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Zero-one laws for functional calculus on operator semigroups

Jonathan R. Partington (Leeds, UK) Lancaster, September 2014 Joint work with Isabelle Chalendar (Lyon) and Jean Esterle (Bordeaux)

Jonathan R. Partington (Leeds, UK) Zero-one laws for functional calculus on operator semigroups

Contents

◮ 1. Some classical results. ◮ 2. Newer dichotomy laws. ◮ 3. Functional calculus.

Jonathan R. Partington (Leeds, UK) Zero-one laws for functional calculus on operator semigroups

One-parameter semigroups

We work with one-parameter families (T(t))0<t<∞ in a Banach algebra A. Often A is the algebra of bounded linear operators on a Banach space X, and indeed given A we can take X = A. As usual, a semigroup satisfies T(s + t) = T(s)T(t) for all t, s > 0. We assume strong continuity for t > 0 but not necessarily at t = 0, that is, the mapping t → T(t)x is continuous in norm for all x ∈ X.

Jonathan R. Partington (Leeds, UK) Zero-one laws for functional calculus on operator semigroups

An example

Consider the semigroup T(t) : x → xt in A = X = C[0, 1]. Even with T(0) ≡ 1, we don’t have strong continuity at 0. Note that T(t) − I = 1 for all t > 0.

Jonathan R. Partington (Leeds, UK) Zero-one laws for functional calculus on operator semigroups

The classical zero-one law

The classical zero-one law, elementary to prove: If L := lim supt→0+ T(t) − I < 1, then T(t) − I → 0 and hence the semigroup is uniformly continuous and so has the form eAt where A, the generator, is bounded. PROOF (Coulhon): The identity 2(x − 1) = x2 − 1 − (x − 1)2 implies that 2(T(t) − I) = T(2t) − I − (T(t) − I)2, so we have 2L ≤ L + L2 and thus L = 0 or L ≥ 1.

Jonathan R. Partington (Leeds, UK) Zero-one laws for functional calculus on operator semigroups

Hille’s theorem

Hille (1950) had an analogous result for differentiable semigroups. Suppose that (T(t))t>0 is an n-times continuously differentiable semigroup. If lim sup

t→0+

tnT (n)(t) < n e n , then the semigroup has a bounded generator. The usual case is n = 1, when we get a zero-1/e law, but the argument works more generally, and these constants are sharp.

Jonathan R. Partington (Leeds, UK) Zero-one laws for functional calculus on operator semigroups

Analytic semigroups

We will also look at semigroups (T(t))t∈Sα, where t lies in a sector Sα := {z ∈ C : | arg z| < α} for some 0 < α < π/2, the semigroup being supposed to be analytic (holomorphic). Typically we examine T(t) in |t| < δ, with t ∈ Sα.

Jonathan R. Partington (Leeds, UK) Zero-one laws for functional calculus on operator semigroups

Beurling’s extension theorem

Beurling (1970) proved that a semigroup defined on R+ has an analytic extension to some sector Sα if and only if lim sup

t→0+

p(T(t)) < sup{|p(z)| : |z| ≤ 1} for some polynomial p. Kato and Neuberger (both 1970) proved that p(z) = z − 1 is sufficient, giving a zero-two law for analyticity, i.e., that lim sup

t→0+

T(t) − I < 2 implies analyticity.

Jonathan R. Partington (Leeds, UK) Zero-one laws for functional calculus on operator semigroups

Mokhtari’s zero-quarter law

Suppose that the semigroup (T(t))t>0 is bounded at the origin; then (T(tn))n forms a bounded approximate identity in the algebra A generated by the semigroup, whenever tn → 0. Moreover, if lim sup

t→0+

T(t) − T(2t) < 1 4, then either T(t) = 0 for t > 0, or else the semigroup has a bounded generator A. Esterle and Mokhtari (2002): similar results for n ≥ 1, with lim sup

t→0+

T(t) − T((n + 1)t) < n (n + 1)1+1/n = sup

[0,1]

|x − xn+1|.

Jonathan R. Partington (Leeds, UK) Zero-one laws for functional calculus on operator semigroups

Quasinilpotent semigroups

Recall that a semigroup is quasinilpotent if the spectral radius satisfies ρ(T(t)) = 0 for all t. Standard examples can be found in the convolution algebra L1(0, 1). Excluding the trivial case, it then turns out that for each γ > 0 there is a δ > 0 such that T(t) − T((γ + 1)t) > γ (γ + 1)1+1/γ for 0 < t < δ (Esterle, 2005), an improvement on the Esterle–Mokhtari result.

