STABILITY AND DICHOTOMY OF

POSITIVE SEMIGROUPS ON L_{p}

Stephen Montgomery-Smith

Abstract. A new proof of a result of Lutz Weis is given, that states that the stability of a positive strongly continuous semigroup (e^{tA})_{t≥0} on L_{p} may be determined by the quantity s(A). We also give an example to show that the dichotomy of the semigroup may not always be determined by the spectrum σ(A).

Consider a strongly continuous semigroup (e^{tA})_{t≥0} acting on a Banach space X with unbounded generator A. It has long been known that the spectral mapping theorem e^{tσ(A)} = σ(e^{tA}) \ {0} does not necessarily hold. (Here σ(A) denotes the spectrum of an operator A.) Indeed, let s(A) = sup Re(σ(A)), and let ω(A) = sup Re(log(σ(e^{A}))) = inf{λ : ∥e^{tA}∥ ≤ M_{λ}e^{λt}}. Then there are examples of semigroups for which s(A) ≠ ω(A) (see [N]).

The purpose of this paper is to give one situation in which it is true that s(A) = ω(A). This next result has already been proved by Lutz Weis [We]. We will give a different, shorter proof. We refer the reader to [We] for a history of the problem.

Theorem 1. Let e^{tA} be a strongly continuous positive semigroup on L_{p}(Ω, F, μ), where (Ω, F, μ) is a sigma-finite measure space, and 1 ≤ p < ∞. Then ω(A) = s(A).

In order to show this result, we will make use of the following lemmas. The first result may be derived from [C], Theorem 7.4 (the reader may like to know that a proof of the ‘Pringsheim-Landau Theorem’ used in [C] may be found on page 59 of [Wi]).

Lemma 2. Let e^{tA} be a strongly continuous positive semigroup on a Banach lattice X, and let g ∈ X. Then for any λ > s(A) we have that

(λ − A)^{−1}g = ∫_{0}^{∞} e^{s(A−λ)}g ds.

Here the right hand side is taken in the sense of an improper integral.

The next result may be found in [LM1] and [LM2].

Typeset by AMS-TEX

1991 Mathematics Subject Classification. Primary 47-02, 47D06, Secondary 35B40..

Research supported in part by N.S.F. Grant D.M.S. 9201357.

Lemma 3. Let e^{tA} be a strongly continuous semigroup on a Banach space X, and let 1 ≤ p < ∞. Then 1 ∉ σ(e^{2πA}) if and only if iℤ ∩ σ(A) = ∅ and there is a constant c > 0 such that for any v_{−n}, v_{−n+1}, …, v_{n} ∈ X we have

∫_{0}^{2π} ∥ ∑_{k=−n}^{n}(ik − A)^{−1}v_{k}e^{ikt}∥^{p} dt ≤ c^{p} ∫_{0}^{2π} ∥ ∑_{k=−n}^{n} v_{k}e^{ikt}∥^{p} dt.

For the next result, we specialize to a Banach lattice of functions on a sigmafinite measure space. In fact, this is really no loss of generality, and the interested reader should find no trouble making sense of this result for a general Banach lattice by applying the ideas in [LT] Chapter 1.4.

Lemma 4. Let P be a positive operator on X, a Banach lattice of functions on a sigma-finite measure space, such that |g| ≤ f ∈ X implies that g ∈ X. Let 1 ≤ p < ∞. If f : [0, 2π] → X is a measurable, simple function, then

(∫_{0}^{2π} |P (f(t))|^{p} dt)^{1/p} ≤ P ((∫_{0}^{2π} |f(t)|^{p} dt)^{1/p}) .

Proof. Let us set f = ∑_{k=1}^{n} v_{k}χ_{A_{k}}, where v_{k} ∈ X, and the sets A_{k} ⊆ [0, 2π] are disjoint. Then, letting f_{k} = v_{k} |A_{k}|^{1/p}, the result reduces to showing that

(∑_{k=1}^{n} |P (f_{k})|^{p})^{1/p} ≤ P ((∑_{k=1}^{n} |f_{k}|^{p})^{1/p}) .

