# Portugaliae Mathematica

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**Volume 66, Issue 2, 2009, pp. 147–158**

**DOI: 10.4171/PM/1837**

Published online: 2009-06-30

A note on common range of a class of co-analytic Toeplitz operators

Romeo Meštrović^{[1]}and Žarko Pavićević (1) University of Montenegro, Kotor, Montenegro

We characterize the intersection of the ranges
of a class of co-analytic Toeplitz operators
by considering this set as the dual space of the Privalov space
`N`^{p}, 1 < `p` < ∞, in a certain topology. For a fixed `p` we
define the class `H`_{p} consisting of those
de Branges spaces ℋ(`b`) such that the function `b` is not an
extreme point of the unit ball of `H`^{∞}, and
the associated measure `μ _{}`

`b`for

`b`satisfies an additional condition. It is proved that the function

`f`analytic on

**D**is a multiplier of every de Branges space from

`H`

_{p}if and only if

`f`is in the intersection of the ranges of all Toeplitz operators belonging to the class

`H`

_{p}. We show that this is true if and only if the Taylor coefficients \hat{

`f`}(

`n`) of

`f`decay like

`O`(exp (−

`cn`

^{1/(p + 1)}))

`c`.

*Keywords: *Toeplitz operator, Privalov class `N`^{p}, Helson topology, weight, de Branges space, multipliers

Meštrović Romeo, Pavićević Žarko: A note on common range of a class of co-analytic Toeplitz operators. *Port. Math.* 66 (2009), 147-158. doi: 10.4171/PM/1837