Jonathan R. Partington (Leeds, UK) Zero-one laws for functional calculus on operator semigroups

The other extreme

Suppose that the algebra A is semi-simple, so no quasinilpotent elements except 0. Theorem (Bendaoud-Chalendar–Esterle–P., 2010). If for some γ > 0 we have ρ(T(t) − T((γ + 1)t)) < γ (γ + 1)1+1/γ for 0 < t < δ (some δ > 0), then A is unital and we have T(t) = etA for some bounded A ∈ A. In general, one can deduce similar properties of A/ Rad A (quotienting out the radical).

Jonathan R. Partington (Leeds, UK) Zero-one laws for functional calculus on operator semigroups

Sectorial semigroups

An easy-stated result for the half-plane C+: Theorem (Bendaoud-Chalendar–Esterle–P., 2010). If sup

t∈C+,|t|<δ

ρ(T(t) − T((γ + 1)t)) < 2 then A/ Rad A is a unital algebra, and the projection of the semigroup onto it has a bounded generator. Our aim now: look at more general expressions, and “explain” the constants.

Jonathan R. Partington (Leeds, UK) Zero-one laws for functional calculus on operator semigroups

More general expressions

Theorem (BCEP, 2010). Let f be a real linear combination of functions zm exp(−zw) with m = 0, 1, 2, . . . and w > 0, such that f (0) = 0 and f (z) → 0 as Re z → ∞. Let (T(t))t∈Sα be analytic and non-quasinilpotent. Define kα = supz∈Sα |f (z)|. If sup

t∈Sα,|t|<δ

ρ(f (−tA)) < kα then A/ Rad A is unital and the projection of the semigoup has a bounded generator.

Jonathan R. Partington (Leeds, UK) Zero-one laws for functional calculus on operator semigroups

Examples

For f (z) we may take p(z) exp(−z), p a suitable polynomial. Or take combinations exp(−z) − exp(−(γ + 1)z), as we did earlier. Thus we may estimate expressions such as tnAnT(t) = tnT (n)(t) and T(t) − T((γ + 1)t). In the first case kα =

• n

e cos α n , recovering and extending the Hille result. In the second, kα ր 2 as α ր π/2. All constants are sharp, as examples in C[0, 1] show.

Jonathan R. Partington (Leeds, UK) Zero-one laws for functional calculus on operator semigroups

A note on techniques

The methods here are largely based on complex analysis ideas. In the quasinilpotent case we make estimates of the resolvent of A (which is an entire function). In the non-quasinilpotent case we have Banach algebra ideas available. In particular there are nontrivial characters χ : A → C. We may check that χ(T(t)) = exp(λt) for some λ ∈ C and proceed from there to show that the Gelfand space A is compact.

Jonathan R. Partington (Leeds, UK) Zero-one laws for functional calculus on operator semigroups

A curiosity about quasinilpotent semigroups

First, an analytic semigroup (T(t))t∈Sα bounded near the origin has an extension to Sα making it strongly continuous at boundary points. Second, if the semigroup is quasinilpotent and bounded on the half-plane C+, then it is trivial. Indeed, if its boundary values satisfy ∞

−∞

log+ T(iy) 1 + y2 < ∞, then T(t) = 0 for t ∈ C+ (Chalendar–Esterle-P., 2010).

Jonathan R. Partington (Leeds, UK) Zero-one laws for functional calculus on operator semigroups

Basic Functional Calculus

We begin with semigroups on R+. If (T(t))t>0 is uniformly bounded and strongly continuous, then we may write (A + λI)−1 = − ∞ eλtT(t) dt, for Re λ < 0 (Bochner integral with respect to strong operator topology). If in addition (T(t))t>0 is quasinilpotent, then we have the above for all λ ∈ C.

Jonathan R. Partington (Leeds, UK) Zero-one laws for functional calculus on operator semigroups

Functions defined by measures

Take µ ∈ Mc(0, ∞), i.e., complex finite Borel measure of compact support. Then its Laplace transform is, as usual, F(s) := Lµ(s) = ∞ e−sξ dµ(ξ). Now we can define a functional calculus for the generator of a semigroup on X by F(−A)x = ∞ T(ξ)x dµ(ξ) (x ∈ X).

Jonathan R. Partington (Leeds, UK) Zero-one laws for functional calculus on operator semigroups

Examples

The results will apply to examples with ∞

0 dµ(t) = 0.

For instance, take µ = δ1 − δ2; then F(s) = e−s − e−2s and F(−tA) = T(t) − T(2t). More exotic examples: dµ(t) = (χ(1,2) − χ(2,3))(t) dt

• r

µ = δ1 − 3δ2 + δ3 + δ4.

Jonathan R. Partington (Leeds, UK) Zero-one laws for functional calculus on operator semigroups

Theorem for quasinilpotent semigroups

Theorem (Chalendar–Esterle–P., 2013) Let µ ∈ Mc(0, ∞) be real with ∞

0 dµ(t) = 0.

