On lower bounds for C 0 -semigroups

Size: px
Start display at page:

Download "On lower bounds for C 0 -semigroups"

Transcription

1 On lower bounds for C 0 -semigroups Yuri Tomilov IM PAN, Warsaw Chemnitz, August, 2017 Yuri Tomilov (IM PAN, Warsaw) On lower bounds for C 0 -semigroups Chemnitz, August, / 17

2 Trivial bounds For f L 1 (R) define its Fourier transform by 1 ˆf (ξ) := e itξ f (t) dt, 2π R and for f L 1 (0, 2π) define its Fourier coefficients (transform) by 1 2π ˆf (n) = e int f (t) dt, n Z. 2π 0 Yuri Tomilov (IM PAN, Warsaw) On lower bounds for C 0 -semigroups Chemnitz, August, / 17

3 Trivial bounds For f L 1 (R) define its Fourier transform by 1 ˆf (ξ) := e itξ f (t) dt, 2π R and for f L 1 (0, 2π) define its Fourier coefficients (transform) by 1 2π ˆf (n) = e int f (t) dt, n Z. 2π 0 By the Riemann-Lebesgue Lemma: ˆf C0 (R), (ˆf (n)) n Z c 0 (Z). From Plancherel s (Parseval) theorem: ˆf L 2 (R) jeśli f L 1 (R) L 2 (R), (ˆf (n)) n Z l 2 dla f L 2 (0, 2π)). Yuri Tomilov (IM PAN, Warsaw) On lower bounds for C 0 -semigroups Chemnitz, August, / 17

4 Problem. QUESTION: How large can the Fourier transform be? Yuri Tomilov (IM PAN, Warsaw) On lower bounds for C 0 -semigroups Chemnitz, August, / 17

5 Problem. QUESTION: How large can the Fourier transform be? ANSWER: The Fourier transform can be as large as possible. Yuri Tomilov (IM PAN, Warsaw) On lower bounds for C 0 -semigroups Chemnitz, August, / 17

6 Problem. QUESTION: How large can the Fourier transform be? ANSWER: The Fourier transform can be as large as possible. OUR AIM: Get the answer in the framework of weak orbits of C 0 -semigroups. Yuri Tomilov (IM PAN, Warsaw) On lower bounds for C 0 -semigroups Chemnitz, August, / 17

7 Fourier transforms of integrable functions Theorem (Kolmogorov-Titchmarsh, 1920s) 1. Given c = (c(n)) n Z c 0 (Z) there exists f L 1 (0, 2π) such that ˆf (n) c(n), n Z. Yuri Tomilov (IM PAN, Warsaw) On lower bounds for C 0 -semigroups Chemnitz, August, / 17

8 Fourier transforms of integrable functions Theorem (Kolmogorov-Titchmarsh, 1920s) 1. Given c = (c(n)) n Z c 0 (Z) there exists f L 1 (0, 2π) such that ˆf (n) c(n), n Z. 2. Given c C 0 (R) there exists f L 1 (R) such that ˆf (ξ) c(ξ), ξ R. Yuri Tomilov (IM PAN, Warsaw) On lower bounds for C 0 -semigroups Chemnitz, August, / 17

9 Fourier transforms of integrable functions Theorem (Kolmogorov-Titchmarsh, 1920s) 1. Given c = (c(n)) n Z c 0 (Z) there exists f L 1 (0, 2π) such that ˆf (n) c(n), n Z. 2. Given c C 0 (R) there exists f L 1 (R) such that ˆf (ξ) c(ξ), ξ R. Generalization[Curtis, Figa-Talamanca, 1966]: The same result is true in the context of locally compact abelian groups. Yuri Tomilov (IM PAN, Warsaw) On lower bounds for C 0 -semigroups Chemnitz, August, / 17

10 Fourier transforms of continuous functions Remark. The Fourier transform is an isometric isomorphism on L 2 (R) (or on L 2 (0, 2π)). Yuri Tomilov (IM PAN, Warsaw) On lower bounds for C 0 -semigroups Chemnitz, August, / 17

11 Fourier transforms of continuous functions Remark. The Fourier transform is an isometric isomorphism on L 2 (R) (or on L 2 (0, 2π)). Theorem (de-leeuw-kahane-katznelson-d ly, ) 1. Given {c n } n Z l 2 (Z) there exists a 2π-periodic function f C([0, 2π]) such that ˆf (n) c n, n Z. Yuri Tomilov (IM PAN, Warsaw) On lower bounds for C 0 -semigroups Chemnitz, August, / 17

12 Fourier transforms of continuous functions Remark. The Fourier transform is an isometric isomorphism on L 2 (R) (or on L 2 (0, 2π)). Theorem (de-leeuw-kahane-katznelson-d ly, ) 1. Given {c n } n Z l 2 (Z) there exists a 2π-periodic function f C([0, 2π]) such that ˆf (n) c n, n Z. 2. Given c L 2 (R) there exists a function f L 2 (R) C 0 (R) such that Many other settings!!! ˆf (ξ) c(ξ) for almost every ξ. Yuri Tomilov (IM PAN, Warsaw) On lower bounds for C 0 -semigroups Chemnitz, August, / 17

13 Abstract setting Theorem (K. Ball, Inventiones M. 1991, BLMS 1994) 1. If {x n : n Z} is a sequence of bounded linear functionals of norm 1 on a Banach space X and a = (a n ) n Z l 1 (Z), a l1 < 1, then there exists x X, x 1, such that x n, x a n, n Z. Yuri Tomilov (IM PAN, Warsaw) On lower bounds for C 0 -semigroups Chemnitz, August, / 17

14 Abstract setting Theorem (K. Ball, Inventiones M. 1991, BLMS 1994) 1. If {x n : n Z} is a sequence of bounded linear functionals of norm 1 on a Banach space X and a = (a n ) n Z l 1 (Z), a l1 < 1, then there exists x X, x 1, such that x n, x a n, n Z. 2. If {x n : n Z} is a sequence of elements of norm 1 in a Hilbert space H and (a n ) n Z l 2 (Z), a l2 < 1, then there exists x H, x 1, such that (x n, x) a n, n Z. Yuri Tomilov (IM PAN, Warsaw) On lower bounds for C 0 -semigroups Chemnitz, August, / 17

15 Implication for the Fourier transform: Define the bounded linear functionals xn, n Z, on L 1 (R) by xn(x) := e int x(t) dt, x L 1 (R). R Yuri Tomilov (IM PAN, Warsaw) On lower bounds for C 0 -semigroups Chemnitz, August, / 17

16 Implication for the Fourier transform: Define the bounded linear functionals xn, n Z, on L 1 (R) by xn(x) := e int x(t) dt, x L 1 (R). R Then x n = 1, n Z, and for every a = (a n ) n Z l 1 (Z), a l1 < 1, there exists x L 1 (R), x L 1 (R) 1, such that x n(x) a n, n Z. Yuri Tomilov (IM PAN, Warsaw) On lower bounds for C 0 -semigroups Chemnitz, August, / 17

17 Implication for the Fourier transform: Define the bounded linear functionals xn, n Z, on L 1 (R) by xn(x) := e int x(t) dt, x L 1 (R). R Then x n = 1, n Z, and for every a = (a n ) n Z l 1 (Z), a l1 < 1, there exists x L 1 (R), x L 1 (R) 1, such that x n(x) a n, n Z. This is still far from the result by Kolmogorov and Titchmarsh where (a n ) n Z c 0 (Z)! Yuri Tomilov (IM PAN, Warsaw) On lower bounds for C 0 -semigroups Chemnitz, August, / 17

18 The Fourier transform via operator (semi-)groups Let (U(t)) t R L(L 2 (R)) be a family of unitary operators on L 2 (R) defined by (U(t)f )(s) = e its f (s), s R Yuri Tomilov (IM PAN, Warsaw) On lower bounds for C 0 -semigroups Chemnitz, August, / 17

