# An extremal problem for functions annihilated by a Toeplitz operator

An extremal problem for functions annihilated by a Toeplitz operator For a bounded function $$\varphi$$ φ on the unit circle $${\mathbb {T}}$$ T , let $$T_{\varphi }$$ T φ be the associated Toeplitz operator on the Hardy space $$H^2$$ H 2 . Assume that the kernel \begin{aligned} K_2(\varphi ):=\left\{ f\in H^2:\,T_{\varphi } f=0\right\} \end{aligned} K 2 ( φ ) : = f ∈ H 2 : T φ f = 0 is nontrivial. Given a unit-norm function f in $$K_2(\varphi )$$ K 2 ( φ ) , we ask whether an identity of the form $$|f|^2=\frac{1}{2}\left( |f_1|^2+|f_2|^2\right)$$ | f | 2 = 1 2 | f 1 | 2 + | f 2 | 2 may hold a.e. on $${\mathbb {T}}$$ T for some $$f_1,f_2\in K_2(\varphi )$$ f 1 , f 2 ∈ K 2 ( φ ) , both of norm 1 and such that $$|f_1|\ne |f_2|$$ | f 1 | ≠ | f 2 | on a set of positive measure. We then show that such a decomposition is possible if and only if either f or $$\overline{z\varphi f}$$ z φ f ¯ has a nonconstant inner factor. The proof relies on an intrinsic characterization of the moduli of functions in $$K_2(\varphi )$$ K 2 ( φ ) , a result which we also extend to $$K_p(\varphi )$$ K p ( φ ) (the kernel of $$T_{\varphi }$$ T φ in $$H^p$$ H p ) with $$1\le p\le \infty$$ 1 ≤ p ≤ ∞ . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Analysis and Mathematical Physics Springer Journals

# An extremal problem for functions annihilated by a Toeplitz operator

, Volume 9 (3) – Mar 4, 2019
11 pages

/lp/springer-journals/an-extremal-problem-for-functions-annihilated-by-a-toeplitz-operator-AZpSD8t7Ub
Publisher
Springer Journals
Subject
Mathematics; Analysis; Mathematical Methods in Physics
ISSN
1664-2368
eISSN
1664-235X
DOI
10.1007/s13324-019-00291-0
Publisher site
See Article on Publisher Site

### Abstract

For a bounded function $$\varphi$$ φ on the unit circle $${\mathbb {T}}$$ T , let $$T_{\varphi }$$ T φ be the associated Toeplitz operator on the Hardy space $$H^2$$ H 2 . Assume that the kernel \begin{aligned} K_2(\varphi ):=\left\{ f\in H^2:\,T_{\varphi } f=0\right\} \end{aligned} K 2 ( φ ) : = f ∈ H 2 : T φ f = 0 is nontrivial. Given a unit-norm function f in $$K_2(\varphi )$$ K 2 ( φ ) , we ask whether an identity of the form $$|f|^2=\frac{1}{2}\left( |f_1|^2+|f_2|^2\right)$$ | f | 2 = 1 2 | f 1 | 2 + | f 2 | 2 may hold a.e. on $${\mathbb {T}}$$ T for some $$f_1,f_2\in K_2(\varphi )$$ f 1 , f 2 ∈ K 2 ( φ ) , both of norm 1 and such that $$|f_1|\ne |f_2|$$ | f 1 | ≠ | f 2 | on a set of positive measure. We then show that such a decomposition is possible if and only if either f or $$\overline{z\varphi f}$$ z φ f ¯ has a nonconstant inner factor. The proof relies on an intrinsic characterization of the moduli of functions in $$K_2(\varphi )$$ K 2 ( φ ) , a result which we also extend to $$K_p(\varphi )$$ K p ( φ ) (the kernel of $$T_{\varphi }$$ T φ in $$H^p$$ H p ) with $$1\le p\le \infty$$ 1 ≤ p ≤ ∞ .

### Journal

Analysis and Mathematical PhysicsSpringer Journals

Published: Mar 4, 2019