Document Zbl 1116.47001

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The <strong>and-operator &</strong> is default and can be omitted.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=Chebyshev+%7C+Tschebyscheff">Chebyshev | Tschebyscheff</a></span> <span class="search-explanation" role="cell">The <strong>or-operator |</strong> allows to search for <em>Chebyshev</em> or <em>Tschebyscheff</em>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=Quasi%2A+map%2A+py%3A+1989">Quasi* map* py: 1989</a></span> <span class="search-explanation" role="cell">The resulting documents have <strong><u>p</u>ublication <u>y</u>ear</strong> <em>1989</em>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=so%3A+Eur%2A+J%2A+Mat%2A+Soc%2A+cc%3A+14">so: Eur* J* Mat* Soc* cc: 14</a></span> <span class="search-explanation" role="cell">Search for publications in a particular <strong><u>so</u>urce</strong> with a <strong>Mathematics Subject <u>C</u>lassification <u>c</u>ode (<u>cc</u>)</strong> in <em>14</em>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=%22Partial+diff%2A+eq%2A%22+%21+elliptic">"Partial diff* eq*" ! elliptic</a></span> <span class="search-explanation" role="cell">The <strong>not</strong>-operator <strong>!</strong> eliminates all results containing the word <em>elliptic</em>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=dt%3A+b+%26+au%3A+Hilbert">dt: b & au: Hilbert</a></span> <span class="search-explanation" role="cell">The <strong><u>d</u>ocument <u>t</u>ype</strong> is set to books; alternatively: <u>j</u> for <strong>journal articles</strong>, <u>a</u> for <strong>book articles</strong>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=py%3A+2000-2015+cc%3A+%2894A+%7C+11T%29">py: 2000-2015 cc: (94A | 11T)</a></span> <span class="search-explanation" role="cell">Number <strong>ranges</strong> are accepted. Terms can be grouped within <strong><u>(</u>parentheses<u>)</u></strong>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=la%3A+chinese">la: chinese</a></span> <span class="search-explanation" role="cell">Find documents in a given <strong><u>la</u>nguage</strong>. <a href="http://en.wikipedia.org/wiki/ISO_639-1">ISO 639-1</a> language codes can also be used.</span> </div> </div> <div id="help-fields"> <h2>Fields</h2> <table> <tr> <td class="nowrap padding" role="rowheader"><strong>any</strong></td> <td class="padding">anywhere</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>an</strong></td> <td class="padding">internal document identifier</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>au</strong></td> <td class="padding">author, editor</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>ai</strong></td> <td class="padding">internal author identifier</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>ti</strong></td> <td class="padding">title</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>la</strong></td> <td class="padding">language</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>so</strong></td> <td class="padding">source</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>ab</strong></td> <td class="padding">review, abstract</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>py</strong></td> <td class="padding">publication year</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>rv</strong></td> <td class="padding">reviewer</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>cc</strong></td> <td class="padding">MSC code</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>ut</strong></td> <td class="padding">uncontrolled term</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>dt</strong></td> <td class="padding" colspan="4">document type (<strong>j</strong>: journal article; <strong>b</strong>: book; <strong>a</strong>: book article)</td> </tr> </table> </div> <div id="help-operators"> <h2>Operators</h2> <table> <tr> <td class="nowrap padding" role="rowheader">a <strong>&</strong> b</td> <td class="padding">logic and</td> </tr> <tr> <td class="nowrap padding" role="rowheader">a <strong>|</strong> b</td> <td class="padding">logic or</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>!</strong>ab</td> <td class="padding">logic not</td> </tr> <tr> <td class="nowrap padding" role="rowheader">abc<strong>*</strong></td> <td class="padding">right wildcard</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>"</strong>ab c<strong>"</strong></td> <td class="padding">phrase</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>(</strong>ab c<strong>)</strong></td> <td class="padding">parentheses</td> </tr> </table> </div> <p> See also our <a href="/general-help/">General Help</a>. </p></div> </div> </form> <div class="clearfix"></div> </div> <div class="content-shadow"></div> </div> </div> <div id="body"> <div id="main"> <div class="messages"> </div> <div id="documents"> <div class="content-main"> <div class="content-item"><div class="item"> <article> <div class="author"><a href="/authors/martinez-avendano.ruben-a" title="Author Profile">Martínez-Avendaño, Rubén A.</a>; <a href="/authors/rosenthal.peter" title="Author Profile">Rosenthal, Peter</a></div> <h2 class="title"> <strong>An introduction to operators on the Hardy–Hilbert space.