www.pdfgrip.com
Operator Theory: Advances and Applications
Vol. 139
Editor:
I. Gohberg
Editorial Office:
School of Mathematical
Sciences
Tel Aviv University
Ramat Aviv, Israel
Editorial Board:
D. Alpay (Beer-Sheva)
J. Arazy (Haifa)
A. Atzmon (Tel Aviv)
J. A. Ball (Blacksburg)
A. Ben-Artzi (Tel Aviv)
H. Bercovici (Bloomington)
A. Böttcher (Chemnitz)
K. Clancey (Athens, USA)
L. A. Coburn (Buffalo)
R. E. Curto (Iowa City)
K. R. Davidson (Waterloo, Ontario)
R. G. Douglas (College Station)
A. Dijksma (Groningen)
H. Dym (Rehovot)
P. A. Fuhrmann (Beer Sheva)
B. Gramsch (Mainz)
J. A. Helton (La Jolla)
M. A. Kaashoek (Amsterdam)
H. G. Kaper (Argonne)
S. T. Kuroda (Tokyo)
P. Lancaster (Calgary)
L. E. Lerer (Haifa)
B. Mityagin (Columbus)
V. Olshevsky (Storrs)
M. Putinar (Santa Barbara)
L. Rodman (Williamsburg)
J. Rovnyak (Charlottesville)
D. E. Sarason (Berkeley)
I. M. Spitkovsky (Williamsburg)
S. Treil (Providence)
H. Upmeier (Marburg)
S. M. Verduyn Lunel (Leiden)
D. Voiculescu (Berkeley)
D. Xia (Nashville)
D. Yafaev (Rennes)
Honorary and Advisory
Editorial Board:
C. Foias (Bloomington)
T. Kailath (Stanford)
H. Langer (Vienna)
P. D. Lax (New York)
H. Widom (Santa Cruz)
Subseries: Advances in
Partial Differential Equations
Subseries editors:
Bert-Wolfgang Schulze
Universität Potsdam
Germany
Sergio Albeverio
Universität Bonn
Germany
www.pdfgrip.com
Vladimir Müller
Spectral Theory of
Linear Operators
and Spectral Systems in
Banach Algebras
Second edition
Birkhäuser
Basel . Boston . Berlin
www.pdfgrip.com
Author:
Vladimir Müller
Institute of Mathematics
Czech Academy of Sciences
Zitna 25
115 67 Praha 1
Czech Republic
e-mail:
0DWKHPDWLFV6XEMHFW&ODVVL¿FDWLRQ$[[ %
Library of Congress Control Number: 2007929011
%LEOLRJUDSKLFLQIRUPDWLRQSXEOLVKHGE\'LH'HXWVFKH%LEOLRWKHN
'LH'HXWVFKH%LEOLRWKHNOLVWVWKLVSXEOLFDWLRQLQWKH'HXWVFKH1DWLRQDOELEOLRJUD¿H GHWDLOHG
bibliographic data is available in the Internet at <>.
,6%1 %LUNKlXVHU9HUODJ$*%DVHO%RVWRQ%HUOLQ
7KLVZRUNLVVXEMHFWWRFRS\ULJKW$OOULJKWVDUHUHVHUYHGZKHWKHUWKHZKROHRUSDUWRIWKH
PDWHULDOLVFRQFHUQHGVSHFL¿FDOO\WKHULJKWVRIWUDQVODWLRQUHSULQWLQJUHXVHRI
LOOXVWUDWLRQVUHFLWDWLRQEURDGFDVWLQJUHSURGXFWLRQRQPLFUR¿OPVRULQRWKHUZD\VDQG
VWRUDJHLQGDWDEDQNV)RUDQ\NLQGRIXVHSHUPLVVLRQRIWKHFRS\ULJKWRZQHUPXVWEH
obtained.
%LUNKlXVHU9HUODJ$*32%R[&+%DVHO6ZLW]HUODQG
3DUWRI6SULQJHU6FLHQFH%XVLQHVV0HGLD
3ULQWHGRQDFLGIUHHSDSHUSURGXFHGIURPFKORULQHIUHHSXOS7&) f
&RYHUGHVLJQ +HLQ]+LOWEUXQQHU%DVHO
3ULQWHGLQ*HUPDQ\
,6%1
H,6%1
ZZZELUNKDXVHUFK
www.pdfgrip.com
Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vii
Preface to the Second Edition . . . . . . . . . . . . . . . . . . . . . . . . . .
ix
I
Banach Algebras
1
Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Commutative Banach algebras . . . . . . . . . . . . . . . . . .
3
Approximate point spectrum in commutative Banach algebras .
4
Permanently singular elements and removability of spectrum .
5
Non-removable ideals . . . . . . . . . . . . . . . . . . . . . . . .
6
Axiomatic theory of spectrum . . . . . . . . . . . . . . . . . . .
7
Spectral systems . . . . . . . . . . . . . . . . . . . . . . . . . .
8
Basic spectral systems in Banach algebras . . . . . . . . . . . .
Comments on Chapter I . . . . . . . . . . . . . . . . . . . . . . . . .
II Operators
9
Spectrum of operators . . . . . . . . . .
10 Operators with closed range . . . . . . .
11 Factorization of vector-valued functions
12 Kato operators . . . . . . . . . . . . . .
13 General inverses and Saphar operators .
14 Local spectrum . . . . . . . . . . . . . .
Comments on Chapter II . . . . . . . . . . .
III Essential Spectrum
15 Compact operators . . . . . . . . . . . .
16 Fredholm and semi-Fredholm operators
17 Construction of Sadovskii/Buoni, Harte,
18 Perturbation properties of Fredholm
and semi-Fredholm operators . . . . . .
