Document Zbl 0148.12601

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Band 132. Berlin-Heidelberg-New York: Springer-Verlag. xx, 592 p. with 3 figures (1966). </div> <div class="abstract">Das vorliegende Buch, eine Kombination von Monographie und Lehrbuch, enth盲lt die erste systematische Darstellung der &bdquo;St枚rungstheorie linearer Operatoren“ (dieser Terminus wird verwendet, um von anderen mathematischen St枚rungstheorien, wie sie z. B. in der Himmelsmechanik oder der nichtlinearen Schwingungslehre vorkommen, zu unterscheiden), die sich im Laufe der letzten 15 Jahre zu einer eigenst盲ndigen mathematischen Disziplin entwickelt hat. Auf diese Entwicklung haben immer wieder Problemstellungen und Begriffsbildungen von physikalischer Seite her, insbesondere aus der Quantentheorie, f枚rdernd eingewirkt. Der Verf. selbst hat an ihr bedeutenden Anteil. Sie ist auch heute noch keineswegs abgeschlossen. Vielmehr stehen gegenw盲rtig besonders die beiden im Buch behandelten Teildisziplinen &bdquo;Asymptotische St枚rungstheorie“ (instabile Eigenwerte, Spektralkonzentration, Kapitel VIII) und &bdquo;St枚rungstheorie des kontinuierlichen Spektrums“ (im Zusammenhang mit der Streutheorie der Quantenmechanik, Kapitel X) im Blickpunkt des Interesses. Man kann sicher sein, da脽 das Erscheinen des vorliegenden Buches dieser Entwicklung weiteren Auftrieb und neue Impulse geben wird. <br class="zbmathjax-paragraph">Das Buch gliedert sich in 10 Kapitel. Da es sich auch an Physiker wendet, werden auch die der Funktionalanalysis angeh枚renden Grundlagen der Theorie in knapper Form dargestellt. Ausz眉ge aus dem Inhalt in Stichworten m枚gen die Stoffverteilung erl盲utern. <br class="zbmathjax-paragraph">Kapitel I. Operatortheorie in endlichdimensionalen Vektorr盲umen. Die grundlegenden Resultate der linearen Algebra meist ohne Beweise, die Operatoranalysis ausf眉hrlicher. Funktionentheoretische Untersuchung der Resolvente liefert die kanonische (Jordansche) Form eines Operators. <br class="zbmathjax-paragraph">Kapitel II. Analytische St枚rungstheorie beliebiger Operatoren in endlichdimensionalen R盲umen. Resolventeneigenschaften liefern die St枚rungsreihen f眉r Eigenprojektoren und den Mittelwert einer A-Gruppe von Eigenwerten des gest枚rten Operators. Reduktionsmethode (f眉r halbeinfache Eigenwerte). Absch盲tzung von Konvergenzradien und Fehlerabsch盲tzungen. Nichtanalytische St枚rungen, St枚rung symmetrischer Operatoren. <br class="zbmathjax-paragraph">Kapitel III. In sich geschlossene Einf眉hrung in sp盲ter ben枚tigte Teile der Theorie der Banachr盲ume und ihrer Operatoren (nur die Beweise einiger fundamentaler S盲tze sind weggelassen). Spektraltheorie mit Hilfe von Resolventeneigenschaften. Spektrumszerlegungssatz, endliche Systeme isolierter Eigenwerte. <br class="zbmathjax-paragraph">Kapitel IV. Stabilit盲tseigenschaften der f眉r die Anwendungen wichtigen relativ beschr盲nkten St枚rungen. Definition des Begriffs einer &bdquo;kleinen“ St枚rung eines abgeschlossenen Operators (mittels einer Metrik im Raum $\mathcal C(X, X)$, die auf dem Begriff der 脰ffnung zweier Teilr盲ume beruht – verallgemeinerte Konvergenz abgeschlossener Operatoren), Untersuchung von Stabilit盲tseigenschaften (z. B. Stabilit盲t der beschr盲nkten Invertierbarkeit, obere Halbstetigkeit des Spektrums, Stetigkeit der Resolvente). Stabilit盲tsaussagen f眉r Fredholmsche und semi-Fredholmsche Operatoren (Indextheorem). <br class="zbmathjax-paragraph">Kapitel V. Hilfsmittel aus der Theorie der Hilbertr盲ume (unter anderem Studium von akkretiven und $m$-akkretiven Operatoren sowie deren Quadratwurzeln). St枚rungen selbstadjungierter Operatoren (Stabilit盲t der Selbstadjungiertheit, St枚rung des Spektrums, nichtselbstadjungierte St枚rung selbstadjungierter Operatoren). Anwendungen auf die Schr枚dinger- und Dirac-Operatoren der Quantenmechanik. <br class="zbmathjax-paragraph">Kapitel VI. Sesquilinearformen und quadratische Formen (speziell sektorielle Formen) und zugeordnete Operatoren (1. und 2. Darstellungssatz). Theorie der (relativ beschr盲nkten) St枚rungen der Formen bzw. der zugeordneten Operatoren und ihrer Resolventen. Anwendungen auf gew枚hnliche Differentialoperatoren sowie den Laplace- und den Schr枚dinger-Operator. Spektralsatz selbstadjungierter Operatoren, Stabilit盲tseigenschaften der Spektralschar gest枚rter selbstadjungierter Operatoren. <br class="zbmathjax-paragraph">Kapitel VII. Analytische St枚rungstheorie. Definition einer holomorphen Operatorschar sowie Herleitung verschiedener hinreichender Kriterien f眉r die Holomorphie (dementsprechend Unterscheidung verschiedener Typen holomorpher Scharen, $A$, $B$, $B_0$, $C$). Im 眉brigen, 脺bertragung der S盲tze aus Kapitel II (Separation endlicher Eigenwertsysteme endlicher algebraischer Vielfachheit, funktionentheoretische Aussagen 眉ber gest枚rte Eigenwerte und Eigenprojektoren (S盲tze von Rellich bzw. Verallgemeinerungen von Kato, Sz.