However, we know that

(∑_{k=1}^{n} |f_{k}|^{p})^{1/p} = l.u.b. ; ∑|a_{k}|^{q}≤1 ∑_{k=1}^{n} Re(a_{k}f_{k}).

Here, l.u.b. denotes the least upper bound in the lattice. Now, since P is positive, we have that

P ( l.u.b. ; ∑|a_{k}|^{q}≤1 ∑_{k=1}^{n} Re(a_{k}f_{k}))

is an upper bound for ∑_{k=1}^{n} Re(a_{k}P (f_{k})) whenever ∑ |a_{k}|^{q} ≤ 1. Hence

(∑_{k=1}^{n} |P (f_{k})|^{p})^{1/p} = l.u.b. ; ∑|a_{k}|^{q}≤1 ∑_{k=1}^{n} Re(a_{k}P (f_{k}))

≤ P ( l.u.b. ; ∑|a_{k}|^{q}≤1 ∑_{k=1}^{n} Re(a_{k}f_{k}))

= P ((∑_{k=1}^{n} |f_{k}|^{p})^{1/p}) .

Proof of Theorem 1. It is well known that s(A) ≤ ω(A) (see [N]). Thus by simple rescaling arguments, we see that it is sufficient to show that if s(A) < 0, then 𝕋 ∩ σ(e^{2πA}) = ∅.

We will show, under the assumption that s(A) < 0, that if f : ℝ → L_{p} is a bounded, measurable function that is periodic with period 2π, then for each N > 0 we have

(∫_{0}^{2π} ∥∫_{0}^{N} e^{sA}f(t − s) ds∥_{L_{p}}^{p} dt)^{1/p} ≤ ∥A^{−1}∥ (∫_{0}^{2π} ∥f(t)∥_{L_{p}}^{p} dt)^{1/p} .

In order to show this, we may assume without loss of generality that f restricted to [0, 2π] is a simple function. Fix N > 0. By the positivity of e^{sA}, and Fubini’s Theorem, we have that

(∫_{0}^{2π} ∥∫_{0}^{N} e^{sA}f(t − s) ds∥_{L_{p}}^{p} dt)^{1/p} ≤ (∫_{0}^{2π} ∥∫_{0}^{N} e^{sA} |f(t − s)| ds∥_{L_{p}}^{p} dt)^{1/p}

= ∥(∫_{0}^{2π} (∫_{0}^{N} e^{sA} |f(t − s)| ds)^{p} dt)^{1/p}∥_{L_{p}} .

By the integral version of Minkowski’s Theorem (see [HLP], Section 203), it follows that for each ω ∈ Ω

(∫_{0}^{2π} (∫_{0}^{N} e^{sA} |f(t − s)(ω)| ds)^{p} dt)^{1/p} ≤ ∫_{0}^{N} (∫_{0}^{2π} (e^{sA} |f(t − s)(ω)|)^{p} dt)^{1/p} ds

= ∫_{0}^{N} (∫_{0}^{2π} (e^{sA} |f(t)(ω)|)^{p} dt)^{1/p} ds.

Finally, from Lemma 4, we see that

(∫_{0}^{2π}(e^{sA} |f(t)|)^{p} dt)^{1/p} ≤ e^{sA} (∫_{0}^{2π} |f(t)|^{p} dt)^{1/p} .

Putting all of these together, and applying Lemma 2, we obtain

(∫_{0}^{2π} ∥∫_{0}^{N} e^{sA}f(t − s) ds∥_{L_{p}}^{p} dt)^{1/p} ≤ ∥∫_{0}^{N} e^{sA} (∫_{0}^{2π} |f(t)|^{p} dt)^{1/p} ds∥_{L_{p}}