19 Essential spectra . . . . . . . . . . . . .
20 Ascent, descent and Browder operators .
www.pdfgrip.com
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
1
18
26
34
42
51
58
68
72
.
.
.
.
.
.
.
.
.
.
.
.
.
.
85
97
106
117
130
135
143
. . . . . . . . . . . . . . . 149
. . . . . . . . . . . . . . . 155
Wickstead . . . . . . . . . 164
. . . . . . . . . . . . . . . 169
. . . . . . . . . . . . . . . 172
. . . . . . . . . . . . . . . 178
vi
Contents
21 Essentially Kato operators . . . . . . . . . . . . . . . . . . .
22 Classes of operators defined by means of kernels and ranges
23 Semiregularities and miscellaneous spectra . . . . . . . . . .
24 Measures of non-compactness and other operator quantities
Comments on Chapter III . . . . . . . . . . . . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
187
197
211
220
228
IV Taylor Spectrum
25 Basic properties . . . . . . . . . . . . . . . . . . .
26 Split spectrum . . . . . . . . . . . . . . . . . . .
27 Some non-linear results . . . . . . . . . . . . . .
28 Taylor functional calculus for the split spectrum
29 Local spectrum for n-tuples of operators . . . . .
30 Taylor functional calculus . . . . . . . . . . . . .
31 Taylor functional calculus in Banach algebras . .
32 k-regular functions . . . . . . . . . . . . . . . . .
33 Stability of index of complexes . . . . . . . . . .
34 Essential Taylor spectrum . . . . . . . . . . . . .
Comments on Chapter IV . . . . . . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
237
245
252
260
266
271
283
285
293
299
305
V Orbits and Capacity
35 Joint spectral radius . . . . . . . . . . . .
36 Capacity . . . . . . . . . . . . . . . . . . .
37 Invariant subset problem and large orbits
38 Hypercyclic vectors . . . . . . . . . . . . .
39 Weak orbits . . . . . . . . . . . . . . . . .
40 Scott Brown technique . . . . . . . . . . .
41 Kaplansky’s type theorems . . . . . . . .
42 Polynomial orbits and local capacity . . .
Comments on Chapter V . . . . . . . . . . . .
Appendix . . . . . . . . . . . . . . . . . . . .
A.1 Banach spaces . . . . . . . . . . . . .
A.2 Analytic vector-valued functions . .
A.3 C ∞ -functions . . . . . . . . . . . . .
A.4 Semicontinuous set-valued functions
A.5 Some geometric properties of Banach
A.6 Basic properties of H ∞ . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
311
321
327
341
348
359
375
378
387
. . . .
. . . .
. . . .
. . . .
. . . .
spaces
. . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
393
393
398
401
403
404
405
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429
List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437
www.pdfgrip.com
Preface
Spectral theory is an important part of functional analysis. It has numerous applications in many parts of mathematics and physics including matrix theory, function theory, complex analysis, differential and integral equations, control theory
and quantum physics.
In recent years, spectral theory has witnessed an explosive development.
There are many types of spectra, both for one or several commuting operators,
with important applications, for example the approximate point spectrum, Taylor
spectrum, local spectrum, essential spectrum, etc.
The present monograph is an attempt to organize the available material
most of which exists only in the form of research papers scattered throughout the
literature. The aim is to present a survey of results concerning various types of
spectra in a unified, axiomatic way.
The central unifying notion is that of a regularity, which in a Banach algebra
is a subset of elements that are considered to be “nice”. A regularity R in a Banach
algebra A defines the corresponding spectrum σR (a) = {λ ∈ C : a − λ ∈
/ R} in
the same way as the ordinary spectrum is defined by means of invertible elements,
σ(a) = {λ ∈ C : a − λ ∈
/ Inv(A)}.
Axioms of a regularity are chosen in such a way that there are many natural
interesting classes satisfying them. At the same time they are strong enough for
non-trivial consequences, for example the spectral mapping theorem.
Spectra of n-tuples of commuting elements of a Banach algebra are described
similarly by means of a notion of joint regularity. This notion is closely related to
˙
the axiomatic spectral theory of Zelazko
and Slodkowski.
The book is organized in five chapters. The first chapter contains spectral
theory in Banach algebras which form a natural frame for spectral theory of operators.
In the second chapter the spectral theory of Banach algebras is applied to
operators. Of particular interest are regular functions – operator-valued functions
whose ranges (kernels) behave continuously. Applied to the function z → T − z
where T is a fixed operator, this gives rise to the important class of Kato operators
and the corresponding Kato spectrum (studied in the literature under many names,
e.g., semi-regular operators, Apostol spectrum etc.).
www.pdfgrip.com
viii
Preface
The third chapter gives a survey of results concerning various types of essential spectra, Fredholm and Browder operators etc.
The next chapter concentrates on the Taylor spectrum, which is by many
experts considered to be the proper generalization of the ordinary spectrum of
single operators. The most important property of the Taylor spectrum is the existence of the functional calculus for functions analytic on a neighbourhood of the
Taylor spectrum. We present the Taylor functional calculus in an elementary way,
without the use of sheaf theory or cohomological methods.
Further we generalize the concept of regular functions. We introduce and
study operator-valued functions that admit finite-dimensional discontinuities of
the kernel and range. This is closely related with stability results for the index of
complexes of Banach spaces.
