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under relatively tame requirements, a continuation of a
Hermitian operator exists which is self-adjoint, the domain of this continuation is contained in the domain of the adjoint, and most importantly it is equal to A* everywhere it is defined. This essentially reduces the question most of the time to a question of what the appropriate domain is. With the common convention of considering continuations of operators to be the same operator(Although this is formally not the case), we have roughly the synonimity in the statement. I don't have a reference, but a necessary and sufficient condition for the self-adjoint continuation to exist is that the defect indices of the quasiregular points of the operator must be equal to each other. This is clearly not always the case (Which is what the word "usally" probably means in this context)--
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don't agree with the merge. In fact, I don't recall ever reading anything that refered to the adjoint of an operator on a
Hilbert Space as the Conjugate Transpose. Anyway keeping the articles seperate helps to keep a distinction between linear algebra and operator theory. There is enough confusion between the two among undergraduate math students as it is.--
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I included the definition of adjoint operators for operators between normed spaces from Brezis' book on functional analysis. Please check, I'm new to this. I will probably include the case where both spaces are
Hilbert spaces as an example tomorrow. Also there is need for some rewriting of the intro
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A is symmetric, A is densely defined, and dom (A) = dom (A^*). Equivalently: A is densely defined, A=A^* on dom(A) and dom (A) = dom (A^*). Equivalently, this is also written simply as A=A^* and this is taken to include the statement about A being densely defined (so that A^* exists) and the domains
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The article states that one can prove the existence of the adjoint operator using the Riesz representation theorem for the dual of
Hilbert spaces. While this is certainly true it glosses over a nice, short, instructive proof which I'd like to add to the article. If there are no objections I'll add a
1345:
Please could someone provide an accessible reference to clarify the statement posted above that "Hermiticity and self-adjoitness are synonymous. (the notation A* = A implicitly implies Dom(A) = Dom(A*), usually.)" In particular, could someone please explain the usage of "usually" in this context.
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I agree. I encounter discussions of operators in quantum mechanics that go back and forth between mention of operators, as such, and matrix representations. BUT operators can be considered quite apart from the matrix representations, and I take for granted manipulations of other representations of
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I included an informal definition to ease into the notion of adjoint operators and mentioned the "mixed" case (operator from
Hilbert to Banach), which is especially interesting when one considers for example A as the inclusion operator from some Hilbert space which is a proper subset of a Banach
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Sometimes when people write A=A* they only mean that the equality is on dom(A). Note that For a
Hermitian operator Dom(A) is always a subset of Dom(A*). But inclusion in the other direction is not necessary. In my understanding of the subject, this implies that the terms are not synonymous, BUT:
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No merge. In order to do this don't we need to "broaden" the definiton of a matrix to include infinite dimensional matrices? Even then this only allows us to consider spaces isomorphic to L and the matrix representation only converges for bounded operators(right? Maybe my memory is faulty). So I
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Adjoints of operators generalize conjugate transposes of square matrices to (possibly) infinite-dimensional situations. If one thinks of operators on a
Hilbert space as "generalized complex numbers", then the adjoint of an operator plays the role of the complex conjugate of a complex
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1295:. A matrix is a matrix, and an operator is an operator. Of course there is a relation between those objects (and indeed an isomorphism between the spaces conataining them), but both articles should definitely remain separated in my opinion --
937:{\displaystyle \displaystyle {\begin{array}{rl}||T^{*}T||&=\sup _{||x||=1}|\langle T^{*}Tx,x\rangle |=\sup _{||x||=1}|\langle Tx,Tx\rangle |\\&=\sup _{||x||=1}||Tx||^{2}=\left(\sup _{||x||=1}||Tx||\right)^{2}=||T||^{2}\end{array}}}
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the operators, and extenstions that cannot be handled so easily by matrices. And I am opposed to the appalling proliferation of articles on interrelated topics and am in favour of many mergers elsewhere.
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Clear specification of the nature of objects represented by symbols and the circumstances needed for a statement containing these to be true in mathematical discourse is not an unreasonable request.
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Do so, I'm interested by that proof. Also I will change the first line: we are implicitely talking about bounded operator. In case of unbounded one has to say something about the domain.
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So doesn't quite make sense to say self-adjointness is not guaranteed by
Hermiticity. on the other hand, being symmetric does not imply an operator is self-adjoint/Hermitian in general.
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Since this is all about inner products, wouldn't it be possible to start the "information definition" section with a simple example involving finite-dimensional vectors and matrices?