</strong> <i>(English)</i> <a class="label nowrap" href="/1116.47001">Zbl 1116.47001</a> </h2> <div class="source"> <a href="/serials/3917" title="Series Profile">Graduate Texts in Mathematics</a> 237. New York, NY: Springer (ISBN 0-387-35418-2/hbk; 0-387-48578-3/hbk). xii, 220 p. (2007). </div> <div class="abstract">This text is a gentle introduction to the “concrete” operator theory on the Hardy-Hilbert space $H^ 2$, an important model space for operator theory on function spaces. The book appears to require only some minimum knowledge from real analysis, measure theory, analytic functions and functional analysis, while at the same time offering a rather complete background needed for doing research on this rich subject. Most of the material presented is either classical or well-known, and all relevant basic theorems are covered. Towards the end of some chapters, one can also notice a touch of the authors’ recent results, as they are both experts in this general field.<br class="zbmathjax-paragraph">The contents are as follows: 1. Introduction (including some basic facts about the Hardy space and Poisson kernel and also about linear operators, their spectrum and numerical range) 2. The Unilateral Shift and Factorization of Functions (including the inner-outer factorization, Blaschke products, the Müntz-Szasz theorem, invariant and reducing subspaces) 3. Toeplitz Operators (with an overview of the Toeplitz matrices and the basic spectral theory of Toeplitz operators) 4. Hankel Operators (with a discussion of Nehari’s theorem, finite rank, compact, self-adjoint and normal Hankel operators and their relationship with Toeplitz operators) 5. Composition Operators (covering the norm bounds and compactness, invertibility, eigenvalues, and operators induced by disk automorphisms) 6. Further Reading.<br class="zbmathjax-paragraph">Each chapter contains a number of exercises. Some are basic, while others have been taken from various recent papers on the subject but have been carefully organized and divided into smaller units. Historical notes and remarks are also included. The bibliography is exhaustive and up to date.<br class="zbmathjax-paragraph">The book seems completely suitable for a first- or second-year graduate course in function spaces and operator theory (with the emphasis on operators rather than on the spaces themselves) and can probably be adapted to both a basic and a more advanced course. It should be possible to cover the entire content in at most two semesters, much of it even in one term. Given that the material is rather focused and goal-oriented, that the proofs are detailed and the examples are numerous, the reader should not have a hard time following the details of the exposition in the present book, which will likely become a popular graduate textbook on the subject.<div class="reviewer"> Reviewer: <a href="/authors/?q=rv%3A10045">Dragan Vukotić (Madrid)</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A5080402">Cited in <strong>104</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A47-01" title="MSC2020">47-01</a> </td> <td class="space"> Introductory exposition (textbooks, tutorial papers, etc.) pertaining to operator theory </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A46E15" title="MSC2020">46E15</a> </td> <td class="space"> Banach spaces of continuous, differentiable or analytic functions </td> </tr><tr> <td class="mono"> 30D55 </td> <td class="space"> $H^p$-classes (MSC2000) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A30H05" title="MSC2020">30H05</a> </td> <td class="space"> Spaces of bounded analytic functions of one complex variable </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A47B33" title="MSC2020">47B33</a> </td> <td class="space"> Linear composition operators </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A47B35" title="MSC2020">47B35</a> </td> <td class="space"> Toeplitz operators, Hankel operators, Wiener-Hopf operators </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3AHardy+space">Hardy space</a>; <a href="/?q=ut%3Aunilateral+shift+operator">unilateral shift operator</a>; <a href="/?q=ut%3AToeplitz+operators">Toeplitz operators</a>; <a href="/?q=ut%3AHankel+operators">Hankel operators</a>; <a href="/?q=ut%3Acomposition+operators">composition operators</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1116.47001" data-ciurl="/ci/05080402" data-biburl="/bibtex/05080402.bib" data-amsurl="/amsrefs/05080402.bib" data-xmlurl="/xml/05080402.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/05080402.pdf" title="Zbl 1116.47001 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1007/978-0-387-48578-2" aria-label="DOI for “An introduction to operators on the Hardy--Hilbert space”" title="10.1007/978-0-387-48578-2">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> </article> </div></div> </div> </div> <div class="clearfix"></div> </div> </div> <div id="foot"><div class="copyright"> © 2025 <a target="fiz" href="https://www.fiz-karlsruhe.de/en">FIZ Karlsruhe GmbH</a> <a href="/privacy-policy/">Privacy Policy</a> <a href="/legal-notices/">Legal Notices</a> <a href="/terms-conditions/">Terms & Conditions</a> <div class="info"> <ul class="nav"> <li class="mastodon"> <a href="https://mathstodon.xyz/@zbMATH" target="_blank" class="no-new-tab-icon"> <img src="/static/mastodon.png" title="zbMATH at Mathstodon (opens in new tab)" alt="Mastodon logo"> </a> </li> </ul> </div> </div> <div class="clearfix" style="height: 0px;"></div> </div> </div> <script src="https://static.zbmath.org/contrib/jquery/1.9.1/jquery.min.js"></script> <script src="https://static.zbmath.org/contrib/jquery-caret/1.5.2/jquery.caret.min.js"></script> <script src="/static/js/jquery-ui-1.10.1.custom.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap/v3.3.7zb1/js/bootstrap.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-lightbox/v0.7.0/bootstrap-lightbox.min.js"></script> <script src="https://static.zbmath.org/contrib/retina/unknown/retina.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-select/v1.13.14/js/bootstrap-select.min.js"></script> <script> var SCRIPT_ROOT = ""; 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