The last chapter is concentrated on the study of orbits of operators. By
an orbit of an operator T we mean a sequence {T n x : n = 0, 1, . . . } where x
is a fixed vector. Similarly, a weak orbit is a sequence of the form { T n x, x∗ :
n = 0, 1, . . . } where x ∈ X and x∗ ∈ X ∗ are fixed, and a polynomial orbit
is a set {p(T )x : p polynomial}. These notions, which originated in the theory
of dynamical systems, are closely related to the invariant subspace problem. We
investigate these notions by means of the essential approximate point spectrum.
All results are presented in an elementary way. We assume only a basic knowledge of functional analysis, topology and complex analysis. Moreover, basic notions
and results from the theory of Banach spaces, analytic and smooth vector-valued
functions and semi-continuous set-valued functions are given in the Appendix.
The author would like to express his gratitude to many experts in the field
who influenced him in various ways. In particular, he would like to thank V. Pt´
ak
for his earlier guidance and later interest in the subject, and A. Soltysiak and
J. Zem´
anek who read parts of the manuscript and made various comments. The au˙
thor is further indebted to V. Kordula, W. Zelazko,
M. Mbekhta, F.-H. Vasilescu,
C. Ambrozie, E. Albrecht, F. Leon and many others for cooperation and useful
discussions over the years. Finally, the author would like to acknowledge that this
book was written while he was partially supported by grant No. 201/00/0208 of
the Grant Agency of the Czech Republic.
Prague
November 2001
V. M.
www.pdfgrip.com
Preface to the Second Edition
Since this book was written several years ago, further progress has been made in
some parts of the theory. I use the opportunity to include some of the new results,
improve the arguments in other places, and also to correct some unfortunate errors
and misprints that appeared in the first edition.
My sincere thanks are due to A. Soltysiak, J. Braˇciˇc and J. Vrˇsovsk´
y who
contributed to the improvement of the text. The work was supported by grant
ˇ
No. 201/06/0128 of GA CR.
Prague
April 2007
V. M.
www.pdfgrip.com
Chapter I
Banach Algebras
In this chapter we study spectral theory in Banach algebras. Basic concepts and
classical results are summarized in the first two sections. In the subsequent sections
we study the approximate point spectrum, which is one of the most important
examples of a spectrum in Banach algebras. The approximate point spectrum is
closely related with the notions of removable and non-removable ideals.
The axiomatic theory of spectrum is introduced in Sections 6 and 7. This
enables us to study various types of spectra, both of single elements and commuting
n-tuples, in a unified way.
All algebras considered here are complex and unital. The field of complex
numbers will be denoted by C.
1 Basic Concepts
This section contains basic definitions and results from the theory of Banach algebras. For more details see the monograph [BD] or some other textbook about
Banach algebras (e.g., [Ric], [Zel6], [Pal]).
Definition 1. An algebra A is a complex linear space A together with a multiplication mapping (x, y) → xy from A × A into A which satisfies the following
conditions (for all x, y, z ∈ A, α ∈ C):
(i)
(ii)
(iii)
(iv)
(xy)z = x(yz);
x(y + z) = xy + xz, (x + y)z = xz + yz;
(αx)y = α(xy) = x(αy);
there exists a unit element e ∈ A such that e = 0 and ex = xe = x for all
x ∈ A.
It is easy to show that the unit element is determined uniquely. Indeed, if
e is another unit element, then e = ee = e . The unit element of an algebra A
www.pdfgrip.com
2
Chapter I. Banach Algebras
will be denoted by 1A , (or simply 1 when no confusion can arise). Similarly, for a
complex number α, the symbol α also denotes the algebra element α · 1A .
Definition 2. Let A be an algebra. An algebra seminorm in A is a function
A → 0, ∞), x → x satisfying (for all x, y ∈ A, α ∈ C):
(i)
(ii)
(iii)
(iv)
·
:
αx = |α| · x ;
x+y ≤ x + y ;
xy ≤ x · y ;
1A = 1.
An algebra norm in A is an algebra seminorm such that
(v) if x = 0, then x = 0.
Definition 3. A normed algebra is a pair (A, · ), where A is an algebra and · is
an algebra norm in A. A Banach algebra is a normed algebra that is complete in
the topology defined by the norm (in other words, (A, · ) considered as a linear
space is a Banach space).
Examples 4. There are many examples of Banach algebras that appear naturally
in functional analysis.
(i) For the purpose of this monograph the most important example of a Banach
algebra is the algebra B(X) of all (bounded linear) operators acting on a
Banach space X, dim X ≥ 1, with naturally defined algebraic operations
and with the operator-norm T = sup{ T x : x ∈ X, x = 1}. The unit
element in B(X) is the identity operator I defined by Ix = x (x ∈ X).
In particular, if dim X = n < ∞, then B(X) can be identified with the
algebra of all n × n complex matrices.
(ii) Let K be a non-empty compact space. Then the algebra C(K) of all continuous complex-valued functions on K with the sup-norm f = sup{|f (z)| :
z ∈ K} is a Banach algebra.
(iii) Let E be a non-empty set. The set of all bounded complex-valued functions
defined on E with pointwise algebraic operations and the sup-norm is a Banach algebra.
Similarly, let L∞ be the set of all bounded measurable complex-valued functions defined on the real line (as usually, we identify two functions that differ
only on a set of Lebesgue measure zero). Then L∞ with pointwise algebraic
operations and the usual L∞ norm f ∞ = ess sup{|f (t)| : t ∈ R} is a
Banach algebra.
(iv) Let D be the open unit disc in the complex plane. Denote by H ∞ the algebra
of all functions that are analytic and bounded on D.
The disc algebra A(D) is the algebra of all functions that are continuous on
D and analytic on D. Both H ∞ and A(D) with the sup-norm are important
examples of Banach algebras.
www.pdfgrip.com
1. Basic Concepts
3
(v) Let S be a semigroup with unit and let 1 (S) be the set of all functions
f : S → C satisfying f = s∈S |f (s)| < ∞. Then 1 (S) with the multiplication defined by (f g)(s) = t1 t2 =s f (t1 )g(t2 ) is a Banach algebra.
