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Author Lata, Sneh ♦ Mittal, Meghna ♦ Singh, Dinesh
Source arXiv.org
Content type Text
File Format PDF
Date of Submission 2009-10-28
Language English
Subject Domain (in DDC) Natural sciences & mathematics ♦ Mathematics
Subject Keyword Mathematics - Functional Analysis ♦ Mathematics - Complex Variables ♦ 47A15 (Primary) ♦ math
Abstract This paper proves two theorems. The first of these simplifies and lends clarity to the previous characterizations of the invariant subspaces of $S$, the operator of multiplication by the coordinate function $z$, on $L^2(\mathbb{T};\mathbb{C}^n)$, where $\mathbb{T}$ is the unit circle, by characterizing the invariant subspaces of $S^n$ on scalar valued $L^p$ ($0<p\le\infty$) thereby eliminating range functions and partial isometries. It also gives precise conditions as to when the operator shall be a pure shift and describes the precise nature of the wandering vectors and the doubly invariant subspaces. The second theorem describes the contractively contained Hilbert spaces in $L^p$ that are simply invariant under $S^n$ thereby generalizing the first theorem.
Educational Use Research
Learning Resource Type Article
Page Count 18


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