Zero-one laws for functional calculus on operator semigroups

Transcription

1 Zero-one laws for functional calculus on operator semigroups Lancaster, September 2014 Joint work with Isabelle Chalendar (Lyon) and Jean Esterle (Bordeaux)

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

3 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.

4 An example Consider the semigroup T (t) : x x t 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.

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

6 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 ( n ) n lim sup t n T (n) (t) <, t 0+ e 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.

7 Analytic semigroups We will also look at semigroups (T (t)) t Sα, where t lies in a sector S α := {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 S α.

8 Beurling s extension theorem Beurling (1970) proved that a semigroup defined on R + has an analytic extension to some sector S α if and only if for some polynomial p. lim sup p(t (t)) < sup{ p(z) : z 1} t 0+ Kato and Neuberger (both 1970) proved that p(z) = z 1 is sufficient, giving a zero-two law for analyticity, i.e., that implies analyticity. lim sup T (t) I < 2 t 0+

9 Mokhtari s zero-quarter law Suppose that the semigroup (T (t)) t>0 is bounded at the origin; then (T (t n )) n forms a bounded approximate identity in the algebra A generated by the semigroup, whenever t n 0. Moreover, if lim sup T (t) T (2t) < 1 t 0+ 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 (t) T ((n + 1)t) < t 0+ n (n + 1) = sup x x n /n [0,1]

10 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 L 1 (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.

11 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) = e ta for some bounded A A. In general, one can deduce similar properties of A/ Rad A (quotienting out the radical).

12 Sectorial semigroups An easy-stated result for the half-plane C + : Theorem (Bendaoud-Chalendar Esterle P., 2010). If sup ρ(t (t) T ((γ + 1)t)) < 2 t C +, t <δ 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.

13 More general expressions Theorem (BCEP, 2010). Let f be a real linear combination of functions z m 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 Define k α = sup z Sα f (z). If be analytic and non-quasinilpotent. sup ρ(f ( ta)) < k α t S α, t <δ then A/ Rad A is unital and the projection of the semigoup has a bounded generator.

14 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 t n A n T (t) = t n T (n) (t) and T (t) T ((γ + 1)t). In the first case ( n ) n k α =, e cos α recovering and extending the Hille result. In the second, k α 2 as α π/2. All constants are sharp, as examples in C[0, 1] show.

15 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  is compact.

16 A curiosity about quasinilpotent semigroups First, an analytic semigroup (T (t)) t Sα bounded near the origin has an extension to S α 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 + y 2 <, then T (t) = 0 for t C + (Chalendar Esterle-P., 2010).

17 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 = 0 e λt T (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.

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

19 Examples The results will apply to examples with 0 dµ(t) = 0. For instance, take µ = δ 1 δ 2 ; then F (s) = e s e 2s and F ( ta) = T (t) T (2t). More exotic examples: or dµ(t) = (χ (1,2) χ (2,3) )(t) dt µ = δ 1 3δ 2 + δ 3 + δ 4.

20 Theorem for quasinilpotent semigroups Theorem (Chalendar Esterle P., 2013) Let µ M c (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 F (x) x 0 (0 < s η). For complex measures we define F = Lµ, so F (z) = F (z). Then F ( sa) F ( sa) > max F (x) 2 (0 x 0 < s η).

21 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 t k 0 with F ( t k A) < sup F (x), x>0 then there are idempotents P n A (i.e., P 2 n = P n ) such that n=1 P na is dense in A and each semigroup (P n T (t)) has a bounded generator.

22 Analytic semigroups For analytic semigroups on S α we can replace measures by distributions. Take H(S α ) to be the Frechet space of analytic functions on S α with topology of local uniform convergence. Now take (K n ) compact increasing, with n=1 K n = S α. Our distributions are ϕ : H(S α ) C, such that for some M > 0 and n 1. f, ϕ M sup{ f (z) : z K n }

23 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 K n, i.e., f, ϕ = f (ξ) dµ(ξ). K n For example, f, ϕ := f (1) = 1 2πi C where C is a small circle surrounding 1. f (z) dz (z 1) 2,

24 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) = e zξ dµ(ξ). K n Then we define F ( A) = T, ϕ = T (ξ) dµ(ξ). K n as a Bochner integral, and independent of the choice of µ.

25 Theorem for non-quasinilpotent semigroups Theorem (CEP 2013). Take S α for 0 < α < π/2, and ϕ induced by a symmetric measure, i.e., µ(s) = µ(s), supported on S β with 0 β < α, such that S α dµ(z) = 0. Let F = FB(ϕ). If there exists δ > 0 with sup ρ(f ( za)) < sup F (z), z S α β, z δ z S α β then A/ Rad A is unital and the quotient semigroup has bounded generator. Note that a priori F ( za) only makes sense for z S α β.

26 The case β = 0 and A semisimple A related result holds for the case (T (t)) t Sα nontrivial quasnilpotent elements). If there exists δ > 0 with semisimple (so no sup 0<t δ F ( ta) < sup F (t), t>0 then the semigroup has a bounded generator. For example, F (t) = e t e 2t and the sup is 1 4.

27 The case C[0, 1] Consider the universal example T (t) : x x t in C[0, 1]. For F = FB(ϕ) it is easy to check that and Thus F ( ta)(x) = F ( t log x), ρ(f ( ta)) = F ( ta) = sup F ( t log x) = sup F (tr). x>0 r>0 sup 0<t<δ and there is no bounded generator. F ( ta) = sup F (t), t>0

28 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 (z 1, z 2 ) = z 1 z There are many other zero-one laws. Today we have restricted ourselves to estimates near the origin.

29 Very final comment The end. Thank you.