More generally, if α : S → (0, ∞) is a submultiplicative function and
|f (s)|α(s) < ∞} is a Banach algebra
α(1S ) = 1, then {f : S → C :
with the norm f = |f (s)|α(s) and the above-defined multiplication.
(vi) Let L1 be the set of all integrable functions f : R → C. Define the multiplication and norm in L1 by
(f ∗ g)(s) =
and
∞
−∞
∞
f =
−∞
f (t)g(s − t) dt
|f (t)| dt.
Then L1 satisfies all axioms of Banach algebras except of the existence of the
unit element. The unitization (see C.1.1) L1 (R) ⊕ C with the norm f ⊕ λ =
f + |λ| and multiplication (f ⊕ λ) · (g ⊕ µ) = f ∗ g + λg + µf ⊕ λµ is a
Banach algebra.
Remark 5. Sometimes little bit different definitions of Banach algebras are used.
Frequently the existence of the unit element is not assumed or condition (iii) of
Definition 2 is replaced by a weaker condition of continuity of the multiplication.
However, it is possible to reduce these more general definitions of Banach algebras
to the present definition, see C.1.1 and C.1.3. Many results get a more natural
formulation in this way and the proofs are not obscured by technical difficulties.
Definition 6. Let A, B be algebras. A linear mapping ρ : A → B is called a
homomorphism if ρ(xy) = ρ(x)ρ(y) for all x, y ∈ A and ρ(1A ) = 1B .
Let A and B be normed algebras. A homomorphism ρ : A → B is continuous
if ρ := sup{ ρ(x) : x ∈ A, x = 1} < ∞. A continuous homomorphism ρ
satisfying inf{ ρ(x) : x ∈ A, x = 1} > 0 is called an isomorphism. A homomorphism ρ is called isometrical if ρ(x) = x for all x ∈ A.
A subset M of an algebra A is called a subalgebra if it is closed under the
algebraic operations (i.e., M is a linear subspace of A, 1A ∈ M and x, y ∈ M ⇒
xy ∈ M ).
If A is a closed subalgebra of a Banach algebra B, then A with the restricted
norm is again a Banach algebra.
Each normed algebra A has a completion – the uniquely determined (up to
an isometrical isomorphism) Banach algebra B such that A is a dense subalgebra
of B.
Remark 7. Let A be a Banach algebra. For a ∈ A define the operator La : A → A
by La x = ax. It is easy to verify that the mapping a → La is an isometrical
www.pdfgrip.com
4
Chapter I. Banach Algebras
homomorphism A → B(A). If we identify A with the image of this homomorphism,
then we can consider A as a closed subalgebra of B(A).
This simple but important construction enables us often to generalize results
from operator theory to Banach algebras.
Definition 8. Let A be a Banach algebra. A set J ⊂ A is called a left (right) ideal
in A if J is a subspace of A and ax ∈ J (xa ∈ J) for all x ∈ J, a ∈ A. J is a
two-sided ideal in A if J is both a left and right ideal in A. An ideal J ⊂ A (left,
right or two-sided) is called proper if J = A. Equivalently, J is proper if and only
if 1A ∈
/ J.
Let ρ : A → B be a continuous homomorphism from a Banach algebra A to
a Banach algebra B. It is easy to see that Ker ρ = {x ∈ A : ρ(x) = 0} is a closed
two-sided ideal in A.
Conversely, if J ⊂ A is a closed proper two-sided ideal in A, then we can
define a multiplication in the quotient space A/J by (x + J)(y + J) = xy + J. The
space A/J then becomes a Banach algebra with the unit 1A + J. For the canonical
homomorphism π : A → A/J defined by πx = x + J (x ∈ A) we have Ker π = J.
Invertible elements
Definition 9. Let x, y be elements of an algebra A. Then y is called a left (right)
inverse of x if yx = 1 (xy = 1). If y is both a left and right inverse of x, then it is
called an inverse of x. If x has a left inverse y and a right inverse z, then y = z.
Indeed, we have y = y(xz) = (yx)z = z. In particular, an element has at most one
inverse.
An element of A for which there exists an inverse (left inverse, right inverse)
will be called invertible (left invertible, right invertible). The unique inverse of an
invertible element x will be denoted by x−1 . The set of all invertible elements in an
algebra A will be denoted by Inv(A). Similarly, the set of all left (right) invertible
elements in A will be denoted by Invl (A) and Invr (A),
Invl (A) = {x ∈ A : there exists y ∈ A such that yx = 1},
Invr (A) = {x ∈ A : there exists y ∈ A such that xy = 1}.
Obviously, Inv(A) = Invl (A) ∩ Invr (A).
It is easy to see that an element a ∈ A is left (right) invertible if and only if
there is no proper left (right) ideal containing a.
Remark 10. The left and right properties in Banach algebras are perfectly symmetrical. The simplest way how to give an exact meaning to this statement is to
consider the following construction: for a Banach algebra A consider the reversed
multiplication a b = ba for all a, b ∈ A. In this way we obtain the Banach
algebra rev A, and the left ideals, left inverses etc. in A correspond to the right
www.pdfgrip.com
1. Basic Concepts
5
objects in the algebra rev A. Using this construction, each one-sided result implies
immediately the corresponding symmetrical result.
Theorem 11. Let A be a Banach algebra. Then:
(i) if a ∈ A, a < 1, then 1 − a ∈ Inv(A);
(ii) the sets Invl (A), Invr (A) and Inv(A) are open;
(iii) the mapping x → x−1 is continuous in Inv(A).
