let F be a Banach space, and let T = {T_{t}}_{t∈ℝ_{+}} be a bounded strongly continuous semigroup on F with generator B. If x ∈ Fav(T), then there exists a sequence {x_{n}}_{n∈ℕ} in D_{B} such that lim_{n→∞} x_{n} = x and {Bx_{n}}_{n∈ℕ} is bounded.

This proposition is well known (cf. [5, p. 69]), but we give a short proof for the convenience of the reader.

Proof of Proposition 1. Let {λ_{n}}_{n∈ℕ} be a sequence of positive numbers such that lim_{n→∞} λ_{n} = +∞. Given x ∈ Fav(T), set x_{n} = λ_{n}R_{λ_{n}}(B)x for each n ∈ ℕ. Clearly, {x_{n}}_{n∈ℕ} is sequence in D_{B} such that x = lim_{n→∞} x_{n}. We shall show that {Bx_{n}}_{n∈ℕ} is bounded. Let C be a positive constant such that h^{−1}∥T_{h}x − x∥ ≤ C for all h ∈ ℝ_{+}^{•}. In view of (3), for each n ∈ ℕ and each h ∈ ℝ_{+}^{•},

h^{−1}(T_{h}x_{n} − x_{n}) = ∫_{ℝ_{+}} h^{−1}λ_{n}e^{−λ_{n}t}T_{t}(T_{h}x − x) dt,

whence ∥h^{−1}(T_{h}x_{n} − x_{n})∥ ≤ Cm_{T} , and finally ∥Bx_{n}∥ ≤ Cm_{T}. □

We can now pass to proving Theorem 4.

Proof of Theorem 4. In view of Theorem 1, all that we need is to show that Fav(S^{+}) ⊂ R_{R′}. Let x′ ∈ Fav(S^{+}). By Proposition 1, there exists a sequence {x′_{n}}_{n∈ℕ} in D_{A^{+}} such that sup_{n∈ℕ} ∥A^{+}x′_{n}∥ < +∞ and lim_{n→∞} ∥x′_{n}−x′∥ = 0. Fix λ ∈ ℝ_{+}^{•} arbitrarily. Clearly, for each n ∈ ℕ,

x′_{n} = R′_{λ}(λ − A^{+})x′_{n}.

By the relative weak^{∗} compactness of bounded sets in E′, there exists a (generalised) subsequence {A^{+}x′_{n_{i}}}_{i∈I} and y′ ∈ E′ such that

w^{*}-lim_{i∈I} A^{+}x′_{n_{i}} = y′.

Since R′_{λ} is weakly^{∗} continuous, it follows that

(11)  x′ = R′_{λ}(λx′ − y′).

Hence x′ ∈ R_{R′}, establishing the desired inclusion. □

A simple modification of the above proof leads to the following result:

Theorem 5. If E is reflexive, then R_{r(R)} = R_{R} = Fav(S).

Proof. Assume that E is reflexive. We shall prove that

(12)  R_{r(R′)} = R_{R′} = Fav(S^{+}).

This will be a sufficient step: the theorem follows immediately upon applying (12) to the Favard class of the semigroup S^{++} associated with the bidual pseudo-resolvent R′′, modulo the observation that E′ is reflexive (so that the suggested application of (12) makes sense) and the remark that R′′ and S^{++} are naturally identifiable with R and S, respectively.

By virtue of Theorem 1, the proof of (12) reduces to showing that Fav(S^{+}) ⊂ R_{r(R′)}. Let x′ ∈ Fav(S^{+}). Retaining the notation from the proof of the foregoing theorem, note that y′ appearing in (11) belongs to the weak^{∗} closure of {A^{+}x′_{n}}_{n∈ℕ}, which, in view of the reflexivity of E, coincides with the weak closure of {A^{+}x′_{n}}_{n∈ℕ}. Remembering that D_{A^{+}} = R_{r(R′)} ⊂ R_{R′} and noting that, being norm closed, \overline{R_{R′}} is also weakly closed, we see that y′ ∈ \overline{R_{R′}}. In view of Theorem 4, we have x′ ∈ R_{R′}. Thus λx′ − y′ falls into \overline{R_{R′}}. Now (11) implies that x′ ∈ R_{r(R′)}, which establishes the desired inclusion. □