19 The Fourier transform via operator (semi-)groups Let (U(t)) t R L(L 2 (R)) be a family of unitary operators on L 2 (R) defined by (U(t)f )(s) = e its f (s), s R Observe that for fixed f, g L 2 (R) : (U(t)f, g) = e its f (t)g(t) dt = e its ϕ(t) dt, R R ϕ := f ḡ L 1 (R). (U(t)f, g) = (U( t)f, g), t 0. it is enough to study bounds for one-sided weak orbits of (U(t)) t R. NOTE: (U(t)) t R is a strongly continuous operator (semi-)group. Yuri Tomilov (IM PAN, Warsaw) On lower bounds for C 0 -semigroups Chemnitz, August, / 17

20 The Fourier transform via operator (semi-)groups Let (U(t)) t R L(L 2 (R)) be a family of unitary operators on L 2 (R) defined by (U(t)f )(s) = e its f (s), s R Observe that for fixed f, g L 2 (R) : (U(t)f, g) = e its f (t)g(t) dt = e its ϕ(t) dt, R R ϕ := f ḡ L 1 (R). (U(t)f, g) = (U( t)f, g), t 0. it is enough to study bounds for one-sided weak orbits of (U(t)) t R. NOTE: (U(t)) t R is a strongly continuous operator (semi-)group. NOTE: The weak orbit (U(t)f, g) of (U(t)) t R is the Fourier transform of the L 1 (R)-function f ḡ. Yuri Tomilov (IM PAN, Warsaw) On lower bounds for C 0 -semigroups Chemnitz, August, / 17

21 A bit of theory: Yuri Tomilov (IM PAN, Warsaw) On lower bounds for C 0 -semigroups Chemnitz, August, / 17

22 A bit of theory: For a C 0 -sem. (T (t)) t 0 on a Ban. space X with generator A define s(a) := sup{re λ : λ σ(a)} s b (A) := inf{ω R : R(λ, A) is uniformly bounded for Re λ ω} ω 0 := ln T (t) lim sup t t Clearly, s(a) s b (A) ω 0. Yuri Tomilov (IM PAN, Warsaw) On lower bounds for C 0 -semigroups Chemnitz, August, / 17

23 A bit of theory: For a C 0 -sem. (T (t)) t 0 on a Ban. space X with generator A define s(a) := sup{re λ : λ σ(a)} s b (A) := inf{ω R : R(λ, A) is uniformly bounded for Re λ ω} ω 0 := ln T (t) lim sup t t Clearly, s(a) s b (A) ω 0. Main Problem: Lack of the spectral mapping theorem for (T (t)) t 0 Yuri Tomilov (IM PAN, Warsaw) On lower bounds for C 0 -semigroups Chemnitz, August, / 17

24 A bit of theory: For a C 0 -sem. (T (t)) t 0 on a Ban. space X with generator A define s(a) := sup{re λ : λ σ(a)} s b (A) := inf{ω R : R(λ, A) is uniformly bounded for Re λ ω} ω 0 := ln T (t) lim sup t t Clearly, s(a) s b (A) ω 0. Main Problem: Lack of the spectral mapping theorem for (T (t)) t 0 For any a b c there exist a reflexive Banach space X and a C 0 -semigroup (T (t)) t 0 on X : s(a) = a, s b (A) = b, w 0 = c. Yuri Tomilov (IM PAN, Warsaw) On lower bounds for C 0 -semigroups Chemnitz, August, / 17

25 A bit of theory: For a C 0 -sem. (T (t)) t 0 on a Ban. space X with generator A define s(a) := sup{re λ : λ σ(a)} s b (A) := inf{ω R : R(λ, A) is uniformly bounded for Re λ ω} ω 0 := ln T (t) lim sup t t Clearly, s(a) s b (A) ω 0. Main Problem: Lack of the spectral mapping theorem for (T (t)) t 0 For any a b c there exist a reflexive Banach space X and a C 0 -semigroup (T (t)) t 0 on X : s(a) = a, s b (A) = b, w 0 = c. If X is Hilbert, then ω 0 = s b. Yuri Tomilov (IM PAN, Warsaw) On lower bounds for C 0 -semigroups Chemnitz, August, / 17

26 What can we say about the size of weak orbits for C 0 -semigroups (T (t)) t 0? (T (t)x, x ), t 0, x X, x X, Yuri Tomilov (IM PAN, Warsaw) On lower bounds for C 0 -semigroups Chemnitz, August, / 17

27 What can we say about the size of weak orbits for C 0 -semigroups (T (t)) t 0? (T (t)x, x ), t 0, x X, x X, History:... essentially, no history exists, apart from sporadic papers... Semigroups framework: van Neerven, Weiss, Nikolski,... Operator framework: Beauzamy, Müller, Radjavi, Rosenthal, Nordgren,... Yuri Tomilov (IM PAN, Warsaw) On lower bounds for C 0 -semigroups Chemnitz, August, / 17

28 What can we say about the size of weak orbits for C 0 -semigroups (T (t)) t 0? (T (t)x, x ), t 0, x X, x X, History:... essentially, no history exists, apart from sporadic papers... Semigroups framework: van Neerven, Weiss, Nikolski,... Operator framework: Beauzamy, Müller, Radjavi, Rosenthal, Nordgren,... A bit more notation: Let A : D(A) X X be a densely defined closed operator with ρ(a). Define C (A) := D(A n ) and note C (A) = X. n=1 Yuri Tomilov (IM PAN, Warsaw) On lower bounds for C 0 -semigroups Chemnitz, August, / 17

29 Decay of weak orbits, spectral conditions Theorem (Müller-T.) Let (T (t) t 0 be a weakly stable C 0 -semigroup a Hilbert space H with generator A (T (t) 0, t in WOT). Yuri Tomilov (IM PAN, Warsaw) On lower bounds for C 0 -semigroups Chemnitz, August, / 17

30 Decay of weak orbits, spectral conditions Theorem (Müller-T.) Let (T (t) t 0 be a weakly stable C 0 -semigroup a Hilbert space H with generator A (T (t) 0, t in WOT). Let f : [0, ) (0, ) be a bounded function such that lim t f (t) = 0 and let ɛ > 0. Yuri Tomilov (IM PAN, Warsaw) On lower bounds for C 0 -semigroups Chemnitz, August, / 17

31 Decay of weak orbits, spectral conditions Theorem (Müller-T.) Let (T (t) t 0 be a weakly stable C 0 -semigroup a Hilbert space H with generator A (T (t) 0, t in WOT). Let f : [0, ) (0, ) be a bounded function such that lim t f (t) = 0 and let ɛ > 0. a) Suppose that 0 σ(a). Then there exists x C (A) such that x < sup{f (t) : t 0} + ɛ and for all t 0. Re T (t)x, x f (t) Yuri Tomilov (IM PAN, Warsaw) On lower bounds for C 0 -semigroups Chemnitz, August, / 17

32 Decay of weak orbits, spectral conditions Theorem (Müller-T.) Let (T (t) t 0 be a weakly stable C 0 -semigroup a Hilbert space H with generator A (T (t) 0, t in WOT). Let f : [0, ) (0, ) be a bounded function such that lim t f (t) = 0 and let ɛ > 0. a) Suppose that 0 σ(a). Then there exists x C (A) such that x < sup{f (t) : t 0} + ɛ and for all t 0. Re T (t)x, x f (t) b) Suppose that σ(a) ir. Then there exists x C (A) such that x < sup{f (t) : t 0} + ɛ and for all t 0. T (t)x, x f (t) Yuri Tomilov (IM PAN, Warsaw) On lower bounds for C 0 -semigroups Chemnitz, August, / 17

33 Decay of weak orbits, resolvent conditions Theorem (Müller-T, 13) Let (T (t) t 0 be a weakly stable C 0 -semigroup on a Hilbert space H with generator A. Yuri Tomilov (IM PAN, Warsaw) On lower bounds for C 0 -semigroups Chemnitz, August, / 17