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In particular, the concepts of "symmetric operator" and "self-adjoint operator" are equally valid, and have formally the same definition, both for real and complex vector spaces.
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Sometimes a solution to a problem involving an adjoint converts into a solution to the original problem as for example in the case of some second order differential equations.
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1733:(leading to Hermitian inner product): B is sesquilinear and B(x,y) = bar (B(y,x)). Defined for complex vector spaces and given by a self-adjoint (or Hermitian) matrix.
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Is the terminology "self-adjoint" only defined for complex
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It is somewhat different for operators (linear transformations) than for matrices and bilinear/sesquilinear forms). That is perhaps why it is confusing.
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If this is still too vague, perhaps we could add a line about switching over linear operators from vector spaces to their dual (like with vectors
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This holds in both finite and infinite dimensions, and for both bounded and unbounded operators. In the case of unbounded operators,
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378:{\displaystyle \displaystyle {\begin{array}{rl}A^{*}&=A,{\text{ and}}\\{\text{dom}}(A^{*})&={\text{dom}}(A).\end{array}}}
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Neither I easily found a link from this article to adjoint of general lin. bound. operator nor is it coevered in this article.
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A matrix is symmetric if it is equal to its own transpose. It is self-adjoint, or
Hermitian, if it is equal to its own adjoint.
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is perfectly admissible) and I believe your change is useless. I believe I saw more often (say 60% to 40%) the sentence that
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This page explains what adjoints are, but just reading this page it isn't clear why it might be useful to define "adjoint".
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space (as in Cameron-Martin spaces). I got most of the input from Brezis' book. Will probably include some examples soon.
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section that contains a proof of the existence of adjoint operators using sesquilinear forms, as in the book by Kreyszig.
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So for operators (in contrast to matrices!), the distinction between symmetric and self-adjoint has to do with the
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B is bilinear B(x,y) = B(y,x). Defined for both real and complex vector spaces and is given by a symmetric matrix.
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I will try to draft a generalization for operators between Banach spaces and distinct Hilbert spaces tomorrow.
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I don't really see how this follows from the properties above it. Can someone provide a simple proof for this?
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symbol or else define it? With the prevalence of ambiguous notation concerning conjugates and adjoints etc (
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So for real matrices, self-adjoint and symmetric are the same. For complex matrices, they are different.
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Notation was A, A* in matrix notation. I changed them to linear maps, so it is like T*(x) T(x) etc.
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at least one concrete example would be nice! otherwise it seems A*=A^{-1} as in A*A=AA*=I... anyone?
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of the operators, not the use of the conjugate. It is an analytic difference, not an algebraic one.
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on Knowledge. If you would like to participate, please visit the project page, where you can join
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operator. (This was the question asked above.) But there is one difference: as soon as you say
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This seems like a serious omission. How can I flag an article for having a too-narrow scope?
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strange thing about this article: isn't the adjoint also defined when the two spaces are
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Note that if A is symmetric and A is densely defined, then dom(A) ⊆ dom(A^*) always.
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for matrices and for bilinear/sequilinear forms than it is for operators.
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Hermiticity does not necessarily guarantee the latter statement.
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How about adding a simple concrete nontrivial example, or two?
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http://www.encyclopediaofmath.org/index.php/Symmetric_operator
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without parentheses for the action of an operator on a vector
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Changed A and A* to T and T* as functions rather than matrices
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Adjoint of a bounded linear operator between normed spaces
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is a positive operator on the Hilbert space, compared to
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for appropriate definitions and clarifying statements.
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The adjoint of a matrix is its transpose conjugate.
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201:{\displaystyle \langle Ax,y\rangle }
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1826:Mid-priority mathematics articles
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1132:I think you are wrong. Notation
115:Knowledge:WikiProject Mathematics
1821:Start-Class mathematics articles
1718:For bilinear/sequilinear forms:
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109:and see a list of open tasks.