We now examine more closely the case in which \overline{R_{R}} = E. The associated semigroup S acts then on the whole of E, its generator A is densely defined on E, and R_{λ} = R_{λ}(A) for each λ ∈ ℝ_{+}^{•}. The dual semigroup S′ = {S′_{t}}_{t∈ℝ_{+}} on E′ may fail to be strongly continuous; it is, however, weakly^{∗} continuous in the sense that, for each x′ ∈ E′, the map ℝ_{+} ∋ t ↦ S′_{t}x′ ∈ E′ is continuous under the weak^{∗} topology on E′. The operator A′ is the weak^{∗} generator of S′ (see [3, Prop. 1.4.4], [8, Thm. 1.34], [12, p. 335–336]). The sets ρ_{A} and ρ_{A′} coincide and R_{λ}(A′) = R_{λ}(A)′ for each λ ∈ ρ_{A}. In particular, R′_{λ} = R_{λ}(A′) for each λ ∈ ℝ_{+}^{•}, which can emphatically be written as

(13)  R′ = R(A′).

The set

E^{⊙} = {x′ ∈ E′ | lim_{t→0+} ∥S′_{t}x′ − x′∥ = 0}

is a norm closed subspace of E′. According to a theorem of Phillips [14] (see also [3, Props. 1.4.69(b) and 1.4.7(a)], [5, Thm. 3.14], [12, p. 344–345]), D_{A′} is a norm dense subspace of E^{⊙}. Hence in particular E^{⊙} is weakly^{∗} dense in E′. Coupled with (13) and D_{A′} = R_{R(A′)}, the norm denseness of D_{A′} implies also that

(14)  E^{⊙} = \overline{R_{R′}}.

It is easily seen that the space E^{⊙} is invariant for all the S′_{t} and that setting

S_{t}^{⊙} = S′_{t} ↾ E^{⊙} (t ∈ ℝ_{+})

defines a strongly continuous semigroup S^{⊙} = {S_{t}^{⊙}}_{t∈ℝ_{+}} on E^{⊙}. Let A^{⊙} be the generator of S^{⊙}. Another part of the theorem of Phillips mentioned above asserts that

D_{A^{⊙}} = {x′ ∈ D_{A′} | A′x′ ∈ E^{⊙}}

and

A^{⊙} = A′ ↾ D_{A^{⊙}}.

(cf. [3, Prop. 1.4.7(b)]). Since m_{S^{⊙}} ≤ m_{S′} = m_{S}, we have ℝ_{+}^{•} ⊂ ρ_{A^{⊙}}, which together with the above characterisation of A^{⊙} immediately implies that, for each λ ∈ ℝ_{+}^{•},

(15)  R_{λ}(A^{⊙}) = R_{λ}(A′) ↾ E^{⊙}.

Now, taking into account (13) and (14), we see that

(16)  r(R′) = R(A^{⊙}).

It is also clear that A^{⊙} coincides with A^{+}, and that S^{⊙} coincides with S^{+}.

We are now prepared to state our next result.

Theorem 6. If \overline{R_{R}} = E, then R_{R′} = R_{r(R′)} if and only if \overline{R_{R′}} = E′.

Proof. Assume that \overline{R_{R}} = E. We first prove the necessity part. Suppose that R_{R′} = R_{r(R′)}. Fix λ ∈ ℝ_{+}^{•} arbitrarily. If x′ ∈ E′, then, in view of (13), R_{λ}(A′)x′ falls into R_{R′}, and hence also into R_{r(R′)}. On account of (15) and (16),

(17)  R_{r(R′)} = \{R_{λ}(A′)y′ | y′ ∈ \overline{R_{R′}}\}.

Therefore there exists y′ ∈ \overline{R_{R′}} such that R_{λ}(A′)x′ = R_{λ}(A′)y′. But R_{λ}(A′) is one-to-one, so x′ = y′, which proves that E′ ⊂ \overline{R_{R′}}. Since the reverse inclusion is obvious, we obtain \overline{R_{R′}} = E′.

