From 84da9b7cd865b6d9ed5fab3106fbaddb2da00952 Mon Sep 17 00:00:00 2001 From: "Stephen A. Crowley" Date: Sun, 21 Jan 2024 03:40:46 -0600 Subject: [PATCH] Spectrum of the Real Part of a Weighted Shift Operator MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit by Stephen Crowley January 21, 2024 1 Introduction This document presents an overview of the spectrum of the real part of a weighted shift operator and explains why it is equal to the closed interval [-1,1]. The explanation is grounded in the principles of functional analysis and operator theory. 2 Preliminaries 2.1 Weighted Shift Operator A weighted shift operator, denoted as T, is a type of linear operator acting on a Hilbert space H. It shifts the elements of a sequence or a function space and multiplies each element by a corresponding weight. The spectral properties of this operator are significantly influenced by these weights. 2.2 Spectrum of an Operator The spectrum of a linear operator T, denoted by σ(T), comprises the set of complex numbers λ for which T-λ*I is not invertible, where I is the identity operator. The spectrum includes eigenvalues and other values where the operator is not invertible. 2.3 Real Part of an Operator The real part of an operator T, denoted as ℜ(T), is given by 1/2*(T+T^∗), where T^∗ is the adjoint of T. This operator is self-adjoint, even if T itself is not. 3 Spectrum of the Real Part of a Weighted Shift Operator The statement in question is that the spectrum of the real part of a weighted shift operator is the closed interval [-1,1]. This can be understood through the following considerations: • The weights of the shift operator, bounded by 1 in absolute value, limit the operator norms of T and T^∗ to 1. • The real part of T combines T and T^∗ and inherits their spectral characteristics. • Consequently, the spectrum of the real part lies within [-1,1], as the operator norms are bounded and the spectrum of a self-adjoint operator is always within the interval defined by its norm. 4 Conclusion The spectrum of the real part of a weighted shift operator being within the interval [-1,1] is a result derived from the bounded nature of the operator and its adjoint, as well as the spectral theorem for self-adjoint operators. This theorem asserts that a self-adjoint operator on a Hilbert space has a real spectrum, comprising eigenvalues or continuous spectrum intervals, which in this case is the interval [-1,1].[1] Bibliography [1] J. DOMBROWSKI. Spectral properties of real parts of weighted shift operators. 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||<\author-affiliation> + + >>> + + + + This document presents an overview of the spectrum of the real part of a + weighted shift operator and explains why it is equal to the closed interval + >. The explanation is grounded in the principles of + functional analysis and operator theory. + + + + + + A weighted shift operator, denoted as , is a type of linear + operator acting on a Hilbert space . It shifts the elements of a + sequence or a function space and multiplies each element by a corresponding + weight. The spectral properties of this operator are significantly + influenced by these weights. + + + + The spectrum of a linear operator , denoted by + >, comprises the set of complex numbers + > for which *I> is not invertible, where + is the identity operator. The spectrum includes eigenvalues and + other values where the operator is not invertible. + + + + The real part of an operator , denoted as + >, is given by + *>|)>>, where + >> is the adjoint of . This operator is + self-adjoint, even if itself is not. + + + + The statement in question is that the spectrum of the real part of a + weighted shift operator is the closed interval >. + This can be understood through the following considerations: + + <\itemize> + The weights of the shift operator, bounded by 1 in absolute value, + limit the operator norms of and >> to 1. + + The real part of combines and + >> and inherits their spectral characteristics. + + Consequently, the spectrum of the real part lies within + >, as the operator norms are bounded and the + spectrum of a self-adjoint operator is always within the interval defined + by its norm. + + + + + The spectrum of the real part of a weighted shift operator being within the + interval > is a result derived from the bounded + nature of the operator and its adjoint, as well as the spectral theorem for + self-adjoint operators. This theorem asserts that a self-adjoint operator + on a Hilbert space has a real spectrum, comprising eigenvalues or + continuous spectrum intervals, which in this case is the interval + >. + + <\bibliography|bib|tm-plain|refs> + <\bib-list|1> + J.DOMBROWSKI. + Spectral properties of real parts of weighted shift + operators. , 29(2):249\U259, 1980. + + + + +<\initial> + <\collection> + + + + + + + + +<\references> + <\collection> + > + > + > + > + > + > + > + > + > + + + +<\auxiliary> + <\collection> + <\associate|bib> + RealWeightedShiftOperatorSpectra + + <\associate|toc> + |math-font-series||1Introduction> + |.>>>>|> + + + |math-font-series||2Preliminaries> + |.>>>>|> + + + |2.1Weighted Shift Operator + |.>>>>|> + > + + |2.2Spectrum of an Operator + |.>>>>|> + > + + |2.3Real Part of an Operator + |.>>>>|> + > + + |math-font-series||3Spectrum + of the Real Part of a Weighted Shift Operator> + |.>>>>|> + + + |math-font-series||4Conclusion> + |.>>>>|> + + + |math-font-series||Bibliography> + |.>>>>|> + + + + \ No newline at end of file diff --git a/docs/refs.bib b/docs/refs.bib index ce5167d40..8c8708cbf 100644 --- a/docs/refs.bib +++ b/docs/refs.bib @@ -24,7 +24,7 @@ @Book{yaglom1987 lccn = {86010167}, url = {https://books.google.com/books?id=HaAZAQAAIAAJ}, } -@article{c4864bb7-afee-3559-82e7-373ab33458a1, +@article{RealWeightedShiftOperatorSpectra, ISSN = {00222518, 19435258}, URL = {http://www.jstor.org/stable/24892893}, author = {J. DOMBROWSKI}, diff --git a/docs/refs.bib~ b/docs/refs.bib~ deleted file mode 100644 index f074cec81..000000000 --- a/docs/refs.bib~ +++ /dev/null @@ -1,26 +0,0 @@ -% Encoding: UTF-8 - -@Book{RieszNagyFA, - title = {Functional Analysis}, - publisher = {F. Ungar Pub. Co.}, - year = {1955}, - author = {Riesz, Frigyes and Szőkefalvi-Nagy, Béla}, - editor = {Translated from the 2nd French ed. by Leo F. Boron.}, - address = {New York}, - booktitle = {Functional analysis}, - keywords = {Harmonic analysis}, - language = {eng}, - lccn = {55008437 /L}, -} - -@Comment{jabref-meta: databaseType:bibtex;} -@Book{yaglom1987correlation, - title = {Correlation Theory of Stationary and Related Random Functions: Volume I: Basic Results}, - publisher = {Springer New York}, - year = {1987}, - author = {Yaglom, A.M.}, - series = {Applied Probability}, - isbn = {9783540962687}, - lccn = {86010167}, - url = {https://books.google.com/books?id=HaAZAQAAIAAJ}, -}