34 Decay of weak orbits, resolvent conditions Theorem (Müller-T, 13) Let (T (t) t 0 be a weakly stable C 0 -semigroup on a Hilbert space H with generator A. Suppose that ω 0 (= s b ) = 0. Let f : [0, ) (0, ) be a bounded function such that lim t f (t) = 0 and let ɛ > 0. Yuri Tomilov (IM PAN, Warsaw) On lower bounds for C 0 -semigroups Chemnitz, August, / 17

35 Decay of weak orbits, resolvent conditions Theorem (Müller-T, 13) Let (T (t) t 0 be a weakly stable C 0 -semigroup on a Hilbert space H with generator A. Suppose that ω 0 (= s b ) = 0. Let f : [0, ) (0, ) be a bounded function such that lim t f (t) = 0 and let ɛ > 0. There exists x H such that x < sup{f (s) : s 0} + ɛ and for all t 0. T (t)x, x f (t) Yuri Tomilov (IM PAN, Warsaw) On lower bounds for C 0 -semigroups Chemnitz, August, / 17

36 Decay of weak orbits, resolvent conditions Theorem (Müller-T, 13) Let (T (t) t 0 be a weakly stable C 0 -semigroup on a Hilbert space H with generator A. Suppose that ω 0 (= s b ) = 0. Let f : [0, ) (0, ) be a bounded function such that lim t f (t) = 0 and let ɛ > 0. There exists x H such that x < sup{f (s) : s 0} + ɛ and for all t 0. T (t)x, x f (t) Weak stability of (T (t)) t 0 is essential! Yuri Tomilov (IM PAN, Warsaw) On lower bounds for C 0 -semigroups Chemnitz, August, / 17

37 For S R + define its density d(s) as d(s) = lim t mes(s [0, t]) t whenever the limit exists. Yuri Tomilov (IM PAN, Warsaw) On lower bounds for C 0 -semigroups Chemnitz, August, / 17

38 For S R + define its density d(s) as d(s) = lim t mes(s [0, t]) t whenever the limit exists. Theorem (Müller-T, 13) Let (T (t) t 0 be a bdd C 0 -semigroup a Hilbert space H with generator A. Yuri Tomilov (IM PAN, Warsaw) On lower bounds for C 0 -semigroups Chemnitz, August, / 17

39 For S R + define its density d(s) as d(s) = lim t mes(s [0, t]) t whenever the limit exists. Theorem (Müller-T, 13) Let (T (t) t 0 be a bdd C 0 -semigroup a Hilbert space H with generator A. Let f : [0, ) (0, ) be a bounded function such that lim t f (t) = 0 and let ɛ > 0. Yuri Tomilov (IM PAN, Warsaw) On lower bounds for C 0 -semigroups Chemnitz, August, / 17

40 For S R + define its density d(s) as d(s) = lim t mes(s [0, t]) t whenever the limit exists. Theorem (Müller-T, 13) Let (T (t) t 0 be a bdd C 0 -semigroup a Hilbert space H with generator A. Let f : [0, ) (0, ) be a bounded function such that lim t f (t) = 0 and let ɛ > 0. a) Suppose that 0 σ(a). Then there exists x C (A) and a measurable B R +, d(b) = 1, such that x < sup{f (t) : t 0} + ɛ and Re T (t)x, x f (t), t B. Yuri Tomilov (IM PAN, Warsaw) On lower bounds for C 0 -semigroups Chemnitz, August, / 17

41 b) Suppose that σ(a) ir. Then there exists x C (A) and a measurable B R +, d(b) = 1, such that x < sup{f (t) : t 0} + ɛ and Yuri Tomilov (IM PAN, Warsaw) T (t)x, On lower x bounds f for (t), C 0 -semigroups t B. Chemnitz, August, / 17 For S R + define its density d(s) as d(s) = lim t mes(s [0, t]) t whenever the limit exists. Theorem (Müller-T, 13) Let (T (t) t 0 be a bdd C 0 -semigroup a Hilbert space H with generator A. Let f : [0, ) (0, ) be a bounded function such that lim t f (t) = 0 and let ɛ > 0. a) Suppose that 0 σ(a). Then there exists x C (A) and a measurable B R +, d(b) = 1, such that x < sup{f (t) : t 0} + ɛ and Re T (t)x, x f (t), t B.

42 Applications to function theory: back to Kolmogorov-Titchmarsh Let µ be a finite (positive) Borel measure on R such that ˆµ C 0 (R). Yuri Tomilov (IM PAN, Warsaw) On lower bounds for C 0 -semigroups Chemnitz, August, / 17

43 Applications to function theory: back to Kolmogorov-Titchmarsh Let µ be a finite (positive) Borel measure on R such that ˆµ C 0 (R). Theorem Given a bounded f : R [0, ) satisfying lim t f (t) = 0 there exist a positive g L 1 (R, dµ) : Yuri Tomilov (IM PAN, Warsaw) On lower bounds for C 0 -semigroups Chemnitz, August, / 17

44 Applications to function theory: back to Kolmogorov-Titchmarsh Let µ be a finite (positive) Borel measure on R such that ˆµ C 0 (R). Theorem Given a bounded f : R [0, ) satisfying lim t f (t) = 0 there exist a positive g L 1 (R, dµ) : (i) ĝ C (R); Yuri Tomilov (IM PAN, Warsaw) On lower bounds for C 0 -semigroups Chemnitz, August, / 17

45 Applications to function theory: back to Kolmogorov-Titchmarsh Let µ be a finite (positive) Borel measure on R such that ˆµ C 0 (R). Theorem Given a bounded f : R [0, ) satisfying lim t f (t) = 0 there exist a positive g L 1 (R, dµ) : (i) ĝ C (R); (ii) ĝ(t) f (t), t R. Yuri Tomilov (IM PAN, Warsaw) On lower bounds for C 0 -semigroups Chemnitz, August, / 17

46 The general case: Let µ be a finite (positive) Borel measure on R. Yuri Tomilov (IM PAN, Warsaw) On lower bounds for C 0 -semigroups Chemnitz, August, / 17

47 The general case: Let µ be a finite (positive) Borel measure on R. Theorem Given a bounded f : R [0, ) satisfying lim t f (t) = 0 there exist a positive g L 1 (R, dµ) and a measurable B R, Dens (B) = 1 : Yuri Tomilov (IM PAN, Warsaw) On lower bounds for C 0 -semigroups Chemnitz, August, / 17

48 The general case: Let µ be a finite (positive) Borel measure on R. Theorem Given a bounded f : R [0, ) satisfying lim t f (t) = 0 there exist a positive g L 1 (R, dµ) and a measurable B R, Dens (B) = 1 : (i) ĝ C (R); Yuri Tomilov (IM PAN, Warsaw) On lower bounds for C 0 -semigroups Chemnitz, August, / 17

49 The general case: Let µ be a finite (positive) Borel measure on R. Theorem Given a bounded f : R [0, ) satisfying lim t f (t) = 0 there exist a positive g L 1 (R, dµ) and a measurable B R, Dens (B) = 1 : (i) ĝ C (R); (ii) ĝ(t) f (t), t B. Recall, that by Wiener s theorem, if µ has no atoms, then µ(t) 0, when t along B with dens(b) = 1. Yuri Tomilov (IM PAN, Warsaw) On lower bounds for C 0 -semigroups Chemnitz, August, / 17

50 What if the boundedness of (T (t)) t 0 is dropped? Yuri Tomilov (IM PAN, Warsaw) On lower bounds for C 0 -semigroups Chemnitz, August, / 17

51 What if the boundedness of (T (t)) t 0 is dropped? For S R + define its upper density d(s) as d(s) = lim sup t mes(s [0, t]). t Yuri Tomilov (IM PAN, Warsaw) On lower bounds for C 0 -semigroups Chemnitz, August, / 17