≤ ∥∫_{0}^{∞} e^{sA} (∫_{0}^{2π} |f(t)|^{p} dt)^{1/p} ds∥_{L_{p}}

= ∥A^{−1} (∫_{0}^{2π} |f(t)|^{p} dt)^{1/p}∥_{L_{p}}

≤ ∥A^{−1}∥ ∥(∫_{0}^{2π} |f(t)|^{p} dt)^{1/p}∥_{L_{p}}

= ∥A^{−1}∥ (∫_{0}^{2π} ∥f(t)∥_{L_{p}}^{p} dt)^{1/p} ,

where the last equality uses Fubini’s theorem. Now, if f(t) = e^{iβt} ∑_{k=−n}^{n} v_{k}e^{ikt} for some β ∈ ℝ, then by Lemma 2, we see that

∫_{0}^{N} e^{sA}f(t − s) ds → ∑_{k=−n}^{n}(ik + iβ − A)^{−1}v_{k}e^{ikt}

uniformly in t as N → ∞. Hence by Lemma 3 it follows that e^{iβ} ∉ σ(e^{2πA}).

One might conjecture that the spectrum of the generator of a positive semigroup e^{tD} on an L_{p} space might characterize the dichotomy of the semigroup, that is, if a is any real number, then (a + iℝ) ∩ σ(D) = ∅ if and only if e^{ta}𝕋 ∩ σ(e^{tD}) = ∅. However, this is not the case, as the next result shows.

Theorem 5. There is a positive semigroup e^{tD} acting on an L_{2} space such that (1 + iℝ) ∩ σ(D) = ∅, but e^{2π} ∈ σ(e^{2πD}).

Proof. For each M ∈ ℕ, let C_{M} be the contraction acting on ℓ_{2}^{M} by the matrix

C_{M} = [ \begin{matrix}0 & 1 & 0 & 0 & ⋯ & 0 \\ 0 & 0 & 1 & 0 & ⋯ & 0 \\ 0 & 0 & 0 & 1 & ⋯ & 0 \\ 0 & 0 & 0 & 0 & ⋯ & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & & ⋮ \\ 0 & 0 & 0 & 0 & ⋯ & 0 \end{matrix} ] .

Note that if λ ≠ 0, then

(λ − C_{M})^{−1} = ∑_{j=0}^{M−1} λ^{−1−j}C_{M}^{j} = [ \begin{matrix}λ^{−1} & λ^{−2} & λ^{−3} & λ^{−4} & ⋯ & λ^{−M} \\ 0 & λ^{−1} & λ^{−2} & λ^{−3} & ⋯ & λ^{−M+1} \\ 0 & 0 & λ^{−1} & λ^{−2} & ⋯ & λ^{−M+2} \\ 0 & 0 & 0 & λ^{−1} & ⋯ & λ^{−M+3} \\ ⋮ & ⋮ & ⋮ & ⋮ & & ⋮ \\ 0 & 0 & 0 & 0 & ⋯ & λ^{−1} \end{matrix} ] .

Thus, if |λ| = 1, then ∥(λ − C_{M})^{−1}∥ ≥ \sqrt{M}. Also, if |λ| > 1, then ∥(λ − C_{M})^{−1}∥ ≤ ∑_{j=0}^{M−1} |λ|^{−1−j} ≤ 1/(|λ| − 1). In particular, if |λ| ≥ 2, then ∥(λ − C_{M})^{−1}∥ ≤ 1.

Also note that

e^{tC_{M}} = [ \begin{matrix}1 & t & t^{2}/2 & t^{3}/6 & ⋯ & t^{M−1}/(M − 1)! \\ 0 & 1 & t & t^{2}/2 & ⋯ & t^{M−2}/(M − 2)! \\ 0 & 0 & 1 & t & ⋯ & t^{M−3}/(M − 3)! \\ 0 & 0 & 0 & 1 & ⋯ & t^{M−4}/(M − 4)! \\ ⋮ & ⋮ & ⋮ & ⋮ & & ⋮ \\ 0 & 0 & 0 & 0 & ⋯ & 1 \end{matrix} ] .

Thus we see that e^{tC_{M}} is a positive operator. Clearly ∥e^{tC_{M}} ∥ ≤ e^{t∥C_{M} ∥} ≤ e^{t}. Consider the positive semigroup acting on L_{2}([0, 2π]) by

e^{tA_{M}} f(x) = (e^{4t} − 1) ∫_{0}^{2π} f(x) \frac{dx}{2π} + f(x + Mt),

so that its generator is the closure of

A_{M}f(x) = 4 ∫_{0}^{2π} f(x) \frac{dx}{2π} + M \frac{d}{dx}f(x).