The proof of the sufficiency part is simple: If \overline{R_{R′}} = E′, then the equality R_{r(R′)} = R_{R′} is a consequence of (13) and (17). □

Bearing in mind (14), from Theorem 6 we immediately deduce the following result:

Corollary 1. Let E be a Banach space, let S be a bounded strongly continuous oneparameter semigroup on E with the property that E^{⊙} ⊊ E′, and let R be the resolvent of the generator of S, indexed by ℝ_{+}^{•}. Then R_{r(R′)} ⊊ R_{R′}.

We finally note that Banach spaces and semigroups satisfying the hypothesis of this corollary exist and are the subject of many expositions (see [3, p. 52–55], [8, p. 24–26], [12, p. 344–347]).

4. Strict inclusions

In this section we present two examples showing that both inclusions in (5) may simultaneously be strict.

Example 1. Let C_{b}(ℝ_{+}) be the Banach space of all complex-valued bounded continuous functions on ℝ_{+}, and let C_{bu}(ℝ_{+}) be the Banach space of all complex-valued bounded uniformly continuous functions on ℝ_{+}. It is understood that both these spaces are endowed with the supremum norm. We regard C_{bu}(ℝ_{+}) as being isometrically embedded into C_{b}(ℝ_{+}). Let L^{∞}(ℝ_{+}) be the Banach space of (the equivalence classes of) Lebesgue measurable essentially bounded functions on ℝ_{+}, equipped with the norm ∥x∥ = ess sup_{t∈ℝ_{+}} |x(t)|.

For each t ∈ ℝ_{+}, let T_{t}^{−}: C_{b}(ℝ_{+}) → C_{b}(ℝ_{+}) be the backward shift by t given by

(T_{t}^{−}f)(x) = f(x + t) (f ∈ C_{b}(ℝ_{+}), x ∈ ℝ_{+}).

The family T^{−} = {T_{t}^{−}}_{t∈ℝ_{+}} is a semigroup of linear contractions on C_{b}(ℝ_{+}). This semigroup is not strongly continuous. Nonetheless, it can be used to define a pseudoresolvent in L(C_{b}(ℝ_{+})), tempered at infinity. Guided by (3), we let, for each λ ∈ ℝ_{+}^{•}, each f ∈ C_{b}(ℝ_{+}), and each x ∈ ℝ_{+},

(R_{λ}f)(x) = ∫_{0}^{∞} e^{−λt}(T_{t}^{−}f)(x) dt = e^{λx} ∫_{x}^{∞} e^{−λt}f(t) dt.

Clearly, the function R_{λ}: x ↦ (R_{λ}f)(x) is bounded and uniformly continuous; the mapping R_{λ}: f ↦ R_{λ}f is a bounded linear operator on C_{b}(ℝ_{+}) of norm equal to λ^{−1}; and the family R = {R_{λ}}_{λ∈ℝ_{+}^{•}} is a pseudo-resolvent in L(C_{b}(ℝ_{+})) with c_{R} = 1.

We claim that

(18)  R_{R} = {f ∈ C_{bu}(ℝ_{+}) | f′ ∈ C_{b}(ℝ_{+})}.

Fix λ ∈ ℝ_{+}^{•} arbitrarily and let f ∈ C_{b}(ℝ_{+}). We have already noticed that R_{λ}f ∈ C_{bu}(ℝ_{+}). It is also clear that R_{λ}f is differentiable, and that (R_{λ}f)′ = λR_{λ}f − f. Thus (R_{λ}f)′ belongs to C_{b}(ℝ_{+}), and so

R_{R} ⊂ {f ∈ C_{bu}(ℝ_{+}) | f′ ∈ C_{b}(ℝ_{+})}.

To show the reverse inclusion, let f be a differentiable function in C_{bu}(ℝ_{+}) such that f′ ∈ C_{b}(ℝ_{+}). It is directly verified that R_{λ}(λf − f′) = f. Hence f ∈ R_{R}, establishing the inclusion in question and thereby also proving (18).

It immediately follows from (18) that \overline{R_{R}} = C_{bu}(ℝ_{+}). Let S = \{S_{t}\}_{t∈ℝ_{+}} be the semigroup associated with R. It is clear that, for each t ∈ ℝ_{+},

S_{t} = T_{t}^{−} ↾ C_{bu}(ℝ_{+}).