52 What if the boundedness of (T (t)) t 0 is dropped? For S R + define its upper density d(s) as d(s) = lim sup t mes(s [0, t]). t Theorem (Müller-T., 17) Let (T (t)) t 0 be a C 0 -semigroup on a Banach space X with generator A, such that s b (A) 0. Let f : [0, ) (0, ) be a bounded function such that lim t f (t) = 0. Then there exist x X, x X and a measurable B [0, ) such that Dens B = 1 and T (t)x, x f (t), t B. Yuri Tomilov (IM PAN, Warsaw) On lower bounds for C 0 -semigroups Chemnitz, August, / 17

53 What if the boundedness of (T (t)) t 0 is dropped? For S R + define its upper density d(s) as d(s) = lim sup t mes(s [0, t]). t Theorem (Müller-T., 17) Let (T (t)) t 0 be a C 0 -semigroup on a Banach space X with generator A, such that s b (A) 0. Let f : [0, ) (0, ) be a bounded function such that lim t f (t) = 0. Then there exist x X, x X and a measurable B [0, ) such that Dens B = 1 and T (t)x, x f (t), t B. Lemma. Let A, B be finite sets, A B 2. Let M A B, M A + B 1. Then there exist a, a A, b, b B such that a a, b b and (a, b), (a, b ), (a, b) M. Yuri Tomilov (IM PAN, Warsaw) On lower bounds for C 0 -semigroups Chemnitz, August, / 17

54 Theorem (Müller-T., 17) Let (T (t)) t 0 be a C 0 -semigroup on a reflexive Banach space X with generator A, such that s b (A) 0. Let ɛ > 0 be fixed. Then (i) there exist x X, x X and a measurable B [1, ) with Dens B = 1 such that T (t)x, x 1, t B. t1/2+ɛ Yuri Tomilov (IM PAN, Warsaw) On lower bounds for C 0 -semigroups Chemnitz, August, / 17

55 Theorem (Müller-T., 17) Let (T (t)) t 0 be a C 0 -semigroup on a reflexive Banach space X with generator A, such that s b (A) 0. Let ɛ > 0 be fixed. Then (i) there exist x X, x X and a measurable B [1, ) with Dens B = 1 such that T (t)x, x 1, t B. t1/2+ɛ (ii) there exist x X, x X and a measurable set B [1, ) with meas([1, ) \ B) < ɛ such that T (t)x, x 1, t B. t2+ɛ Yuri Tomilov (IM PAN, Warsaw) On lower bounds for C 0 -semigroups Chemnitz, August, / 17

56 Theorem (Müller-T., 17) Let (T (t)) t 0 be a C 0 -semigroup on a reflexive Banach space X with generator A, such that s b (A) 0. Let ɛ > 0 be fixed. Then (i) there exist x X, x X and a measurable B [1, ) with Dens B = 1 such that T (t)x, x 1, t B. t1/2+ɛ (ii) there exist x X, x X and a measurable set B [1, ) with meas([1, ) \ B) < ɛ such that T (t)x, x 1, t B. t2+ɛ Optimality is widely open... Yuri Tomilov (IM PAN, Warsaw) On lower bounds for C 0 -semigroups Chemnitz, August, / 17

Analysis Preliminary Exam Workshop: Hilbert Spaces

Analysis Preliminary Exam Workshop: Hilbert Spaces Analysis Preliminary Exam Workshop: Hilbert Spaces 1. Hilbert spaces A Hilbert space H is a complete real or complex inner product space. Consider complex Hilbert spaces for definiteness. If (, ) : H H

More information

16 1 Basic Facts from Functional Analysis and Banach Lattices

16 1 Basic Facts from Functional Analysis and Banach Lattices 16 1 Basic Facts from Functional Analysis and Banach Lattices 1.2.3 Banach Steinhaus Theorem Another fundamental theorem of functional analysis is the Banach Steinhaus theorem, or the Uniform Boundedness

More information

FUNCTIONAL ANALYSIS LECTURE NOTES: WEAK AND WEAK* CONVERGENCE

FUNCTIONAL ANALYSIS LECTURE NOTES: WEAK AND WEAK* CONVERGENCE FUNCTIONAL ANALYSIS LECTURE NOTES: WEAK AND WEAK* CONVERGENCE CHRISTOPHER HEIL 1. Weak and Weak* Convergence of Vectors Definition 1.1. Let X be a normed linear space, and let x n, x X. a. We say that

More information

Laplace Transforms, Non-Analytic Growth Bounds and C 0 -Semigroups

Laplace Transforms, Non-Analytic Growth Bounds and C 0 -Semigroups Laplace Transforms, Non-Analytic Growth Bounds and C -Semigroups Sachi Srivastava St. John s College University of Oxford A thesis submitted for the degree of Doctor of Philosophy Hilary 22 Laplace Transforms,

More information

Exponential stability of operators and operator semigroups

Exponential stability of operators and operator semigroups Exponential stability of operators and operator semigroups 1 J.M.A.M. van Neerven California Institute of Technology Alfred P. Sloan Laboratory of Mathematics Pasadena, CA 91125, U.S.A. Extending earlier

More information

MAT 578 FUNCTIONAL ANALYSIS EXERCISES

MAT 578 FUNCTIONAL ANALYSIS EXERCISES MAT 578 FUNCTIONAL ANALYSIS EXERCISES JOHN QUIGG Exercise 1. Prove that if A is bounded in a topological vector space, then for every neighborhood V of 0 there exists c > 0 such that tv A for all t > c.

More information

Fine scales of decay rates of operator semigroups

Fine scales of decay rates of operator semigroups Operator Semigroups meet everything else Herrnhut, 5 June 2013 A damped wave equation 2 u u + 2a(x) u t2 t = 0 (t > 0, x Ω) u(x, t) = 0 (t > 0, x Ω) u(, 0) = u 0 H0 1 (Ω), u t (, 0) = u 1 L 2 (Ω). Here,

More information

Chapter 6 Integral Transform Functional Calculi

Chapter 6 Integral Transform Functional Calculi Chapter 6 Integral Transform Functional Calculi In this chapter we continue our investigations from the previous one and encounter functional calculi associated with various semigroup representations.

More information

PROBLEMS. (b) (Polarization Identity) Show that in any inner product space

PROBLEMS. (b) (Polarization Identity) Show that in any inner product space 1 Professor Carl Cowen Math 54600 Fall 09 PROBLEMS 1. (Geometry in Inner Product Spaces) (a) (Parallelogram Law) Show that in any inner product space x + y 2 + x y 2 = 2( x 2 + y 2 ). (b) (Polarization

More information

Functional Analysis F3/F4/NVP (2005) Homework assignment 3

Functional Analysis F3/F4/NVP (2005) Homework assignment 3 Functional Analysis F3/F4/NVP (005 Homework assignment 3 All students should solve the following problems: 1. Section 4.8: Problem 8.. Section 4.9: Problem 4. 3. Let T : l l be the operator defined by

More information

A Concise Course on Stochastic Partial Differential Equations

A Concise Course on Stochastic Partial Differential Equations A Concise Course on Stochastic Partial Differential Equations Michael Röckner Reference: C. Prevot, M. Röckner: Springer LN in Math. 1905, Berlin (2007) And see the references therein for the original

More information

Uniform polynomial stability of C 0 -Semigroups

Uniform polynomial stability of C 0 -Semigroups Uniform polynomial stability of C 0 -Semigroups LMDP - UMMISCO Departement of Mathematics Cadi Ayyad University Faculty of Sciences Semlalia Marrakech 14 February 2012 Outline 1 2 Uniform polynomial stability

More information

MATH MEASURE THEORY AND FOURIER ANALYSIS. Contents

MATH MEASURE THEORY AND FOURIER ANALYSIS. Contents MATH 3969 - MEASURE THEORY AND FOURIER ANALYSIS ANDREW TULLOCH Contents 1. Measure Theory 2 1.1. Properties of Measures 3 1.2. Constructing σ-algebras and measures 3 1.3. Properties of the Lebesgue measure

More information

1 Math 241A-B Homework Problem List for F2015 and W2016

1 Math 241A-B Homework Problem List for F2015 and W2016 1 Math 241A-B Homework Problem List for F2015 W2016 1.1 Homework 1. Due Wednesday, October 7, 2015 Notation 1.1 Let U be any set, g be a positive function on U, Y be a normed space. For any f : U Y let

More information

Holomorphic functions which preserve holomorphic semigroups

Holomorphic functions which preserve holomorphic semigroups Holomorphic functions which preserve holomorphic semigroups University of Oxford London Mathematical Society Regional Meeting Birmingham, 15 September 2016 Heat equation u t = xu (x Ω R d, t 0), u(t, x)

More information

Left invertible semigroups on Hilbert spaces.

