A finite multiplicity Helson–Lowdenslager–de Branges theorem

Sneh Lata ¹   ; Meghna Mittal ¹   ; Dinesh Singh ²

¹ Department of Mathematics University of Houston Houston, TX 77204-3476, U.S.A.
² Department of Mathematics University of Delhi Delhi 110007, India

Studia Mathematica, Tome 200 (2010) no. 3, pp. 247-266

Résumé

We prove two theorems. The first theorem reduces to a scalar situation the well known vector-valued generalization of the Helson–Lowdenslager theorem that characterizes the invariant subspaces of the operator of multiplication by the coordinate function $z$ on the vector-valued Lebesgue space $L^2(\mathbb T;\mathbb C^n)$. Our approach allows us to prove an equivalent version of the vector-valued Helson–Lowdenslager theorem in a completely scalar setting, thereby eliminating the use of range functions and partial isometries. The other three major advantages provided by our characterization are: (i) we provide precise necessary and sufficient conditions for the presence of reducing subspaces inside simply invariant subspaces; (ii) we give a complete description of the wandering vectors; (iii) we prove the theorem in the setting of all the Lebesgue spaces $L^p$ $(0 p \le \infty)$. Our second theorem generalizes the first theorem along the lines of de Branges' generalization of Beurling's theorem by characterizing those Hilbert spaces that are simply invariant under multiplication by $z^n$ and which are contractively contained in $L^p$ $( 1\le p \le \infty)$. This also generalizes a theorem of Paulsen and Singh [Proc. Amer. Math. Soc. 129 (2000)] as well as the main theorem of Redett [Bull. London Math. Soc. 37 (2005)].

Détail

DOI : 10.4064/sm200-3-3

Keywords: prove theorems first theorem reduces scalar situation known vector valued generalization helson lowdenslager theorem characterizes invariant subspaces operator multiplication coordinate function vector valued lebesgue space mathbb mathbb approach allows prove equivalent version vector valued helson lowdenslager theorem completely scalar setting thereby eliminating range functions partial isometries other three major advantages provided characterization provide precise necessary sufficient conditions presence reducing subspaces inside simply invariant subspaces complete description wandering vectors iii prove theorem setting lebesgue spaces nbsp infty second theorem generalizes first theorem along lines branges generalization beurlings theorem characterizing those hilbert spaces simply invariant under multiplication which contractively contained nbsp infty generalizes theorem paulsen singh proc amer math soc main theorem redett bull london math soc