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1586:must be densely defined for
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1723:Symmetric bilinear form:
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141:project's priority scale
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1196:{\displaystyle T^{*}T.}
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98:WikiProject Mathematics
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1600:
1599:178.38.97.101
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1568:
1564:
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1556:
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1510:178.38.97.101
1507:
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1471:
1470:
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1467:
1463:
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1387:
1386:178.38.97.101
1383:
1379:
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1373:
1369:
1365:
1364:129.69.21.121
1360:
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1044:
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1039:131.111.8.103
1036:
1026:
1020:
1017:
1013:
1012:
1011:
1008:
1004:
1000:
997:
993:
989:
987:
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165:
164:inner product
158:Section title
157:
142:
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129:
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108:
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100:
99:
91:
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80:
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60:
57:
54:
50:
45:
41:
35:
27:
23:
18:
17:
1798:62.80.108.37
1792:— Preceding
1789:
1773:Valiantrider
1769:
1758:Valiantrider
1754:
1736:
1730:
1722:
1717:
1697:
1694:
1688:
1685:self-adjoint
1684:
1680:
1676:
1663:
1652:being equal.
1648:
1640:<.,.: -->
1637:
1632:
1629:
1613:Valiantrider
1610:
1596:
1587:
1583:
1581:
1573:
1566:
1562:
1558:
1554:
1550:
1548:
1539:
1532:
1528:
1524:
1508:
1505:
1478:
1458:93.96.22.178
1405:
1344:
1324:
1309:
1289:
1273:
1251:
1247:
1137:
1133:
1103:
1075:— Preceding
1071:
1056:
1030:
1002:
998:
995:
991:
985:
983:
979:
975:
967:
951:
503:
423:
390:
387:
261:
253:
247:
208:rather than
172:
161:
137:Mid-priority
136:
96:
62:Mid‑priority
40:WikiProjects
1557:? That is,
1488:Incnis Mrsi
1107:—Preceding
1060:Compsonheir
1033:—Preceding
249:PJ.de.Bruin
112:Mathematics
103:mathematics
59:Mathematics
30:Start-class
1815:Categories
428:Moreover,
1698:different
1689:Hermitian
1681:symmetric
1283:Ladsgroup
1113:Negi(afk)
1007:CompuChip
413:See redux
1794:unsigned
1555:distinct
1549:Another
1531:or from
1468:Erikpan
1297:Vilietha
1121:contribs
1109:unsigned
1089:contribs
1077:unsigned
1035:unsigned
990:, where
1664:domains
969:number.
402:Mct mht
139:on the
1254:s. --
1081:Noix07
420:Proof?
36:scale.
1270:Merge
948:Uses?
173:I.e.
1802:talk
1777:talk
1762:talk
1742:talk
1617:talk
1603:talk
1551:very
1514:talk
1492:talk
1481:Done
1462:talk
1390:talk
1380:and
1376:See
1368:talk
1352:talk
1332:talk
1317:talk
1301:talk
1260:talk
1256:Bdmy
1117:talk
1085:talk
1064:talk
1043:talk
1005:? --
986:and
980:bras
976:kets
392:Dirc
1594:.
1545:???
1538:to
1527:to
978:to
825:sup
747:sup
671:sup
594:sup
355:dom
329:dom
320:and
131:Mid
1817::
1804:)
1779:)
1764:)
1744:)
1619:)
1605:)
1579:.
1516:)
1494:)
1486:.
1464:)
1443:⊥
1414:⊥
1392:)
1370:)
1354:)
1334:)
1319:)
1307:.
1303:)
1280::)
1278:.
1262:)
1183:∗
1153:∗
1123:)
1119:•
1091:)
1087:•
1066:)
1045:)
994:=
984:v'
729:⟩
711:⟨
659:⟩
642:∗
634:⟨
570:∗
523:∗
475:‖
468:‖
453:‖
444:∗
436:‖
341:∗
303:∗
245:.
233:⟩
216:⟨
196:⟩
181:⟨
170:?
1800:(
1775:(
1760:(
1740:(
1615:(
1601:(
1588:A
1584:A
1577:2
1574:H
1572:→
1570:1
1567:H
1565::
1563:A
1559:A
1543:2
1540:H
1536:1
1533:H
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1525:H
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1234:x
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1188:T
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1158:A
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1003:A
999:x
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988:w
925:2
920:|
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910:T
906:|
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892:2
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882:|
877:|
873:x
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848:|
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663:|
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653:,
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630:|
624:1
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617:|
612:|
608:x
604:|
599:|
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584:|
579:|
575:T
566:T
561:|
556:|
528:A
519:A
487:2
482:p
479:o
471:A
465:=
460:p
457:o
449:A
440:A
368:.
365:)
362:A
359:(
351:=
346:)
337:A
333:(
316:,
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310:=
299:A
270:A
230:y
226:|
222:x
219:A
193:y
190:,
187:x
184:A
143:.
42::
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