Spectral properties of noncommuting operators

186 Pages·2004·1.723 MB·English

by  Brian R. Jefferies

1843
Lecture Notes in Mathematics
Editors:
J.--M.Morel,Cachan
F.Takens,Groningen
B.Teissier,Paris
3
Berlin
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Brian Jefferies
Spectral Properties of
Noncommuting Operators
1 3
Author
BrianJefferies
SchoolofMathematics
UniversityofNewSouthWales
Sydney,NSW,2052
Australia
e-mail:[email protected]
http://www.maths.unsw.edu.au/˜brianj
LibraryofCongressControlNumber:2004104471
MathematicsSubjectClassification(2000):
47A13,47A60,30G35,42B20
ISSN0075-8434
ISBN3-540-21923-4Springer-VerlagBerlinHeidelbergNewYork
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Preface
The work described in these notes has had a long gestation. It grew out of
my sojourn at Macquarie University, Sydney, 1986-87 and 1989-90, during
which time Alan McIntosh was applying Clifford analysis techniques to the
study of singular integral operators and irregular boundary value problems.
Hisresearchgroupprovidedastimulatingandconvivialenvironmentoverthe
years.Iwouldliketothankmycollaboratorsinthisenterprise:JerryJohnson,
Alan McIntosh,Susumu Okada,JamesPicton-Warlow,Werner Ricker,Frank
SommenandBerndStraub.Theworkwassupportedbytwolargegrantsfrom
the Australian Research Council.
Sydney, March 2004 Brian Jefferies