Left invertible semigroups on Hilbert spaces. Left invertible semigroups on Hilbert spaces. Hans Zwart Department of Applied Mathematics, Faculty of Electrical Engineering, Mathematics and Computer Science, University of Twente, P.O. Box 217, 75 AE

More information

Functional Analysis I

Functional Analysis I Functional Analysis I Course Notes by Stefan Richter Transcribed and Annotated by Gregory Zitelli Polar Decomposition Definition. An operator W B(H) is called a partial isometry if W x = X for all x (ker

More information

Almost periodic functionals

Almost periodic functionals Almost periodic functionals Matthew Daws Leeds Warsaw, July 2013 Matthew Daws (Leeds) Almost periodic functionals Warsaw, July 2013 1 / 22 Dual Banach algebras; personal history A Banach algebra A Banach

More information

Overview of normed linear spaces

Overview of normed linear spaces 20 Chapter 2 Overview of normed linear spaces Starting from this chapter, we begin examining linear spaces with at least one extra structure (topology or geometry). We assume linearity; this is a natural

More information

Stabilization of Distributed Parameter Systems by State Feedback with Positivity Constraints

Stabilization of Distributed Parameter Systems by State Feedback with Positivity Constraints Stabilization of Distributed Parameter Systems by State Feedback with Positivity Constraints Joseph Winkin Namur Center of Complex Systems (naxys) and Dept. of Mathematics, University of Namur, Belgium

More information

Spectral Measures, the Spectral Theorem, and Ergodic Theory

Spectral Measures, the Spectral Theorem, and Ergodic Theory Spectral Measures, the Spectral Theorem, and Ergodic Theory Sam Ziegler The spectral theorem for unitary operators The presentation given here largely follows [4]. will refer to the unit circle throughout.

More information

Notions such as convergent sequence and Cauchy sequence make sense for any metric space. Convergent Sequences are Cauchy

Notions such as convergent sequence and Cauchy sequence make sense for any metric space. Convergent Sequences are Cauchy Banach Spaces These notes provide an introduction to Banach spaces, which are complete normed vector spaces. For the purposes of these notes, all vector spaces are assumed to be over the real numbers.

More information

Spectral theory for compact operators on Banach spaces

Spectral theory for compact operators on Banach spaces 68 Chapter 9 Spectral theory for compact operators on Banach spaces Recall that a subset S of a metric space X is precompact if its closure is compact, or equivalently every sequence contains a Cauchy

More information

l(y j ) = 0 for all y j (1)

l(y j ) = 0 for all y j (1) Problem 1. The closed linear span of a subset {y j } of a normed vector space is defined as the intersection of all closed subspaces containing all y j and thus the smallest such subspace. 1 Show that

More information

Bernstein s inequality and Nikolsky s inequality for R d

Bernstein s inequality and Nikolsky s inequality for R d Bernstein s inequality and Nikolsky s inequality for d Jordan Bell jordan.bell@gmail.com Department of athematics University of Toronto February 6 25 Complex Borel measures and the Fourier transform Let

More information

REGULARIZATION AND FREQUENCY DOMAIN STABILITY OF WELL POSED SYSTEMS

REGULARIZATION AND FREQUENCY DOMAIN STABILITY OF WELL POSED SYSTEMS REGULARIZATION AND FREQUENCY DOMAIN STABILITY OF WELL POSED SYSTEMS YURI LATUSHKIN, TIMOTHY RANDOLPH, AND ROLAND SCHNAUBELT Abstract. We study linear control systems with unbounded control and observation

More information

The Wiener algebra and Wiener s lemma

The Wiener algebra and Wiener s lemma he Wiener algebra and Wiener s lemma Jordan Bell jordan.bell@gmail.com Department of Mathematics, University of oronto January 17, 2015 1 Introduction Let = R/Z. For f L 1 () we define f L 1 () = 1 For

More information

3. (a) What is a simple function? What is an integrable function? How is f dµ defined? Define it first

3. (a) What is a simple function? What is an integrable function? How is f dµ defined? Define it first Math 632/6321: Theory of Functions of a Real Variable Sample Preinary Exam Questions 1. Let (, M, µ) be a measure space. (a) Prove that if µ() < and if 1 p < q

More information

We denote the space of distributions on Ω by D ( Ω) 2.

We denote the space of distributions on Ω by D ( Ω) 2. Sep. 1 0, 008 Distributions Distributions are generalized functions. Some familiarity with the theory of distributions helps understanding of various function spaces which play important roles in the study

More information

Chapter 4 Optimal Control Problems in Infinite Dimensional Function Space

Chapter 4 Optimal Control Problems in Infinite Dimensional Function Space Chapter 4 Optimal Control Problems in Infinite Dimensional Function Space 4.1 Introduction In this chapter, we will consider optimal control problems in function space where we will restrict ourselves

More information

Functional Analysis, Stein-Shakarchi Chapter 1

Functional Analysis, Stein-Shakarchi Chapter 1 Functional Analysis, Stein-Shakarchi Chapter 1 L p spaces and Banach Spaces Yung-Hsiang Huang 018.05.1 Abstract Many problems are cited to my solution files for Folland [4] and Rudin [6] post here. 1 Exercises

More information

Corollary A linear operator A is the generator of a C 0 (G(t)) t 0 satisfying G(t) e ωt if and only if (i) A is closed and D(A) = X;

Corollary A linear operator A is the generator of a C 0 (G(t)) t 0 satisfying G(t) e ωt if and only if (i) A is closed and D(A) = X; 2.2 Rudiments 71 Corollary 2.12. A linear operator A is the generator of a C 0 (G(t)) t 0 satisfying G(t) e ωt if and only if (i) A is closed and D(A) = X; (ii) ρ(a) (ω, ) and for such λ semigroup R(λ,

More information

Chapter 6 Integral Transform Functional Calculi

Chapter 6 Integral Transform Functional Calculi Chapter 6 Integral Transform Functional Calculi In this chapter we continue our investigations from the previous one and encounter functional calculi associated with various semigroup representations.

More information

OBSERVATION OPERATORS FOR EVOLUTIONARY INTEGRAL EQUATIONS

OBSERVATION OPERATORS FOR EVOLUTIONARY INTEGRAL EQUATIONS OBSERVATION OPERATORS FOR EVOLUTIONARY INTEGRAL EQUATIONS MICHAEL JUNG Abstract. We analyze admissibility and exactness of observation operators arising in control theory for Volterra integral equations.