Note that ∥e^{tA_{M}} ∥ ≤ e^{4t}.

Now consider the positive semigroup e^{tB_{M}} = e^{tA_{M}} ⊗ e^{tC_{M}} acting on

X_{M} = L_{2}([0, 2π]) ⊗ ℓ_{2}^{M} = L_{2}([0, 2π] × {1, 2, …, M}),

We see that this semigroup is generated by B_{M} = A_{M} ⊗ I + I ⊗ C_{M}. Also, ∥e^{tB_{M}} ∥ ≤ e^{5t}.

Consider a typical element of X_{M} given by f(x) = ∑_{n=−∞}^{∞} v_{n}e^{inx} ∈ X_{M}, where v_{n} ∈ ℓ_{2}^{M}, and ∥f∥_{X_{M}}^{2} = 2π ∑_{n=−∞}^{∞} ∥v_{n}∥_{2}^{2}. If λ ≠ 4 and λ ∉ Mℤ \ {0}, then λ ∉ σ(B_{M}), and

(λ − B_{M})^{−1}f(x) = (λ − 4 − C_{M})^{−1}v_{0} + ∑_{n≠0}(λ − inM − C_{M})^{−1}v_{n}e^{inx}.

Thus

∥(λ − B_{M})^{−1}∥ = max {∥(λ − 4 − C_{M})^{−1}∥ , sup_{n≠0} ∥(λ − inM − C_{M})^{−1}∥} .

In particular, if Re(λ) = 1 and |λ| ≤ M − 2, then ∥(λ − B_{M})^{−1}∥ ≤ 1, whereas if λ = 1 + iM, then ∥(λ − B_{M})^{−1}∥ ≥ \sqrt{M}.

Now consider the semigroup e^{tD} = ⨁_{M=1}^{∞} e^{tB_{M}} acting on

⨁_{M=1}^{∞} X_{M} = L_{2} ( ⋁_{M=1}^{∞}([0, 2π] × {1, 2, …, M})) .

Note that e^{tD} really is a strongly continuous semigroup, with ∥e^{tD}∥ ≤ e^{5t}. The

generator D is the closure of ⨁_{M=1}^{∞} B_{M}, and hence its resolvent set consists of those

λ such that

∥(λ − D)^{−1}∥ = sup_{M≥1} ∥(λ − B_{M})^{−1}∥ < ∞,

that is, σ(D) ⊆ {z : |z − 4| ≤ 1} ∪ iℤ \ {0}. In particular, if Re(λ) = 1, then λ ∉ σ(D). However, sup_{λ∈1+iℤ} ∥(λ − D)^{−1}∥ = ∞, and hence, by Gerhard’s Theorem (see [N], p. 95), e^{2π} ∈ σ(e^{2πD}).

References

[C] Ph. Clement, H.J.A.M. Heijmans, et al., One Parameter Semigroups, North-Holland, 1987.

[HLP] G.H. Hardy, J.E. Littlewood and G. Pólya, Inequalities, Cambridge University Press, 1952.

[LT] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces Volume II, Springer-Verlag, 1979.

[LM1] Y. Latushkin and S.J. Montgomery-Smith, Lyapunov theorems for Banach spaces, Bull. A.M.S. 31 (1994), 44–49.

[LM2] Y. Latushkin and S.J. Montgomery-Smith, Evolutionary semigroups and Lyapunov theorems in Banach spaces, J. Func. Anal. 127 (1995), 173–197.

[N] R. Nagel (ed.), One Parameter Semigroups of Positive Operators, Springer-Verlag, 1984.

[We] L. Weis, The stability of positive semigroups on L_{p} spaces, Proc. A.M.S. (to appear).

[Wi] D. Widder, The Laplace Transform, vol. 6, Princeton Math Series, 1946.

Math. Dept., University of Missouri, Columbia MO 65211, U.S.A.