Contents
1 Introduction............................................... 1
2 Weyl Calculus ............................................. 13
2.1 Background............................................. 13
2.2 Operators of Paley-Wiener Type s......................... 16
2.3 The Joint Spectrum ..................................... 21
3 Clifford Analysis........................................... 27
3.1 Clifford Algebras ........................................ 27
3.2 Banach Modules......................................... 29
3.3 Cauchy Formula......................................... 30
3.4 Vector Valued Functions.................................. 32
3.5 Monogenic Expansions ................................... 33
3.6 Monogenic Representation of Distributions ................. 35
3.7 Plane Wave Decomposition ............................... 35
3.8 Approximation.......................................... 36
4 Functional Calculus for Noncommuting Operators ......... 39
4.1 The Weyl Calculus and the Cauchy Kernel ................. 39
4.2 The Joint Spectrum and the Cauchy Kernel ................ 44
4.3 The Monogenic Functional Calculus ....................... 52
4.4 Spectral Decomposition .................................. 60
5 The Joint Spectrum of Matrices ........................... 67
5.1 Nelson’s Formula for Hermitian Matrices ................... 67
5.2 Exponential Bounds for Matrices .......................... 73
5.2.1 Perturbation...................................... 74
5.2.2 The Exponential Bound............................ 77
5.3 The Joint Spectrum of Pairs of Hermitian Matrices .......... 80
5.3.1 The Numerical Range of Matrices ................... 81
5.3.2 Examples ........................................ 88
5.3.3 The Joint Spectrum of Two Hermitian Matrices....... 88
VIII Contents
5.4 Simultaneously Triangularisable Matrices...................104
5.4.1 Disintegration of Measures .........................106
5.4.2 The Image of Simplicial Measure ....................107
5.4.3 Joint Spectrum of Triangularisable Matrices ..........110
5.5 Systems of Matrices .....................................111
6 The Monogenic Calculus for Sectorial Operators...........123
6.1 The H∞-Functional Calculus for a Single Operator ..........124
6.2 The Cauchy Kernel for n Sectorial Operators ...............126
6.3 Monogenic and Holomorphic Functions in Sectors ...........130
6.3.1 Joint Spectral Theory in the Algebra C(n) ............130
6.3.2 Plane Wave Decompositions ........................135
6.3.3 Bounded Monogenic Functions in a Sector............137
6.4 Bounded Holomorphic Functions in Sectors .................140
6.4.1 Sectors in Cn .....................................140
6.4.2 Fourier Analysis in Sectors .........................142
6.5 The Monogenic Calculus for n Sectorial Operators...........153
7 Feynman’s Operational Calculus...........................157
7.1 Operants for the Weyl Calculus ...........................157
7.2 Feynman’s µ-Operational Calculus for n Operators ..........160
7.3 The µ-Monogenic Calculus for n Operators .................165
References.....................................................173
Index of Notation .............................................181
Index..........................................................183
1
Introduction
The subject of these notes is the spectral theory of systems of operators.
Because ‘spectral theory’ means different things to different workers in func-
tional analysis, it is worthwhile to first set down how the term is used in the
presentcontextandthe relationshipitbearsto the spectraltheoryofa single
selfadjoint operator.
Thespectrumσ(A)ofasinglematrixAisthefinitesetofalleigenvalues of
A, that is, complex numbers λ for whichthe equationAv =λv has a nonzero
vector v as a solution. In order to treat linear operators A acting on some
function space, it is preferable to take σ(A) to mean the set of all λ ∈ C for
whichλI−Aisnotinvertible.Themostcompletespectralanalysisisavailable
forselfadjointoperatorsAactinginHilbertspace,forthenthelinearoperator
A has a spectral decomposition
(cid:1)
A= λdP (λ) (1.1)
A
σ(A)
with respect to a spectral measure P associated with A. In the case that A
A
is anhermitianmatrix,the integralrepresentation(1.1)becomesa finite sum
(cid:2)
A= λP ({λ}) (1.2)
A
λ∈σ(A)
in which P ({λ}) is the orthogonal projection onto the eigenspace of the
A
eigenvalueλ.Thespectraltheoryofselfadjointoperatorsliesatthefoundation
of quantum physics.
Thesolutionoflinearoperatorequations,suchasthosethatariseinquan-
tum mechanics, often requires the formation of functions of operators. For
example, in order to solve the linear equation
du(t)
+Au(t)=0, u(0)=u0,
dt
B.Jefferies:LNM1843,pp.1–11,2004.
(cid:1)c Springer-VerlagBerlinHeidelberg2004
2 1 Introduction
we needtoformthe exponentiale−tA,t≥0,ofA.Becauseofthe importance
oflinearevolutionequations,thetheoryofexponentiatinganoperatoriswell-
understood, but in general, the spectral properties of A determine the types
of functions f(A) of A that can be formed in a reasonable manner.
In the case of a selfadjoint operator A, we can take
(cid:1)
f(A)= f(λ)dP (λ) (1.3)
A
σ(A)
foranyP -essentiallyboundedBorelmeasurablefunctionf :σ(A)→C.The
A
pleasant spectral properties of a selfadjoint operator A are reflected in the
rich class of functions f(A) of A that can be formed.
A basic task of quantum mechanics is to find a quantum representa-
tion f(P,Q) of a classical observable (p,q) (cid:4)−→ f(p,q) on phase space. Here
(cid:1) d
P = is the momentum operator and Q is the position operator of ‘mul-
i dx
tiplication by x’. They satisfy the commutation relation QP −PQ = i(cid:1)I.
For example, if H(p,q) = p2 +V(q) is the classical hamiltonian of the sys-
2m
tem, then H(P,Q) = P2 +V(Q) is the corresponding quantum observable,
2m
provided that the sum of the two unbounded operators is interpreted appro-
priately. Although it is known that the structure of classical observables is
not preserved in the quantum setting for an extensive class of observables f,
we are left with the problem of forming a function f(P,Q) of a pair (P,Q) of
operators which do not commute with each other.
In another context, symmetric hyperbolic systems
(cid:2)n
∂u ∂u
+ A =0 (1.4)
j
∂t ∂x
j=1 j
of partial differential equations arise in the linearised equations of magneto-
hydrodynamics [15]. In the case that the matrices A1,...,An are hermitian,
the fundamental solution is the matrix-valued distribution
(cid:3) (cid:5)
1 eit(cid:4)nj=1Ajξj ˆ.
(2π)n
HeretheFouriertransformˆistakeninthesenseofdistributionswithrespect
to the variable ξ ∈Rn.
Then the fundamental solutionf (cid:4)−→f(A1,...,An) of(1.4) at time t=1
may be viewed as a mapping that forms functions f(A1,...,An) of the n
matricesA1,...,An.Thesnapshotofthesupportofthefundamentalsolution
at time t = 1 determines the propagation cone of solutions of the initial
value problem for the symmetric hyperbolic system (1.4). A mapping such
as f (cid:4)−→ f(A1,...,An) will be termed a functional calculus in this work.
Although the expressionis used somewhatloosely,the idea is common to the
areas in functional analysis just mentioned.
Inthe traditionalsettingofasingleoperatorA,adecentfunctionalcalcu-
lus f (cid:4)−→f(A) is a homomorphism of Banach algebras: (fg)(A)=f(A)g(A)