More information

Finite-dimensional spaces. C n is the space of n-tuples x = (x 1,..., x n ) of complex numbers. It is a Hilbert space with the inner product

Finite-dimensional spaces. C n is the space of n-tuples x = (x 1,..., x n ) of complex numbers. It is a Hilbert space with the inner product Chapter 4 Hilbert Spaces 4.1 Inner Product Spaces Inner Product Space. A complex vector space E is called an inner product space (or a pre-hilbert space, or a unitary space) if there is a mapping (, )

More information

Errata Applied Analysis

Errata Applied Analysis Errata Applied Analysis p. 9: line 2 from the bottom: 2 instead of 2. p. 10: Last sentence should read: The lim sup of a sequence whose terms are bounded from above is finite or, and the lim inf of a sequence

More information

SPECTRAL THEOREM FOR SYMMETRIC OPERATORS WITH COMPACT RESOLVENT

SPECTRAL THEOREM FOR SYMMETRIC OPERATORS WITH COMPACT RESOLVENT SPECTRAL THEOREM FOR SYMMETRIC OPERATORS WITH COMPACT RESOLVENT Abstract. These are the letcure notes prepared for the workshop on Functional Analysis and Operator Algebras to be held at NIT-Karnataka,

More information

An Operator Theoretical Approach to Nonlocal Differential Equations

An Operator Theoretical Approach to Nonlocal Differential Equations An Operator Theoretical Approach to Nonlocal Differential Equations Joshua Lee Padgett Department of Mathematics and Statistics Texas Tech University Analysis Seminar November 27, 2017 Joshua Lee Padgett

More information

Patryk Pagacz. Characterization of strong stability of power-bounded operators. Uniwersytet Jagielloński

Patryk Pagacz. Characterization of strong stability of power-bounded operators. Uniwersytet Jagielloński Patryk Pagacz Uniwersytet Jagielloński Characterization of strong stability of power-bounded operators Praca semestralna nr 3 (semestr zimowy 2011/12) Opiekun pracy: Jaroslav Zemanek CHARACTERIZATION OF

More information

Bochner s Theorem on the Fourier Transform on R

Bochner s Theorem on the Fourier Transform on R Bochner s heorem on the Fourier ransform on Yitao Lei October 203 Introduction ypically, the Fourier transformation sends suitable functions on to functions on. his can be defined on the space L ) + L

More information

Lecture Notes in Functional Analysis

Lecture Notes in Functional Analysis Lecture Notes in Functional Analysis Please report any mistakes, misprints or comments to aulikowska@mimuw.edu.pl LECTURE 1 Banach spaces 1.1. Introducton to Banach Spaces Definition 1.1. Let X be a K

More information

Harmonic Analysis on the Cube and Parseval s Identity

Harmonic Analysis on the Cube and Parseval s Identity Lecture 3 Harmonic Analysis on the Cube and Parseval s Identity Jan 28, 2005 Lecturer: Nati Linial Notes: Pete Couperus and Neva Cherniavsky 3. Where we can use this During the past weeks, we developed

More information

ALMOST PERIODIC SOLUTIONS OF HIGHER ORDER DIFFERENTIAL EQUATIONS ON HILBERT SPACES

ALMOST PERIODIC SOLUTIONS OF HIGHER ORDER DIFFERENTIAL EQUATIONS ON HILBERT SPACES Electronic Journal of Differential Equations, Vol. 21(21, No. 72, pp. 1 12. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu ALMOST PERIODIC SOLUTIONS

More information

Cores for generators of some Markov semigroups

Cores for generators of some Markov semigroups Cores for generators of some Markov semigroups Giuseppe Da Prato, Scuola Normale Superiore di Pisa, Italy and Michael Röckner Faculty of Mathematics, University of Bielefeld, Germany and Department of

More information

Optimal series representations of continuous Gaussian random fields

Optimal series representations of continuous Gaussian random fields Optimal series representations of continuous Gaussian random fields Antoine AYACHE Université Lille 1 - Laboratoire Paul Painlevé A. Ayache (Lille 1) Optimality of continuous Gaussian series 04/25/2012

More information

1.2 Fundamental Theorems of Functional Analysis

1.2 Fundamental Theorems of Functional Analysis 1.2 Fundamental Theorems of Functional Analysis 15 Indeed, h = ω ψ ωdx is continuous compactly supported with R hdx = 0 R and thus it has a unique compactly supported primitive. Hence fφ dx = f(ω ψ ωdy)dx

More information

ANALYSIS QUALIFYING EXAM FALL 2016: SOLUTIONS. = lim. F n

ANALYSIS QUALIFYING EXAM FALL 2016: SOLUTIONS. = lim. F n ANALYSIS QUALIFYING EXAM FALL 206: SOLUTIONS Problem. Let m be Lebesgue measure on R. For a subset E R and r (0, ), define E r = { x R: dist(x, E) < r}. Let E R be compact. Prove that m(e) = lim m(e /n).

More information

THEOREMS, ETC., FOR MATH 516

THEOREMS, ETC., FOR MATH 516 THEOREMS, ETC., FOR MATH 516 Results labeled Theorem Ea.b.c (or Proposition Ea.b.c, etc.) refer to Theorem c from section a.b of Evans book (Partial Differential Equations). Proposition 1 (=Proposition

More information

Analysis Comprehensive Exam Questions Fall F(x) = 1 x. f(t)dt. t 1 2. tf 2 (t)dt. and g(t, x) = 2 t. 2 t

Analysis Comprehensive Exam Questions Fall F(x) = 1 x. f(t)dt. t 1 2. tf 2 (t)dt. and g(t, x) = 2 t. 2 t Analysis Comprehensive Exam Questions Fall 2. Let f L 2 (, ) be given. (a) Prove that ( x 2 f(t) dt) 2 x x t f(t) 2 dt. (b) Given part (a), prove that F L 2 (, ) 2 f L 2 (, ), where F(x) = x (a) Using

More information

Applied Analysis (APPM 5440): Final exam 1:30pm 4:00pm, Dec. 14, Closed books.

Applied Analysis (APPM 5440): Final exam 1:30pm 4:00pm, Dec. 14, Closed books. Applied Analysis APPM 44: Final exam 1:3pm 4:pm, Dec. 14, 29. Closed books. Problem 1: 2p Set I = [, 1]. Prove that there is a continuous function u on I such that 1 ux 1 x sin ut 2 dt = cosx, x I. Define

More information

On the asymptotic behaviour of a semigroup of linear operators

On the asymptotic behaviour of a semigroup of linear operators On the asymptotic behaviour of a semigroup of linear operators J.M.A.M. van Neerven California Institute of Technology Alfred P. Sloan Laboratory of Mathematics Pasadena, CA 925, U.S.A. B. Straub 2 and

More information

Nash Type Inequalities for Fractional Powers of Non-Negative Self-adjoint Operators. ( Wroclaw 2006) P.Maheux (Orléans. France)

Nash Type Inequalities for Fractional Powers of Non-Negative Self-adjoint Operators. ( Wroclaw 2006) P.Maheux (Orléans. France) Nash Type Inequalities for Fractional Powers of Non-Negative Self-adjoint Operators ( Wroclaw 006) P.Maheux (Orléans. France) joint work with A.Bendikov. European Network (HARP) (to appear in T.A.M.S)

More information

Part II Probability and Measure

Part II Probability and Measure Part II Probability and Measure Theorems Based on lectures by J. Miller Notes taken by Dexter Chua Michaelmas 2016 These notes are not endorsed by the lecturers, and I have modified them (often significantly)

More information

Topics in Harmonic Analysis Lecture 1: The Fourier transform

Topics in Harmonic Analysis Lecture 1: The Fourier transform Topics in Harmonic Analysis Lecture 1: The Fourier transform Po-Lam Yung The Chinese University of Hong Kong Outline Fourier series on T: L 2 theory Convolutions The Dirichlet and Fejer kernels Pointwise

More information

THEOREMS, ETC., FOR MATH 515

THEOREMS, ETC., FOR MATH 515 THEOREMS, ETC., FOR MATH 515 Proposition 1 (=comment on page 17). If A is an algebra, then any finite union or finite intersection of sets in A is also in A. Proposition 2 (=Proposition 1.1). For every

More information

arxiv: v1 [math.gr] 28 Oct 2016

arxiv: v1 [math.gr] 28 Oct 2016 ROUP 1-COHOMOLOY IS COMPLEMENTED arxiv:1610.09188v1 [math.r] 28 Oct 2016 PIOTR W. NOWAK ABSTRACT. We show a structural property of cohomology with coefficients in an isometric representation on a uniformly

More information

ADJOINT FOR OPERATORS IN BANACH SPACES

ADJOINT FOR OPERATORS IN BANACH SPACES ADJOINT FOR OPERATORS IN BANACH SPACES T. L. GILL, S. BASU, W. W. ZACHARY, AND V. STEADMAN Abstract. In this paper we show that a result of Gross and Kuelbs, used to study Gaussian measures on Banach spaces,

