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Universal multipliers for Sub-Hardy Hilbert spaces

Bartosz Malman, Daniel Seco

Abstract

To every non-extreme point $b$ of the unit ball of $\hil^\infty$ of the unit disk there corresponds a Pythagorean mate, a bounded outer function $a$ satisfying the equation $|a|^2 + |b|^2 = 1$ on the boundary of the disk. We study universal, i.e., simultaneous multipliers for families of de Branges-Rovnyak spaces $\hb$, and develop a general framework for this purpose. Our main results include a new proof of the Davis-McCarthy universal multiplier theorem for the class of all non-extreme spaces $\hb$, a characterization of the Lipschitz classes as the universal multipliers for spaces $\hb$ for which the quotient $b/a$ is contained in a Hardy space, and a similar characterization of the Gevrey classes as the universal multipliers for spaces $\hb$ for which $b/a$ is contained in a Privalov class.

Universal multipliers for Sub-Hardy Hilbert spaces

Abstract

To every non-extreme point of the unit ball of of the unit disk there corresponds a Pythagorean mate, a bounded outer function satisfying the equation on the boundary of the disk. We study universal, i.e., simultaneous multipliers for families of de Branges-Rovnyak spaces , and develop a general framework for this purpose. Our main results include a new proof of the Davis-McCarthy universal multiplier theorem for the class of all non-extreme spaces , a characterization of the Lipschitz classes as the universal multipliers for spaces for which the quotient is contained in a Hardy space, and a similar characterization of the Gevrey classes as the universal multipliers for spaces for which is contained in a Privalov class.

Paper Structure

This paper contains 28 sections, 17 theorems, 87 equations.

Table of Contents

  1. Introduction
  2. De Branges-Rovnyak spaces
  3. Universal multipliers and the Davis-McCarthy theorem
  4. Main result
  5. Other multiplier results and comments
  6. More on the Davis-McCarthy theorem and its variants
  7. Explicit characterization of multipliers for rational $b$
  8. All bounded functions as multipliers
  9. Outline of the paper and the methods
  10. Research of Lotto and Sarason
  11. Hankel operators
  12. Lotto-Sarason characterization
  13. Hankel operators and their continuity
  14. Hankel operators with Lipschitz symbols
  15. Hankel operators with Gevrey symbols
  16. ...and 13 more sections

Key Result

Theorem 1

We have

Theorems & Definitions (24)

  • Theorem : Davis-McCarthy
  • Theorem A
  • Theorem B
  • Theorem 2.1
  • Corollary 2.2
  • Theorem 2.3
  • Corollary 2.4
  • proof
  • Proposition 3.1
  • Lemma 3.2: Duren-Romberg-Shields, Hardy-Littlewood
  • ...and 14 more