The generator of S is easily identified with the differentiation operator f ↦ f′ having {f ∈ C_{bu}(ℝ_{+}) | f′ ∈ C_{bu}(ℝ_{+})} as a domain. Thus

R_{r(R)} = {f ∈ C_{bu}(ℝ_{+}) | f′ ∈ C_{bu}(ℝ_{+})}.

Finally, an application of Lebesgue’s Differentiation Theorem reveals that

Fav(S) = {f ∈ C_{bu}(ℝ_{+}) | f is differentiable λ_{ℝ_{+}}-a. e. and f′ ∈ L^{∞}(ℝ_{+})}.

It is now clear that

R_{r(R)} ⊊ R_{R} ⊊ Fav(S).

Example 2. Let 𝔅(ℝ_{+}) be the σ-algebra of all Borel subsets of ℝ_{+}. Let M(ℝ_{+}) be the collection of all complex bounded regular Borel measures on ℝ_{+}. With addition and scalar multiplication defined pointwise on 𝔅(ℝ_{+}), and with the norm ∥μ∥ = |μ|(ℝ_{+}), where |μ| denotes the total variation of the measure μ ∈ M(ℝ_{+}), M(ℝ_{+}) is a Banach space. With the multiplication defined by the convolution of measures

(μ ∗ ν)(A) = ∫_{ℝ_{+}} μ(A ⊖ t) dν(t) (A ∈ 𝔅(ℝ_{+})),

where

A ⊖ t = {x ∈ ℝ_{+} | x = y − t for some y ∈ A},

M(ℝ_{+}) becomes a Banach algebra. The Dirac measure concentrated at 0, δ_{0}, is the identity of this algebra.

For each f ∈ L^{1}(ℝ_{+}), let ν_{f} be a measure in M(ℝ_{+}) defined as

ν_{f}(A) = ∫_{A} f(t) dt (A ∈ 𝔅(ℝ_{+})).

The mapping f ↦ ν_{f} is a Banach algebra isomorphism of L^{1}(ℝ_{+}) onto the algebra of all measures in M(ℝ_{+}) that are absolutely continuous with respect to Lebesgue measure. We identify L^{1}(ℝ_{+}) with its image via f ↦ ν_{f}. Under this identification, L^{1}(ℝ_{+}) becomes an ideal of M(ℝ_{+}). If f ∈ L^{1}(ℝ_{+}) and μ ∈ M(ℝ_{+}), then f ∗ μ is a member of L^{1}(ℝ_{+}) determined by ν_{f} ∗ μ = ν_{f∗μ} and given by

(19) (f ∗ μ)(x) = ∫_{[0,x]} f(x − t) dμ(t) = ∫_{ℝ_{+}}(T_{t}^{+}f)(x) dμ(t)  (λ_{ℝ_{+}}-a. e. x ∈ ℝ_{+}).

Consider a homomorphism H: L^{1}(ℝ_{+}) → L(M(ℝ_{+})) defined by

H(f)μ = f ∗ μ (f ∈ L^{1}(ℝ_{+}), μ ∈ M(ℝ_{+})).

It is clear that ∥H∥ = 1. Since L^{1}(ℝ_{+}) is an ideal of M(ℝ_{+}), we have R_{H} ⊂ L^{1}(ℝ_{+}). As H(f)δ_{0} = f for any f ∈ L^{1}(ℝ_{+}), we see that

(20)  R_{H} = L^{1}(ℝ_{+}).

For each λ ∈ ℝ_{+}^{•}, set R_{λ} = H(ϵ_{−λ}). Clearly, R = {R_{λ}}_{λ∈ℝ_{+}^{•}} is a pseudo-resolvent in L(M(ℝ_{+})) with c_{R} = 1. We shall show that

R_{r(R)} ⊊ R_{R} ⊊ Fav(S).

In view of (7) and (20), we have \overline{R_{R}} = L^{1}(ℝ_{+}). Using (19) (or alternatively utilising (8)), we verify at once that, for each λ ∈ ℝ_{+}^{•} and each f ∈ L^{1}(ℝ_{+}),

(21)  r(R)_{λ}f = ∫_{ℝ_{+}} e^{−λt}T_{t}^{+}f dt.

Thus the semigroup associated with R coincides with the semigroup of forward shifts on L^{1}(ℝ_{+}).