More information

A Smooth Operator, Operated Correctly

A Smooth Operator, Operated Correctly Clark Department of Mathematics Bard College at Simon s Rock March 6, 2014 Abstract By looking at familiar smooth functions in new ways, we can make sense of matrix-valued infinite series and operator-valued

More information

ON THE FAVARD CLASSES OF SEMIGROUPS ASSOCIATED WITH PSEUDO-RESOLVENTS

ON THE FAVARD CLASSES OF SEMIGROUPS ASSOCIATED WITH PSEUDO-RESOLVENTS 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,

More information

Zero-one laws for functional calculus on operator semigroups

Zero-one laws for functional calculus on operator semigroups Zero-one laws for functional calculus on operator semigroups Lancaster, September 2014 Joint work with Isabelle Chalendar (Lyon) and Jean Esterle (Bordeaux) Contents 1. Some classical results. 2. Newer

More information

Lax Solution Part 4. October 27, 2016

Lax Solution Part 4.   October 27, 2016 Lax Solution Part 4 www.mathtuition88.com October 27, 2016 Textbook: Functional Analysis by Peter D. Lax Exercises: Ch 16: Q2 4. Ch 21: Q1, 2, 9, 10. Ch 28: 1, 5, 9, 10. 1 Chapter 16 Exercise 2 Let h =

More information

Spectral theorems for bounded self-adjoint operators on a Hilbert space

Spectral theorems for bounded self-adjoint operators on a Hilbert space Chapter 10 Spectral theorems for bounded self-adjoint operators on a Hilbert space Let H be a Hilbert space. For a bounded operator A : H H its Hilbert space adjoint is an operator A : H H such that Ax,

More information

Recall that any inner product space V has an associated norm defined by

Recall that any inner product space V has an associated norm defined by Hilbert Spaces Recall that any inner product space V has an associated norm defined by v = v v. Thus an inner product space can be viewed as a special kind of normed vector space. In particular every inner

More information

285K Homework #1. Sangchul Lee. April 28, 2017

285K Homework #1. Sangchul Lee. April 28, 2017 285K Homework #1 Sangchul Lee April 28, 2017 Problem 1. Suppose that X is a Banach space with respect to two norms: 1 and 2. Prove that if there is c (0, such that x 1 c x 2 for each x X, then there is

More information

Strongly continuous semigroups

Strongly continuous semigroups Capter 2 Strongly continuous semigroups Te main application of te teory developed in tis capter is related to PDE systems. Tese systems can provide te strong continuity properties only. 2.1 Closed operators

More information

PROBABILITY: LIMIT THEOREMS II, SPRING HOMEWORK PROBLEMS

PROBABILITY: LIMIT THEOREMS II, SPRING HOMEWORK PROBLEMS PROBABILITY: LIMIT THEOREMS II, SPRING 15. HOMEWORK PROBLEMS PROF. YURI BAKHTIN Instructions. You are allowed to work on solutions in groups, but you are required to write up solutions on your own. Please

More information

NOTES ON FRAMES. Damir Bakić University of Zagreb. June 6, 2017

NOTES ON FRAMES. Damir Bakić University of Zagreb. June 6, 2017 NOTES ON FRAMES Damir Bakić University of Zagreb June 6, 017 Contents 1 Unconditional convergence, Riesz bases, and Bessel sequences 1 1.1 Unconditional convergence of series in Banach spaces...............

More information

Riesz Representation Theorems

Riesz Representation Theorems Chapter 6 Riesz Representation Theorems 6.1 Dual Spaces Definition 6.1.1. Let V and W be vector spaces over R. We let L(V, W ) = {T : V W T is linear}. The space L(V, R) is denoted by V and elements of

More information

Let R be the line parameterized by x. Let f be a complex function on R that is integrable. The Fourier transform ˆf = F f is. e ikx f(x) dx. (1.

Let R be the line parameterized by x. Let f be a complex function on R that is integrable. The Fourier transform ˆf = F f is. e ikx f(x) dx. (1. Chapter 1 Fourier transforms 1.1 Introduction Let R be the line parameterized by x. Let f be a complex function on R that is integrable. The Fourier transform ˆf = F f is ˆf(k) = e ikx f(x) dx. (1.1) It

More information

x n x or x = T -limx n, if every open neighbourhood U of x contains x n for all but finitely many values of n

x n x or x = T -limx n, if every open neighbourhood U of x contains x n for all but finitely many values of n Note: This document was written by Prof. Richard Haydon, a previous lecturer for this course. 0. Preliminaries This preliminary chapter is an attempt to bring together important results from earlier courses

More information

ANALYSIS QUALIFYING EXAM FALL 2017: SOLUTIONS. 1 cos(nx) lim. n 2 x 2. g n (x) = 1 cos(nx) n 2 x 2. x 2.

ANALYSIS QUALIFYING EXAM FALL 2017: SOLUTIONS. 1 cos(nx) lim. n 2 x 2. g n (x) = 1 cos(nx) n 2 x 2. x 2. ANALYSIS QUALIFYING EXAM FALL 27: SOLUTIONS Problem. Determine, with justification, the it cos(nx) n 2 x 2 dx. Solution. For an integer n >, define g n : (, ) R by Also define g : (, ) R by g(x) = g n

More information

Perturbations of the H -calculus

Perturbations of the H -calculus Collect. Math. 58, 3 (2007), 291 325 c 2007 Universitat de Barcelona Perturbations of the H -calculus N.J. Kalton Department of Mathematics, University of Missouri-Columbia Columbia, MO 65211 E-mail: nigel@math.missouri.edu

More information

Functional Analysis II held by Prof. Dr. Moritz Weber in summer 18

Functional Analysis II held by Prof. Dr. Moritz Weber in summer 18 Functional Analysis II held by Prof. Dr. Moritz Weber in summer 18 General information on organisation Tutorials and admission for the final exam To take part in the final exam of this course, 50 % of

More information

Measure Theory on Topological Spaces. Course: Prof. Tony Dorlas 2010 Typset: Cathal Ormond

Measure Theory on Topological Spaces. Course: Prof. Tony Dorlas 2010 Typset: Cathal Ormond Measure Theory on Topological Spaces Course: Prof. Tony Dorlas 2010 Typset: Cathal Ormond May 22, 2011 Contents 1 Introduction 2 1.1 The Riemann Integral........................................ 2 1.2 Measurable..............................................

More information

be the set of complex valued 2π-periodic functions f on R such that

be the set of complex valued 2π-periodic functions f on R such that . Fourier series. Definition.. Given a real number P, we say a complex valued function f on R is P -periodic if f(x + P ) f(x) for all x R. We let be the set of complex valued -periodic functions f on

More information

CESARO OPERATORS ON THE HARDY SPACES OF THE HALF-PLANE

CESARO OPERATORS ON THE HARDY SPACES OF THE HALF-PLANE CESARO OPERATORS ON THE HARDY SPACES OF THE HALF-PLANE ATHANASIOS G. ARVANITIDIS AND ARISTOMENIS G. SISKAKIS Abstract. In this article we study the Cesàro operator C(f)() = d, and its companion operator

More information

Combinatorics in Banach space theory Lecture 12

Combinatorics in Banach space theory Lecture 12 Combinatorics in Banach space theory Lecture The next lemma considerably strengthens the assertion of Lemma.6(b). Lemma.9. For every Banach space X and any n N, either all the numbers n b n (X), c n (X)

More information

4 Linear operators and linear functionals

4 Linear operators and linear functionals 4 Linear operators and linear functionals The next section is devoted to studying linear operators between normed spaces. Definition 4.1. Let V and W be normed spaces over a field F. We say that T : V

More information

Functional Norms for Generalized Bakers Transformations.

