Spectral Mapping Theorems: A Bluffer's Guide

131 Pages·2014·1.458 MB·English

by  Robin Harte (auth.)

SPRINGER BRIEFS IN MATHEMATICS
Robin Harte
Spectral Mapping
Theorems
A Bluffer‘s Guide
123
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Robin Harte
Spectral Mapping Theorems
A Bluffer’s Guide
123
Robin Harte
School ofMathematics
TrinityCollege
Dublin
Ireland
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ISSN 2191-8198 ISSN 2191-8201 (electronic)
ISBN 978-3-319-05647-0 ISBN 978-3-319-05648-7 (eBook)
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Foreword
Suppose ab = ba [ A, a commuting pair of Banach algebra elements (a, b) [ A2;
then
rða þ bÞ(cid:2)rðaÞ þ rðbÞ; rðabÞ(cid:2)rðaÞrðbÞ :
thespectrumofthesumandtheproductaresubsetsofthesumandtheproductof
the spectra. One way to prove this is via Gelfand’s theorem: find a ‘‘maximal
abelian’’ subalgebra D ( A for which {a, b} ( D, and argue
r ða þ bÞ ¼ fuða þ bÞ : u2rðDÞg(cid:2)fwðaÞ þ hðbÞ : fw; hg(cid:2)rðDÞg
A
and similarly
r ðabÞ ¼ fuðabÞ : u2rðDÞg(cid:2)fwðaÞhðbÞ : fw; hg(cid:2)rðDÞg:
A
Here r(D) is the ‘‘maximal ideal space’’ of the commutative Banach algebra
D and we need to know that there is implication
c2D ¼)r ðcÞ ¼ r ðcÞ ¼ fuðcÞ : u2rðDÞg:
A D
Much sweeter would be a joint spectrum argument, enabling us to write
rða þ bÞ ¼ fk þ l : ðk; lÞ2rða; bÞg; rðabÞ ¼ fkl : ðk; lÞ2rða; bÞg
with
rða; bÞ(cid:2)rðaÞ (cid:3) rðbÞ:
That, in a nutshell, is what these notes are all about.
In more detail, we set out to describe the spectral mapping theorem in one,
‘‘several’’ and ‘‘many’’ variables. As background we need to introduce the basic
algebraic systems, including semigroups, rings and linear algebras.
Inasense,abstractalgebraisthestoryofacontinuingnegotiationbetweenthe
‘‘invertible’’andthe‘‘singular.’’Theinvertibles,followersof‘‘one,’’treadlightly
in the garden, doing nothing that cannot be undone; the singularshowever follow
‘‘zero,’’ and have an altogether heavier tread. The garden where all these games
are tobe played willbe somethingcalleda Banach algebra, which, like anurban
environment, is rich in interlocking structures. It may therefore be a good idea to
vii
viii Foreword
stepbackalittletoamuchmoreprimitiveruralsetting.Algebraatitsmostbasicis
playedoutinasemigroup,asystemwitha‘‘multiplication’’;however,mostofthe
interesting semigroups are rings, which are semigroups twice over, with multi-
plication and ‘‘addition’’ tied together by a distributive law. In turn most of the
interesting rings are linear algebras, submitting to real or complex number mul-
tiplication. Basic algebra can also be extended by taking limits, whose study is
sometimes called ‘‘point set topology’’: in a ‘‘Banach algebra,’’ this topology is
generatedbyanormandametricstructure,whichhastobe‘‘complete.’’Itwillbe
abonusthatalmosteverythingwesayaboutsemigroups,orrings,remainsvalidin
abstract categories, or additive categories.
Togetherwithalgebraweneedthereforetodiscusstopology,theabstracttheory
of limits. Now topological algebra is about algebraic systems which also have
topology,insuchawaythatthefundamentalalgebraicoperationsarecontinuous.
OurmainfocusistospecializetocomplexBanachalgebras,wherewefindthatthe
spectrum is truly well-behaved: nonempty, compact, and subject to the spectral
mapping theorem.
Even in semigroups and rings the invertible and the singular are locked into a
minuet: usually products of invertibles are again invertible while often sums of
invertible and singular are invertible. Homomorphisms T : A ? B between
semigroupsorringssendinvertiblestoinvertibles,T(A-1)(B-1,whilethereare
twowaysforahomomorphismtobehavewell.One,relatingtoinvertibility,would
be the Gelfand property T-1B-1 ( A-1, while the other, relating to singularity,
would be a Riesz property, which says that for example A-1 + T-1(0) ( A is
‘‘almost’’ in A-1.
In linear algebras, we meet the idea of the spectrum, with which we can draw
real or complex pictures following the progress of the invertible/singular debate;
this then harnesses complex analysis to the theory: rather than being simply an
elegantcommentaryonevents,thecomplexpicturestoaconsiderableextentdrive
the action.
Spectral theory, therefore, is dedicated to the theory of invertibility: the spec-
trum of an algebra element a [ A simply collects those scalars k which give
perturbations a - k which fail to be invertible. Generally, various kinds of
‘‘nonsingularity’’arenecessaryconditionsforinvertibility:eachofthemgenerates
their own subset of the spectrum.
AllthisisforsingleelementsofaBanach algebra;theextensiontontuplesof
elementsgivesspectraconsistingofntuplesofcomplexnumbers:thefundamental
theorem says that for commuting tuples of elements we still have nonempty
spectrumandthespectralmappingtheoremforpolynomials.Our‘‘jointspectrum’’
could be the union of a ‘‘left’’ and a ‘‘right’’ spectrum, but this turns out to be
deficientinthesensethatitdoesnotingeneralsupportafunctionalcalculus:that
needs the more sophisticated ideas, due to Joseph Taylor, based on exactness.
There are canonical extensions, based on compactness, from finite tuples to
infinite systems: what we find interesting here is what happens when there is
additional algebraic or topological structure on the indexing material.