Given μ ∈ M(ℝ_{+}) and s ∈ ℝ_{+}, denote by L_{μ} the Laplace transform of μ defined as

L_{μ}(s) = ∫_{ℝ_{+}} e^{−st} dμ(t).

If f ∈ L^{1}(ℝ_{+}), we write L_{f} instead of L_{ν_{f}}. As is well known, L_{μ} uniquely determines μ, and we also have L_{μ∗ν} = L_{μ}L_{ν} for any μ, ν ∈ M(ℝ_{+}).

We proceed to show that the inclusion R_{r(R)} ⊂ R_{R} is strict. We assert that, for each λ ∈ ℝ_{+}^{•},

(22)  ϵ_{−λ} ∈ R_{R} \ R_{r(R)}.

Fix λ ∈ ℝ_{+}^{•} arbitrarily. Since ϵ_{λ} = R_{λ}(δ_{0}), we see that ϵ_{λ} ∈ R_{R}. Suppose, contrary to our assertion, that ϵ_{λ} ∈ R_{r(R)}. Then ϵ_{−λ} can be represented as ϵ_{−λ} = ϵ_{−λ} ∗ f for some f ∈ L^{1}(ℝ_{+}). Accordingly, L_{ϵ_{−λ}} = L_{ϵ_{−λ}}L_{f}. Since, for each s ∈ ℝ_{+},

L_{ϵ_{−λ}}(s) = \frac{1}{s + λ} ≠ 0,

it follows that L_{f} = 1 = L_{δ_{0}} and hence ν_{f} = δ_{0}. This contradiction establishes (22).

Let C(ℝ_{+}) be the space of all complex-valued (not necessarily bounded) continuous functions on ℝ_{+}. We claim that

(23)  R_{R} ⊂ C(ℝ_{+}).

In fact, if we fix λ ∈ ℝ_{+} arbitrarily, then every element of R_{R} can be represented as ϵ_{−λ} ∗ μ for an appropriate μ ∈ M(ℝ_{+}). Clearly,

(ϵ_{−λ} ∗ μ)(x) = e^{−λx} ∫_{[0,x)} e^{λt} dμ(t) (λ_{ℝ_{+}}-a. e. x ∈ ℝ_{+}).

By Lebesgue’s Dominated Convergence Theorem, the function

x ↦ ∫_{[0,x)} e^{λt} dμ(t)

is continuous. We thus see that ϵ_{−λ} ∗ μ can be identified with a continuous function on ℝ_{+}. The claim is established.

We now show that the inclusion R_{R} ⊂ Fav(T^{+}) is strict. Choose arbitrarily a, b ∈ ℝ_{+} so that a < b. Set f = 1_{(a,b)}. Clearly, f belongs to L^{1}(ℝ_{+}). We contend that f ∈ Fav(T^{+}) \ R_{R}. As usual, let A △ B denote the symmetric difference of the sets A and B; that is

A △ B = (A \ B) ∪ (B \ A).

Since, for any t ∈ ℝ_{+},

∥T_{t}^{+}f − f∥ = λ_{ℝ_{+}} ((a + t, b + t) △ (a, b)) ≤ 2 min(t, b − a),

we see that f ∈ Fav(T^{+}). Now note that the equivalence class of f in L^{1}(ℝ_{+}) has no representative that is continuous everywhere on ℝ_{+}. Indeed, if g were the (unique) continuous modification of f, then g would equal 1 on (a, b) and 0 off (a, b), which is incompatible with the continuity of g on all of ℝ_{+}. Invoking (23), we see that f ∉ R_{R}. The contention is established.

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(W. Chojnacki) Institute of Applied Mathematics and Mechanics, University of Warsaw, ul. Banacha 2, 02-097 Warszawa, Poland; Department of Computer Science, University of Adelaide, Adelaide, SA 5005, Australia

   E-mail address: wojtekch@appli.mimuw.edu.pl, wojtek@cs.adelaide.edu.au

(J. Kisyński) Department of Electrical Engineering, Technical University of Lublin, GPO Box 189, ul. Nadbystrzycka 38a, 20-618 Lublin, Poland

   E-mail address: kisynski@antenor.pol.lublin.pl