A finite multiplicity Helson-Lowdenslager-de Branges theorem

Sneh Lata, Meghna Mittal, and Dinesh Singh. "A finite multiplicity Helson-Lowdenslager-de Branges theorem." Studia Mathematica 200.3 (2010): 247-266. <http://eudml.org/doc/286089>.

@article{SnehLata2010,
abstract = {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 L²(;ℂⁿ). 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 ≤ ∞). 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 zⁿ and which are contractively contained in $L^\{p\}$ (1 ≤ p ≤ ∞). 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)].},
author = {Sneh Lata, Meghna Mittal, Dinesh Singh},
journal = {Studia Mathematica},
keywords = {invariant subspace; Helson-Lowdenslager theorem; multiplication operator},
language = {eng},
number = {3},
pages = {247-266},
title = {A finite multiplicity Helson-Lowdenslager-de Branges theorem},
url = {http://eudml.org/doc/286089},
volume = {200},
year = {2010},
}

TY - JOUR
AU - Sneh Lata
AU - Meghna Mittal
AU - Dinesh Singh
TI - A finite multiplicity Helson-Lowdenslager-de Branges theorem
JO - Studia Mathematica
PY - 2010
VL - 200
IS - 3
SP - 247
EP - 266
AB - 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 L²(;ℂⁿ). 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 ≤ ∞). 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 zⁿ and which are contractively contained in $L^{p}$ (1 ≤ p ≤ ∞). 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)].
LA - eng
KW - invariant subspace; Helson-Lowdenslager theorem; multiplication operator
UR - http://eudml.org/doc/286089
ER -