Proof. (i) If a < 1, then aj ≤ a
in A and
∞
(1 − a)
for all j, so the series
∞
k
j=0
k→∞
∞
j=0
aj is convergent
k
aj (1 − a) = lim
aj =
j=0
j
aj −
j=0
aj+1 = lim (1 − ak+1 ) = 1.
k→∞
j=0
(ii) Let yx = 1 and let u < y −1 . Then y(x + u) = 1 + yu, where yu ≤
−1
y · u < 1. By (i), y(x + u) is invertible and y(x + u)
y(x + u) = 1. Thus
x + u ∈ Invl (A). Hence Invl (A) is an open set.
Similarly, Invr (A) and Inv(A) = Invl (A) ∩ Invr (A) are open subsets of A.
(iii) Let a ∈ Inv(A) and let x < a−1 −1 . Then the series
convergent in A and one can check directly that
(a − x)−1 = a−1
∞
∞
−1 i
)
i=0 (xa
is
(xa−1 )i .
i=0
Thus
(a − x)−1 − a−1 = a−1
∞
(xa−1 )i
i=1
∞
≤ a−1 ·
x
i
· a−1
i=1
i
=
x · a−1 2
,
1 − x · a−1
and so (a − x)−1 → a−1 for x → 0.
Lemma 12. Let x, xn , yn (n = 1, 2, . . . ) be elements of a Banach algebra A, xn → x
and supn { yn } < ∞. If yn xn = 1 for all n, then x ∈ Invl (A).
If xn yn = 1 for all n, then x ∈ Invr (A).
Proof. Choose n such that x − xn < (sup yn )−1 . Then yn x = yn xn + yn (x −
xn ) = 1 + yn (x − xn ), where yn (x − xn ) < 1, and so yn x ∈ Inv(A). Thus
(yn x)−1 yn x = 1 and x ∈ Invl (A).
The second statement can be proved similarly.
Definition 13. An element a of a Banach algebra A is called a left (right) divisor
of zero if ax = 0 (xa = 0) for some non-zero element x ∈ A.
www.pdfgrip.com
6
if inf
if inf
Chapter I. Banach Algebras
An element a of a Banach algebra A is called a left topological divisor of zero
ax : x ∈ A, x = 1 = 0. Similarly, a is a right topological divisor of zero
xa : x ∈ A, x = 1 = 0.
Topological divisors of zero are closely related to non-invertible elements.
Theorem 14. Let a be an element of a Banach algebra A. Then:
(i) if a is left (right) invertible, then a is not a left (right) topological divisor of
zero;
(ii) if a is invertible, then a is neither a left nor a right topological divisor of zero;
(iii) if a ∈ ∂ Invl (A) (the topological boundary of Invl (A)), then a is a right
topological divisor of zero;
(iv) if a ∈ ∂ Inv(A), then a is both a left and right topological divisor of zero.
Proof. (i) Suppose that b is a left inverse of a, ba = 1. For x ∈ A, x = 1 we have
1 = x = bax ≤ b · ax , so inf ax : x ∈ A, x = 1 ≥ b −1 > 0 and a
is not a left topological divisor of zero.
The right version can be proved similarly.
This implies also (ii).
(iii) Let a ∈ ∂ Invl (A). Then there exist an , bn ∈ A such that limn→∞ an = a and
bn an = 1 for all n. By Lemma 12, lim bn = ∞. Set cn = bbnn . Then cn = 1
for every n and
cn a =
bn
1
(an + (a − an )) ≤
+ a − an ,
bn
bn
so limn→∞ cn a = 0 and a is a right topological divisor of zero.
(iv) Let a ∈ ∂ Inv(A). Then there exist an ∈ Inv(A) with an → a (n → ∞). By
a−1
= ∞. Set cn = an−1 . Then cn = 1 and, as in (iii),
Lemma 12, limn→∞ a−1
n
n
one can get easily that cn a → 0 and acn → 0. Thus a is both a left and right
topological divisor of zero.
Spectrum and spectral radius
Definition 15. Let a be an element of a Banach algebra A. The spectrum of a in
A is the set of all complex numbers λ such that a − λ is not invertible in A. The
spectrum of a in A will be denoted by σ A (a), or σ(a) if the algebra is clear from
the context.
By Theorem 11, σ(a) is a closed subset of C. The function λ → (a − λ)−1
defined in the open set C \ σ(a) is called the resolvent of a.
Theorem 16. Let a be an element of a Banach algebra A. Then the resolvent
λ → (a − λ)−1 is analytic in C \ σ(a).
www.pdfgrip.com
1. Basic Concepts
7
Proof. For λ, µ ∈
/ σ(a) we have
(a − µ)−1 − (a − λ)−1 = (a − µ)−1 (a − λ) − (a − µ) (a − λ)−1
= (µ − λ)(a − µ)−1 (a − λ)−1 ,
and so
(a − µ)−1 − (a − λ)−1
= (a − λ)−2 .
µ→λ
µ−λ
lim
Thus the function λ → (a − λ)−1 is analytic in C \ σ(a).
The following theorem is one of the most important results in the theory of
Banach algebras.
Theorem 17. Let x be an element of a Banach algebra A. Then σ(x) is a non-empty
compact set.
Proof. Let λ ∈ C, |λ| > x . Then the series
∞
(x − λ)
j=0
∞
Similarly
j=0
hence compact.
−xj
λj+1
∞
xj
j=0 λj+1
∞
is convergent in A and
∞
−xj
xj
xj
=
−
+
= 1.
