#### Date of Award

Spring 2017

#### Project Type

Dissertation

#### Program or Major

Mathematics

#### Degree Name

Doctor of Philosophy

#### First Advisor

Junhao Shen

#### Second Advisor

Donald Hadwin

#### Third Advisor

Karen Graham

#### Abstract

We prove Beurling-type theorems for H-invariant spaces in relation to a semifinite von Neu-mann algebra M with a semifinite, faithful, normal tracial weight τ, using an extension of Arveson’s non-commutative Hardy space H-. First we prove a Beurling-Blecher-Labuschagne theorem for H-invariant subspaces of L p (M,τ) when 0 < p ≤ -. We also prove a Beurling-Chen-Hadwin-Shen theorem for H -invariant subspaces of L a (M,τ) where a is a unitarily invariant, locally k 1 -dominating, mutually continuous norm with respect to &\tau;. For a crossed product of a von Neumann algebra M by an action β, M o β Z, we are able to completely characterize all H-invariant subspaces of L a (Mo β Z,t) using our results. As an example, we completely characterize all H-invariant subspaces of the Schatten p-class, S p (H) (0 < p ≤ -), where H - is the lower tri-angular subalgebra of B(H). We also characterize the non-commutative Hardy space H -invariant subspaces in a Banach function space I(τ) on a semifinite von Neumann algebra M.

#### Recommended Citation

Sager, Lauren Beth Meitzler, "A Beurling Theorem for Noncommutative Hardy Spaces Associated with a Semifinite von Neumann Algebra with Various Norms" (2017). *Doctoral Dissertations*. 143.

https://scholars.unh.edu/dissertation/143