Foreword ix
Inseveralvariablestherearevariouswaysinwhichann-tuplea=(a ,a ,...,
1 2
a ) [ An can be considered to be ‘‘invertible,’’ thus generating a ‘‘spectrum’’ r(a)
n
( Cn. The extension to many variables looks at systems a [ AX indexed by
arbitrary sets X which now possibly carry algebraic or topological structure,
capable of being respected by the system a; here the story is that whenever this
happens it continues to happen to k [ r(a) ( CX. However, the true ‘‘multi-
variate’’ extension of the concept of invertibility is the idea of exactness.
One version of the ‘‘joint spectrum’’ of a commuting tuple, and its spectral
mapping theorem, can indeed be derived from ‘‘maximal abelian’’ subalgebras,
andGelfand’s theoremfor commutative Banach algebras; conversely ourspectral
mapping theorem for several variables, and its many variable extension, give an
alternative proof of that same Gelfand theorem.
The first appearance of the definition of the left and the right spectrum of an
ntupleseemstobeinthenumericalrangenotes[26]ofBonsallandDuncan.The
proofofthespectralmappingtheoremforcommutingfinitetupleswasfirstgiven,
for bounded operators on Hilbert space, by John Bunce [32]: the argument
involvedC*algebrasandstates.Anequivalentargument,withoutanydefinitionof
joint spectrum, is part of the Graham Allan paper [3] about holomorphic left
inverses. The spectral mapping theorem for the ‘‘Taylor spectrum’’ is in the first
[267]ofthetwogroundbreakingpapersofJosephTaylor:thesecondofthesegoes
ontoestablishthe‘‘functionalcalculus,’’which successfullydefines f(T)whenf:
U?CisholomorphiconanopenneighborhoodUofthe‘‘Taylorspectrum’’ofa
commuting tuple T = (T , T , . . ., T ) of Banach space operators. As Vladimir
1 2 n
Müller has demonstrated this is significantly easier to describe for functions
holomorphiconthesometimeslarger‘‘Taylorsplitspectrum.’’Thesplitspectrum
hasanaturalextensiontotuplesofBanachalgebraelements;weareasyetunable
to offer the natural extension to Banach algebra elements of the original Taylor
spectrum. Discussion of an abstract idea of spectrum can be traced back to
Wieslaw Zelazko [288]; Vladimir Müller [234] has cast this in the framework of
his concept of a ‘‘regularity.’’ In one variable, this is a refinement of a more
primitive idea contained in our discussion of ‘‘Kato invertibility’’ and nonsingu-
larity [139, 142, 147]. Lucien Waelbroeck [274, 275], and later Vladimir Kisil
[193, 194], have viewed a ‘‘spectrum’’ as the support of a more primitive
‘‘functional calculus.’’
The whole point of the additional subtleties of the Taylor spectrum and split
spectrum is the functional calculus [58–62, 168]. Generations of mathematicians
have been trying to capture and regulate the intuition of the engineer Heaviside,
and for many the test of a ‘‘spectrum’’ will always be whether it supports a
‘‘functional calculus.’’ In these notes, however, we stop short with the spectral
mapping theorem.
In telling our story, we have divided the narrative into six chapters: the first
three, in introductory mode, start with pure algebra and progress, through topol-
ogy, to ‘‘topological algebra,’’ while the remaining three are officially spectral
theory,progressingfromonethroughseveralto‘‘many’’variables.Whatwehave
collected together as pure algebra sometimes, in other versions of the story,
x Foreword
appearson the far side of a build up of topological or metric structures. Similarly
whatwehavecollectedtogetherastopologyissometimesintroducedpiecemealin
themiddleofotherargument.Wehavealsotried,inasense,topresenttopologyas
akindofalgebra:forexample,therelationshipbetweentopologicalboundaryand
connected hull can be given this flavor.
It is clear that there is significant overlap between these notes and our earlier
volume,‘‘InvertibilityandSingularity’’[132].Thetreatmenthereisquitedifferent
however:inourearlierbookwegivecompleteproofsofsuchthingsastheHahn-
Banach, Open Mapping and Liouville theorems, here omitted; we also devote
attention to what happens for bounded operators between incomplete normed
spaces. We believe that strategy offered valuable insight into the open mapping
theorem: between incomplete spaces open, almost open and onto are in general
differentkindsoflinearoperator,whichallcoalescewhenthespacesarecomplete.
SpectralandFredholmtheorycanalsobediscussedinanincompleteenvironment;
withhindsighthowevertheextraeffortprovesdistracting,andinthepresentnotes
weabandonthisdiscussion,lookinginsteadsometimesattheanalagousdiscussion
in the purely linear environment. While there is therefore a good deal of material
in the earlier volume which has not been reproduced here, there is also, thanks to
the passage of time, a great deal of material here which was not available to the
earlier work.
There is also overlap between both this and our earlier work with the book,
‘‘Spectraltheoryoflinearoperators’’[234],ofVladimirMüller.Therethespectral
theory of one and several variables is systematically expounded through the
mediumoftheMüllerconceptof‘‘regularity.’’Thesinglevariableversionofthis
concept,anditsmoreprimitive‘‘noncommutative’’version,canbetracedbackto
our own journal papers [139, 142, 147] of 1992, 1993, and 1996.