-Nagy und Wolf), St枚rungsreihen, Reduktionsverfahren). <br class="zbmathjax-paragraph">Kapitel VIII. Betrachtung allgemeinerer (nicht-holomorpher) St枚rungen (asymptotische St枚rungen). Definition mittels sogenannter verallgemeinerter starker Konvergenz, die auf der starken Konvergenz der Resolvente beruht. M枚glichkeit der Instabilit盲t eines isolierten Eigenwertes endlicher Vielfachheit (obere Halbstetigkeit des Spektrums ist verletzt). Stabilit盲t eines isolierten Eigenwertes f眉hrt zur sogenannten asymptotischen Theorie der gest枚rten Eigenwerte und -vektoren (asymptotische Interpretation der formalen St枚rungsreihen), Instabilit盲t zum Ph盲nomen der Spektralkonzentration und der Theorie der Pseudoeigenwerte und -vektoren (neue Interpretation der formalen St枚rungsterme, soweit sie existieren). <br class="zbmathjax-paragraph">Kapitel IX. Grundlegende Resultate der Operatorhalbgruppentheorie (quasibeschr盲nkte, insbesondere holomorphe Halbgruppen). St枚rungstheorie und Approximationstheorie (nach Trotter) von Halbgruppen. <br class="zbmathjax-paragraph">Kapitel X. Das kontinuierliche Spektrum eines selbstadjungierten Operators, absolutstetiges Spektrum. St枚rung des kontinuierlichen Spektrums (Satz von Weyl–von Neumann). Wellenoperatoren und ihre Eigenschaften. Existenz der Wellenoperatoren bei nuklearen St枚rungen liefert die Stabilit盲t des absolutstetigen Spektrums und die Unit盲r盲quivalenz der absolutstetigen Teile des ungest枚rten und gest枚rten Operators. Invarianz der Wellenoperatoren. Eine station盲re Methode [auf der Grundlage der $\Gamma$-Operationen (Friedrichssche Gleichungen)]. <br class="zbmathjax-paragraph">Die enge Verbindung der entwickelten Theorie zu Anwendungen wird deutlich durch st盲ndig eingestreute, h盲ufig sp盲ter mit neuem Akzent wieder aufgenommene Beispiele (und auch Probleme), oft aus der Theorie linearer gew枚hnlicher Differentialoperatoren zweiter Ordnung. Der Leser ist dadurch angehalten, die allgemeinen Begriffsbildungen und Entwicklungen stets am konkreten Material zu 眉berpr眉fen. Das Arbeiten mit dem Buch ist weiterhin erleichtert durch ein Register der verwendeten Symbole, ein Sach- und ein Autorenregister sowie eine umfangreiche Bibliographie (getrennt f眉r Lehrb眉cher/Monographien und Originalarbeiten). (Interessant ist vielleicht, da脽 von den 281 aufgef眉hrten Titeln 92% nach 1946 und 64% nach 1956 erschienen sind. Das zeigt, welche sprunghafte Entwicklung, jedenfalls gemessen an der Zahl der Publikationen, die Theorie besonders seit 1957 genommen hat.)<div class="reviewer"> Reviewer: <a href="/authors/?q=rv%3A111">Hellmut Baumg盲rtel (Berlin)</a></div> <div class="clearfix"></div><div class="scan"> <span> Page: <a class="goto" href="/scans/148/121.gif">−5</a> <a class="goto" href="/scans/148/122.gif">−4</a> <a class="goto" href="/scans/148/123.gif">−3</a> <a class="goto" href="/scans/148/124.gif">−2</a> <a class="goto" href="/scans/148/125.gif">−1</a> <a class="goto" href="/scans/148/126.gif">±0</a> <a class="goto" href="/scans/148/127.gif">+1</a> <a class="goto" href="/scans/148/128.gif">+2</a> <a class="goto" href="/scans/148/129.gif">+3</a> <a class="goto" href="/scans/148/130.gif">+4</a> <a class="goto" href="/scans/148/131.gif">+5</a></span> <a class="scan" href="/scans/148/126.gif">Show Scanned Page</a><br> <img src="/scans/148/126.gif" alt="scan of Zbl 0148.12601" onclick="$(this).siblings('.scan').click();"> </div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=ci%3A3239038">Cited in <strong>15</strong> Reviews</a></div><div class="clear"><a href="/?q=rf%3A3239038">Cited in <strong>3915</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A47A55" title="MSC2020">47A55</a> </td> <td class="space"> Perturbation theory of linear operators </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A47A10" title="MSC2020">47A10</a> </td> <td class="space"> Spectrum, resolvent </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A47-02" title="MSC2020">47-02</a> </td> <td class="space"> Research exposition (monographs, survey articles) pertaining to operator theory </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A46Cxx" title="MSC2020">46Cxx</a> </td> <td class="space"> Inner product spaces and their generalizations, Hilbert spaces </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A81Q15" title="MSC2020">81Q15</a> </td> <td class="space"> Perturbation theories for operators and differential equations in quantum theory </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Aperturbation+theory">perturbation theory</a>; 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