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## Studia Mathematica

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## The orthogonal complement of $\mathcal{M}(a)$ in $\mathcal{H}(b)$

### Volume 260 / 2021

Studia Mathematica 260 (2021), 327-340 MSC: Primary 47B32, 46E22; Secondary 30H10. DOI: 10.4064/sm200918-5-1 Published online: 22 April 2021

#### Abstract

Let $b$ be a nonextreme function in the unit ball of $H^{\infty }$ on the unit disk $\mathbb D$ and let $a$ be an outer $H^{\infty }$ function such that $|a|^2+|b|^2=1$ almost everywhere on $\partial \mathbb D$. Sufficient and necessary conditions for the orthogonal complement of $\mathcal M (a)$ in $\mathcal H (b)$ to be finite-dimensional were given by D. Sarason (1994). Here we describe this space explicitly.

#### Authors

• Maria T. NowakInstitute of Mathematics
Maria Curie-Skłodowska University
Pl. M. Curie-Skłodowskiej 1
20-031 Lublin, Poland
e-mail
• Paweł SobolewskiInstitute of Mathematics
Maria Curie-Skłodowska University
Pl. M. Curie-Skłodowskiej 1
20-031 Lublin, Poland
e-mail
• Andrzej SołtysiakFaculty of Mathematics and Computer Science
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