λj+1
λj j=0 λj
j=1
(x − λ) = 1, and so λ ∈
/ σ(x). Thus σ(x) is bounded and
Suppose on the contrary that σ(x) = ∅. Consider the function f : C → A
defined by f (λ) = (x − λ)−1 . By Theorem 16, f is an entire function. For |λ| >
j
j
∞
∞
f (λ) ≤ j=0 |λ|xj+1 = |λ|−1 x . Thus
x we have f (λ) = j=0 λ−x
j+1 , and so
f (λ) → 0 for λ → ∞. By the Liouville theorem, f (λ) = 0 for each λ ∈ C. This is
a contradiction, since f (λ) is invertible for each λ.
Remark 18. Let T be an operator on a finite-dimensional Banach space X (i.e., T
is a square matrix). Then σ(T ) is finite and consists of eigenvalues of T .
Since the eigenvalues of a matrix are precisely the roots of its characteristic
polynomial, the non-emptiness of σ(T ) is equivalent to the “fundamental theorem
of algebra” that each complex polynomial has a root. This illustrates how deep is
the previous theorem, and also that operators on finite-dimensional spaces are far
from being trivial.
Corollary 19. (Gelfand, Mazur) Let A be a Banach algebra such that every nonzero element of A is invertible (i.e., A is a field). Then A consists of scalar multiples
of the identity, A = {λ · 1A : λ ∈ C}. Thus A is isometrically isomorphic to the
field of complex numbers C.
Proof. For every x ∈ A there exists λ ∈ σ(x) such that x − λ · 1A ∈
/ Inv(A). Thus
x = λ · 1A .
www.pdfgrip.com
8
Chapter I. Banach Algebras
Definition 20. The spectral radius r(x) of an element x ∈ A is the number
r(x) = max{|λ| : λ ∈ σ(x)}.
Lemma 21. Let s1 , s2 , . . . be non-negative real numbers. Then:
1/n
(i) if sn+m ≤ sn · sm for all m, n ∈ N, then the limit limn→∞ sn
1/n
equal to inf n sn ;
exists and is
1/n
(ii) if sn > 0 and sn+m ≥ sn · sm for all m, n ∈ N, then the limit limn→∞ sn
1/n
exists and is equal to supn sn .
1/n
1/k
Proof. (i) Write t = inf n sn and let ε > 0. Fix k such that sk < t + ε. Any
number n ≥ k can be expressed in the form n = n1 k + r, where 0 ≤ r ≤ k − 1 and
n1 ≥ 1. Then
sn ≤ sr · snk 1 ≤ max{1, s1 , s2 , . . . , sk−1 } · (t + ε)kn1
and
s1/n
≤ max{1, s1 , s2 , . . . , sk−1 }1/n · (t + ε)kn1 /n → t + ε
n
1/n
as n → ∞, since kn1 /n → 1. Thus lim supn→∞ sn
1/n
1/n
arbitrary, we have limn→∞ sn = t = inf n sn .
≤ t + ε and, since ε was
(ii) The second statement can be reduced to (i) by considering the numbers s−1
n .
Theorem 22. (spectral radius formula) Let a be an element of a Banach algebra A.
Then
r(a) = lim an 1/n = inf an 1/n .
n→∞
n
≤ a · a for all m, n, by the previous lemma the limit
Proof. Since a
lim an 1/n exists and is equal to the infimum.
n
∞
Let λ be a complex number with |λ| > lim an 1/n . Then n=0 λan+1 conn
∞
verges and it is easy to verify that (a − λ)−1 = n=0 λ−a
n+1 . Consequently, r(a) ≤
lim an 1/n .
It remains to show that lim an 1/n ≤ r(a). Consider the function f (λ) =
(1 − λa)−1 .
For λ = 0 we have f (λ) = λ−1 (λ−1 − a)−1 . Clearly f is analytic in {λ ∈ C :
0 < |λ| < r(a)−1 } and continuous in {λ ∈ C : |λ| < r(a)−1 }. So f is analytic in
{λ ∈ C : |λ| < r(a)−1 } (if r(a) = 0, then f is analytic in C). For |λ| < a −1 we
∞
∞
can write f (λ) = (1 − λa)−1 = n=0 an λn . Therefore we have f (λ) = n=0 an λn
for all λ, |λ| < r(a)−1 .
For the radius of convergence of the power series
an λn we have (see Theorem A.2.1)
lim inf an −1/n ≥ r(a)−1 ,
m+n
m
n
and so
r(a) ≥ lim sup an
1/n
= lim an
www.pdfgrip.com
1/n
.
1. Basic Concepts
9
Definition 23. Let M be a subset of a Banach algebra A. The commutant of M is
defined by M = a ∈ A : am = ma (m ∈ M ) . We write M instead of (M )
for the second commutant of M . If xy = yx for all x, y ∈ A, then A is called
commutative.
Lemma 24. Let M, N be subsets of a Banach algebra A. Then:
(i) M is a closed subalgebra of A;
(ii) if M ⊂ N , then M ⊃ N and M ⊂ N ;
(iii) M ⊂ M and M = M ;
(iv) if M consists of mutually commuting elements, then M ⊂ M ⊂ M and M
is a commutative Banach algebra.
Proof. The first three statements are clear.
To see (iv), note first that M ⊂ M . Therefore M
means that M is a commutative algebra.
⊂ M = M , which
Lemma 25. Let a ∈ A and let λ ∈ C \ σ(a). Then (a − λ)−1 ∈ {a} . In particular,
σ(a) equals to the spectrum of a in the commutative Banach algebra {a} .
