Multipliers of Hardy spaces, quadratic integrals and Foiaş-Williams-Peller operators

G. Blower

Studia Mathematica (1998)

  • Volume: 131, Issue: 2, page 179-188
  • ISSN: 0039-3223

Abstract

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We obtain a sufficient condition on a B(H)-valued function φ for the operator ⨍ ↦ Γ φ ⨍ ' ( S ) to be completely bounded on H ∞ B ( H ) ; the Foiaş-Williams-Peller operator | St Γφ | Rφ = | | | 0 S | is then similar to a contraction. We show that if ⨍ : D → B(H) is a bounded analytic function for which ( 1 - r ) | | ⨍ ' ( r e i θ ) | | B ( H ) 2 r d r d θ and ( 1 - r ) | | ⨍ " ( r e i θ ) | | B ( H ) r d r d θ are Carleson measures, then ⨍ multiplies ( H 1 c 1 ) ' to itself. Such ⨍ form an algebra A, and when φ’∈ BMO(B(H)), the map ⨍ ↦ Γ φ ⨍ ' ( S ) is bounded A → B ( H 2 ( H ) , L 2 ( H ) ⊖ H 2 ( H ) ) . Thus we construct a functional calculus for operators of Foiaş-Williams-Peller type.

How to cite

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Blower, G.. "Multipliers of Hardy spaces, quadratic integrals and Foiaş-Williams-Peller operators." Studia Mathematica 131.2 (1998): 179-188. <http://eudml.org/doc/216574>.

@article{Blower1998,
abstract = {We obtain a sufficient condition on a B(H)-valued function φ for the operator $⨍ ↦ Γ_φ ⨍^\{\prime \}(S)$ to be completely bounded on $H^∞ B(H)$; the Foiaş-Williams-Peller operator | St Γφ | Rφ = | | | 0 S | is then similar to a contraction. We show that if ⨍ : D → B(H) is a bounded analytic function for which $(1-r) ||⨍^\{\prime \}(re^\{iθ\})||^2_\{B(H)\} rdrdθ$ and $(1-r) ||⨍"(re^\{iθ\})||_\{B(H)\} rdrdθ$ are Carleson measures, then ⨍ multiplies $(H^1c^1)^\{\prime \}$ to itself. Such ⨍ form an algebra A, and when φ’∈ BMO(B(H)), the map $⨍ ↦ Γ_φ ⨍^\{\prime \}(S)$ is bounded $A → B(H^2(H), L^2(H) ⊖ H^2(H))$. Thus we construct a functional calculus for operators of Foiaş-Williams-Peller type.},
author = {Blower, G.},
journal = {Studia Mathematica},
keywords = {polynomially bounded operators; Hankel operators; multipliers; Carleson measures; completely bounded; Foiaş-Williams-Peller operator; functional calculus},
language = {eng},
number = {2},
pages = {179-188},
title = {Multipliers of Hardy spaces, quadratic integrals and Foiaş-Williams-Peller operators},
url = {http://eudml.org/doc/216574},
volume = {131},
year = {1998},
}

TY - JOUR
AU - Blower, G.
TI - Multipliers of Hardy spaces, quadratic integrals and Foiaş-Williams-Peller operators
JO - Studia Mathematica
PY - 1998
VL - 131
IS - 2
SP - 179
EP - 188
AB - We obtain a sufficient condition on a B(H)-valued function φ for the operator $⨍ ↦ Γ_φ ⨍^{\prime }(S)$ to be completely bounded on $H^∞ B(H)$; the Foiaş-Williams-Peller operator | St Γφ | Rφ = | | | 0 S | is then similar to a contraction. We show that if ⨍ : D → B(H) is a bounded analytic function for which $(1-r) ||⨍^{\prime }(re^{iθ})||^2_{B(H)} rdrdθ$ and $(1-r) ||⨍"(re^{iθ})||_{B(H)} rdrdθ$ are Carleson measures, then ⨍ multiplies $(H^1c^1)^{\prime }$ to itself. Such ⨍ form an algebra A, and when φ’∈ BMO(B(H)), the map $⨍ ↦ Γ_φ ⨍^{\prime }(S)$ is bounded $A → B(H^2(H), L^2(H) ⊖ H^2(H))$. Thus we construct a functional calculus for operators of Foiaş-Williams-Peller type.
LA - eng
KW - polynomially bounded operators; Hankel operators; multipliers; Carleson measures; completely bounded; Foiaş-Williams-Peller operator; functional calculus
UR - http://eudml.org/doc/216574
ER -

References

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  14. [14] G. Pisier, A polynomially bounded operator on Hilbert space which is not similar to a contraction, J. Amer. Math. Soc. 10 (1997), 351-369. Zbl0869.47014

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