Let (T(t))t>0 be a nontrivial strongly continuous quasinilpotent

• semigroup. Then there is an η > 0 such that

F(−sA) > max

x≥0 |F(x)|

(0 < s ≤ η). For complex measures we define F = Lµ, so F(z) = F(z). Then F(−sA) F(−sA) > max

x≥0 |F(x)|2

(0 < s ≤ η).

Jonathan R. Partington (Leeds, UK) Zero-one laws for functional calculus on operator semigroups

The non-quasinilpotent case

For non-quasinilpotent semigroups there are various similar results, but they are more technical. For example, in the case of a real measure, if there are tk → 0 with F(−tkA) < sup

x>0

|F(x)|, then there are idempotents Pn ∈ A (i.e., P2

n = Pn) such that

∞

n=1 PnA is dense in A and each semigroup (PnT(t)) has a

bounded generator.

Jonathan R. Partington (Leeds, UK) Zero-one laws for functional calculus on operator semigroups

Analytic semigroups

For analytic semigroups on Sα we can replace measures by distributions. Take H(Sα) to be the Frechet space of analytic functions on Sα with topology of local uniform convergence. Now take (Kn) compact increasing, with ∞

n=1 Kn = Sα.

Our distributions are ϕ : H(Sα) → C, such that |f , ϕ| ≤ M sup{|f (z)| : z ∈ Kn} for some M > 0 and n ≥ 1.

Jonathan R. Partington (Leeds, UK) Zero-one laws for functional calculus on operator semigroups

More on distributions

It’s easy to see (Hahn–Banach) that such a distribution ϕ can be represented by a non-unique Borel measure µ supported on Kn, i.e., f , ϕ =

• Kn

f (ξ) dµ(ξ). For example, f , ϕ := f ′(1) = 1 2πi

• C

f (z) dz (z − 1)2 , where C is a small circle surrounding 1.

Jonathan R. Partington (Leeds, UK) Zero-one laws for functional calculus on operator semigroups

The functional calculus

We need the Fourier–Borel transform of ϕ, given by F(z) := FB(ϕ)(z) = e−z, ϕ, where e−z(ξ) = e−zξ. Thus, F(z) =

• Kn

e−zξ dµ(ξ). Then we define F(−A) = T, ϕ =

• Kn

T(ξ) dµ(ξ). as a Bochner integral, and independent of the choice of µ.

Jonathan R. Partington (Leeds, UK) Zero-one laws for functional calculus on operator semigroups

Theorem for non-quasinilpotent semigroups

Theorem (CEP 2013). Take Sα for 0 < α < π/2, and ϕ induced by a symmetric measure, i.e., µ(S) = µ(S), supported on Sβ with 0 ≤ β < α, such that

• Sα dµ(z) = 0. Let F = FB(ϕ).

If there exists δ > 0 with sup

z∈Sα−β,|z|≤δ

ρ(F(−zA)) < sup

z∈Sα−β

|F(z)|, then A/ Rad A is unital and the quotient semigroup has bounded generator. Note that a priori F(−zA) only makes sense for z ∈ Sα−β.

Jonathan R. Partington (Leeds, UK) Zero-one laws for functional calculus on operator semigroups

The case β = 0 and A semisimple

A related result holds for the case (T(t))t∈Sα semisimple (so no nontrivial quasnilpotent elements). If there exists δ > 0 with sup

0<t≤δ

F(−tA) < sup

t>0

|F(t)|, then the semigroup has a bounded generator. For example, F(t) = e−t − e−2t and the sup is 1 4.

Jonathan R. Partington (Leeds, UK) Zero-one laws for functional calculus on operator semigroups

The case C[0, 1]

Consider the “universal” example T(t) : x → xt in C[0, 1]. For F = FB(ϕ) it is easy to check that F(−tA)(x) = F(−t log x), and ρ(F(−tA)) = F(−tA) = sup

x>0

F(−t log x) = sup

r>0

|F(tr)|. Thus sup

0<t<δ

F(−tA) = sup

t>0

|F(t)|, and there is no bounded generator.

Jonathan R. Partington (Leeds, UK) Zero-one laws for functional calculus on operator semigroups

Final comments

• 1. The general analytic quasinilpotent case is harder, although the

method used to R+ works, with modifications. Again it gives a lower bound on F(−sA) for s near the origin if the semigroup is non-trivial.

• 2. Work in progress deals with multivariable functional calculus

(several complex variables) and a family of commuting semigroups. One complication here is that functions of several variables can vanish on a line, e.g. F(z1, z2) = z1 − z2.

• 3. There are many other zero-one laws. Today we have restricted
• urselves to estimates near the origin.

Jonathan R. Partington (Leeds, UK) Zero-one laws for functional calculus on operator semigroups

Very final comment

The end. Thank you.

Jonathan R. Partington (Leeds, UK) Zero-one laws for functional calculus on operator semigroups