Proof. Let b ∈ A and ab = ba. Then (a − λ)b = b(a − λ) and, by multiplicating
this equality from both sides by (a − λ)−1 , we get b(a − λ)−1 = (a − λ)−1 b. Thus
(a − λ)−1 ∈ {a} .
Theorem 26. Let A, B be Banach algebras, ρ : A → B a homomorphism and let
x ∈ A. Then σ B (ρ(x)) ⊂ σ A (x).
Proof. Let λ ∈ C \ σ A (x) and let y = (x − λ)−1 ∈ A. Then (ρ(x) − λ) · ρ(y) =
ρ(x − λ) · ρ(y) = ρ(1A ) = 1B and similarly ρ(y) · (ρ(x) − λ) = 1B .
Theorem 27. Let A be a subalgebra of a Banach algebra B and let x ∈ A. Then:
(i) if x is a left (right) topological divisor of zero in A, then x is a left (right)
topological divisor of zero in B;
(ii) ∂σ A (x) ⊂ σ B (x) ⊂ σ A (x).
Proof. (i) We have
inf
xb : b ∈ B, b = 1 ≤ inf
xa : a ∈ A, a = 1 = 0.
(ii) If λ ∈ ∂σ A (x), then x − λ is a left topological divisor of zero in A, and so, by
(i), λ ∈ σ B (x).
The inclusion σ B (x) ⊂ σ A (x) follows from the previous theorem.
Consequently, σ A (x) is obtained by filling in some holes in σ B (x).
In the algebra B(X) we have additional information.
www.pdfgrip.com
10
Chapter I. Banach Algebras
Theorem 28. Let X be a Banach space, dim X ≥ 1 and let T ∈ B(X). Then:
(i) T is invertible if and only if T is one-to-one and onto;
(ii) if λ ∈ ∂σ(T ), then (T − λ)X = X and inf (T − λ)x : x ∈ X, x = 1 = 0.
Proof. (i) Follows from the open mapping theorem.
(ii) By Theorem 14, there exist operators Sn ∈ B(X) (n ∈ N) such that Sn = 1
and Sn (T − λ) → 0. For each n there exists xn ∈ X with xn = 1 and
Sn xn ≥ 1/2. Suppose on the contrary that T − λ is onto. By the open mapping
theorem, there exists k > 0 such that (T − λ)BX ⊃ kBX where BX denotes the
closed unit ball in X. Thus there exists yn ∈ X such that (T − λ)yn = xn and
yn ≤ k −1 . Hence
Sn (T − λ) ≥ Sn (T − λ)
yn
yn
=
1
yn
Sn xn ≥
k
,
2
a contradiction with the assumption that Sn (T − λ) → 0.
Similarly, there exist operators Rn ∈ B(X) (n ∈ N) such that Rn = 1 and
(T − λ)Rn → 0. There exist vectors xn ∈ X with xn = 1 and Rn xn ≥ 1/2.
Rn xn
Set yn = R
. Then yn = 1 and
n xn
(T − λ)yn =
Hence inf
(T − λ)Rn xn
≤ 2 (T − λ)Rn → 0.
Rn xn
(T − λ)x : x ∈ X, x = 1 = 0.
Theorem 29. Let a, b ∈ A and let λ be a non-zero complex number. Then ab − λ
is left (right) invertible if and only if ba − λ is left (right) invertible.
Proof. Let c ∈ A, c(ab − λ) = 1. Then
(−λ−1 + λ−1 bca)(ba − λ) = −λ−1 ba + 1 + λ−1 bcaba − bca
= 1 − λ−1 ba + λ−1 bc(ab − λ)a = 1.
Similarly, if (ba − λ)d = 1 for some d ∈ A, then
(ab − λ)(−λ−1 + λ−1 adb) = 1.
Corollary 30. Let x, y be elements of a Banach algebra A. Then
σ(xy) \ {0} = σ(yx) \ {0}.
In general, the spectrum and the spectral radius in a Banach algebra do not
behave continuously, see C.1.14. However, they are always upper semicontinuous
(for definitions and basic properties of semicontinuous set-valued functions see Appendix A.4). Moreover, we prove later in Section 6 that the set of all discontinuity
points of the spectrum is a set of the first category.
www.pdfgrip.com
1. Basic Concepts
11
Theorem 31. (upper semicontinuity of the spectrum) Let A be a Banach algebra,
x ∈ A, let U be an open neighbourhood of σ(x). Then there exists ε > 0 such that
σ(y) ⊂ U for all y ∈ A with y − x < ε. In particular, the function x → r(x) is
upper semicontinuous.
Proof. Suppose on the contrary that for every n there exist xn ∈ A and λn ∈
σ(xn ) \ U such that xn − x < 1/n. Then |λn | ≤ xn ≤ x + 1, and so there
/ Inv(A),
exists a subsequence of (λn ) converging to some λ ∈ C\U . Since xn −λn ∈
we have x − λ ∈
/ Inv(A) by Theorem 11. Thus λ ∈ σ(x) and λ ∈
/ U , which is a
contradiction with the assumption that U is a neighbourhood of σ(x).
Equivalent norms
Two norms · and · on a vector space X are called equivalent if there exists
a positive constant k such that
k −1 x ≤ x
≤k x
for all x ∈ X.
Theorem 32. Let (A, · ) be a Banach algebra and let S ⊂ A be a bounded
semigroup. Then there exists an equivalent algebra norm ·
on A such that
s ≤ 1 for every s ∈ S.
More precisely, there is such a norm · satisfying
k −1 a ≤ a
≤k a
for all a ∈ A, where k = sup{1, s : s ∈ S}.
Proof. Without loss of generality we can assume that 1 ∈ S. Thus k := sup{ s :
s ∈ S} ≥ 1.