Functional Norms for Generalized Bakers Transformations. Functional Norms for Generalized Bakers Transformations. Seth Chart University of Victoria July 14th, 2014 Generalized Bakers Transformations (C. Bose 1989) B φ a = φ Hypothesis on the Cut Function ɛ (0,

More information

CHAPTER X THE SPECTRAL THEOREM OF GELFAND

CHAPTER X THE SPECTRAL THEOREM OF GELFAND CHAPTER X THE SPECTRAL THEOREM OF GELFAND DEFINITION A Banach algebra is a complex Banach space A on which there is defined an associative multiplication for which: (1) x (y + z) = x y + x z and (y + z)

More information

Extreme points of compact convex sets

Extreme points of compact convex sets Extreme points of compact convex sets In this chapter, we are going to show that compact convex sets are determined by a proper subset, the set of its extreme points. Let us start with the main definition.

More information

Functional Analysis F3/F4/NVP (2005) Homework assignment 2

Functional Analysis F3/F4/NVP (2005) Homework assignment 2 Functional Analysis F3/F4/NVP (25) Homework assignment 2 All students should solve the following problems:. Section 2.7: Problem 8. 2. Let x (t) = t 2 e t/2, x 2 (t) = te t/2 and x 3 (t) = e t/2. Orthonormalize

More information

CONTENTS. 4 Hausdorff Measure Introduction The Cantor Set Rectifiable Curves Cantor Set-Like Objects...

CONTENTS. 4 Hausdorff Measure Introduction The Cantor Set Rectifiable Curves Cantor Set-Like Objects... Contents 1 Functional Analysis 1 1.1 Hilbert Spaces................................... 1 1.1.1 Spectral Theorem............................. 4 1.2 Normed Vector Spaces.............................. 7 1.2.1

More information

MA5206 Homework 4. Group 4. April 26, ϕ 1 = 1, ϕ n (x) = 1 n 2 ϕ 1(n 2 x). = 1 and h n C 0. For any ξ ( 1 n, 2 n 2 ), n 3, h n (t) ξ t dt

MA5206 Homework 4. Group 4. April 26, ϕ 1 = 1, ϕ n (x) = 1 n 2 ϕ 1(n 2 x). = 1 and h n C 0. For any ξ ( 1 n, 2 n 2 ), n 3, h n (t) ξ t dt MA526 Homework 4 Group 4 April 26, 26 Qn 6.2 Show that H is not bounded as a map: L L. Deduce from this that H is not bounded as a map L L. Let {ϕ n } be an approximation of the identity s.t. ϕ C, sptϕ

More information

1 Sectorial operators

1 Sectorial operators 1 1 Sectorial operators Definition 1.1 Let X and A : D(A) X X be a Banach space and a linear closed operator, respectively. If the relationships i) ρ(a) Σ φ = {λ C : arg λ < φ}, where φ (π/2, π); ii) R(λ,

More information

THE PERRON PROBLEM FOR C-SEMIGROUPS

THE PERRON PROBLEM FOR C-SEMIGROUPS Math. J. Okayama Univ. 46 (24), 141 151 THE PERRON PROBLEM FOR C-SEMIGROUPS Petre PREDA, Alin POGAN and Ciprian PREDA Abstract. Characterizations of Perron-type for the exponential stability of exponentially

More information

Semigroups and Linear Partial Differential Equations with Delay

Semigroups and Linear Partial Differential Equations with Delay Journal of Mathematical Analysis and Applications 264, 1 2 (21 doi:1.16/jmaa.21.675, available online at http://www.idealibrary.com on Semigroups and Linear Partial Differential Equations with Delay András

More information

Outline of Fourier Series: Math 201B

Outline of Fourier Series: Math 201B Outline of Fourier Series: Math 201B February 24, 2011 1 Functions and convolutions 1.1 Periodic functions Periodic functions. Let = R/(2πZ) denote the circle, or onedimensional torus. A function f : C

More information

Frames of exponentials on small sets

Frames of exponentials on small sets Frames of exponentials on small sets J.-P. Gabardo McMaster University Department of Mathematics and Statistics gabardo@mcmaster.ca Joint work with Chun-Kit Lai (San Francisco State Univ.) Workshop on

More information

The Role of Exosystems in Output Regulation

The Role of Exosystems in Output Regulation 1 The Role of Exosystems in Output Regulation Lassi Paunonen In this paper we study the role of the exosystem in the theory of output regulation for linear infinite-dimensional systems. The main result

More information

Analysis Qualifying Exam

Analysis Qualifying Exam Analysis Qualifying Exam Spring 2017 Problem 1: Let f be differentiable on R. Suppose that there exists M > 0 such that f(k) M for each integer k, and f (x) M for all x R. Show that f is bounded, i.e.,

More information

ANALYTIC SEMIGROUPS AND APPLICATIONS. 1. Introduction

ANALYTIC SEMIGROUPS AND APPLICATIONS. 1. Introduction ANALYTIC SEMIGROUPS AND APPLICATIONS KELLER VANDEBOGERT. Introduction Consider a Banach space X and let f : D X and u : G X, where D and G are real intervals. A is a bounded or unbounded linear operator

More information

Homework If the inverse T 1 of a closed linear operator exists, show that T 1 is a closed linear operator.

Homework If the inverse T 1 of a closed linear operator exists, show that T 1 is a closed linear operator. Homework 3 1 If the inverse T 1 of a closed linear operator exists, show that T 1 is a closed linear operator Solution: Assuming that the inverse of T were defined, then we will have to have that D(T 1

More information

ON THE FRACTIONAL CAUCHY PROBLEM ASSOCIATED WITH A FELLER SEMIGROUP

ON THE FRACTIONAL CAUCHY PROBLEM ASSOCIATED WITH A FELLER SEMIGROUP Dedicated to Professor Gheorghe Bucur on the occasion of his 7th birthday ON THE FRACTIONAL CAUCHY PROBLEM ASSOCIATED WITH A FELLER SEMIGROUP EMIL POPESCU Starting from the usual Cauchy problem, we give

More information

Jae Gil Choi and Young Seo Park

Jae Gil Choi and Young Seo Park Kangweon-Kyungki Math. Jour. 11 (23), No. 1, pp. 17 3 TRANSLATION THEOREM ON FUNCTION SPACE Jae Gil Choi and Young Seo Park Abstract. In this paper, we use a generalized Brownian motion process to define

More information

Spectral Continuity Properties of Graph Laplacians

Spectral Continuity Properties of Graph Laplacians Spectral Continuity Properties of Graph Laplacians David Jekel May 24, 2017 Overview Spectral invariants of the graph Laplacian depend continuously on the graph. We consider triples (G, x, T ), where G

More information

Lecture 5. Ch. 5, Norms for vectors and matrices. Norms for vectors and matrices Why?

Lecture 5. Ch. 5, Norms for vectors and matrices. Norms for vectors and matrices Why? KTH ROYAL INSTITUTE OF TECHNOLOGY Norms for vectors and matrices Why? Lecture 5 Ch. 5, Norms for vectors and matrices Emil Björnson/Magnus Jansson/Mats Bengtsson April 27, 2016 Problem: Measure size of

More information

Math General Topology Fall 2012 Homework 11 Solutions

Math General Topology Fall 2012 Homework 11 Solutions Math 535 - General Topology Fall 2012 Homework 11 Solutions Problem 1. Let X be a topological space. a. Show that the following properties of a subset A X are equivalent. 1. The closure of A in X has empty

More information

Solutions: Problem Set 4 Math 201B, Winter 2007

Solutions: Problem Set 4 Math 201B, Winter 2007 Solutions: Problem Set 4 Math 2B, Winter 27 Problem. (a Define f : by { x /2 if < x

More information

Weakly Compact Composition Operators on Hardy Spaces of the Upper Half-Plane 1

Weakly Compact Composition Operators on Hardy Spaces of the Upper Half-Plane 1 Int. Journal of Math. Analysis, Vol. 4, 2010, no. 37, 1851-1856 Weakly Compact Composition Operators on Hardy Spaces of the Upper Half-Plane 1 Hong Bin Bai School of Science Sichuan University of Science

More information