For a ∈ A define q(a) = sup{ sa : s ∈ S}. Since 1 ∈ S, we have q(a) ≥ a .
Thus
a ≤ q(a) ≤ k a
for all a ∈ A.
Clearly, q is a norm. We have q(1) = k and q(s) ≤ k for every s ∈ S.
For a ∈ A and s ∈ S we have
q(sa) = sup{ s sa : s ∈ S} ≤ q(a).
Further, for a1 , a2 ∈ A we have
q(a1 a2 ) = sup{ sa1 a2 : s ∈ S} ≤ sup{ sa1 · a2 : s ∈ S}
≤ q(a1 ) a2 ≤ q(a1 )q(a2 ).
www.pdfgrip.com
12
Chapter I. Banach Algebras
Define now
a
= sup q(ax) : x ∈ A, q(x) ≤ 1 = sup
q(ax)
:x=0 .
q(x)
Since q(ax) ≤ q(a)q(x), we have a ≤ q(a) ≤ k a and a
Hence
k −1 a ≤ a ≤ k a
for every a ∈ A.
Clearly · is a norm and 1 = 1.
Let a1 , a2 ∈ A. We show that a1 a2 ≤ a1
If a1 a2 = 0, then
a1 a2
≥
q(a)
q(1)
≥ k −1 a .
a2 . This is clear if a1 a2 = 0.
q(a1 a2 x)
q(a1 a2 x)
: x = 0 = sup
: x = 0, q(a1 a2 x) = 0
q(x)
q(x)
q(a1 a2 x) q(a2 x)
·
: q(a1 a2 x) = 0 ≤ a1 a2 .
= sup
q(a2 x)
q(x)
= sup
Finally, for s ∈ S we have
s
= sup
q(sx)
: x = 0 ≤ 1.
q(x)
Corollary 33. Let (A, · ) be a Banach algebra and let x ∈ A. Then
r(x) = inf
x
:
·
is an equivalent algebra norm on A .
Proof. Since r(x) does not depend on the choice of an equivalent algebra norm,
we have r(x) ≤ x for every equivalent algebra norm · .
Conversely, let ε > 0. Consider the semigroup
S=
x
r(x) + ε
n
: n = 0, 1, . . . .
Then S is a bounded semigroup and, by Theorem 32, there exists an equivalent
algebra norm · on A such that s ≤ 1 for each s ∈ S. In particular, x ≤
r(x) + ε. This completes the proof.
Functional calculus
n
Let p(z) = i=0 αi z i be a polynomial with coefficients αi ∈ C. For x ∈ A we
write p(x) = ni=0 αi xi . It is clear that the mapping p → p(x) is a homomorphism
from the algebra of all polynomials to A. The spectra of x and p(x) and related
in the following way:
www.pdfgrip.com
1. Basic Concepts
13
Theorem 34. (spectral mapping theorem) Let x be an element of a Banach algebra
A and let p be a polynomial. Then σ(p(x)) = p(σ(x)).
Proof. The equality is clear if p is a constant polynomial. Suppose p is non-constant
and let λ ∈ C. Then we can write p(z) − λ = β(z − α1 ) · · · (z − αn ) for some
β, α1 , . . . , αn ∈ C, β = 0, n ≥ 1. Clearly, p(x) − λ = β(x − α1 ) · · · (x − αn ) and
p(x) − λ is non-invertible if and only if at least one of the factors x − αi is noninvertible, i.e., if αi ∈ σ(x) for some i. Thus λ ∈ σ(p(x)) if and only if p(z) − λ = 0
for some z ∈ σ(x). Hence σ(p(x)) = p(σ(x)).
In Banach algebras we can substitute an element a ∈ A not only to polynomials but also to functions analytic on a neighbourhood of the spectrum σ(a).
If f is a function analytic on a disc {z : |z| < R} where R > r(a) and
∞
∞
f (z) = i=0 αi z i is the Taylor expansion of f , then the series i=0 αi ai converges
in A to an element denoted by f (a).
If f is a function analytic only on a neighbourhood U of σ(a), then we can
define f (a) by means of a Cauchy integral. We define
f (a) =
1
2πi
f (z)(z − a)−1 dz,
Γ
where Γ is a contour surrounding σ(a) in U , see Appendix A.2. The integral is
well defined since the mapping z → (z − a)−1 is continuous on Γ by Theorem 11.
By the Cauchy formula, the integral does not depend on the choice of Γ. The
definition coincides with the previous definition for polynomials:
Proposition 35. Let a be an element of a Banach algebra A, let Γ be a contour
surrounding σ(a). Let p(z) = nj=0 αj z j be a polynomial with complex coefficients
αj . Then
n
1
p(z)(z − a)−1 dz =
αj aj .
2πi Γ
j=0
Proof. It is sufficient to show that
1
2πi
z k (z − a)−1 dz = ak
Γ
for all k ≥ 0. For R > r(a) we have
1
2πi
=
z k (z − a)−1 dz =
Γ
1
2πi
1
2πi
|z|=R
z k (z − a)−1 dz =
1
2πi
∞
|z|=R
z k−1 + az k−2 + · · · + ak−2 z + ak−1 +
j=0
by the residue theorem.
www.pdfgrip.com
∞
zk
|z|=R
j=0
aj
z j+1
ak+j
dz = ak ,
z j+1
dz
14
Chapter I. Banach Algebras
Proposition 36. Let a ∈ A, let Γ be a contour surrounding σ(a) and let
rational function such that no zero of q is surrounded by Γ. Then
1
2πi
Γ
p(z)
q(z)
be a
p(z)
(z − a)−1 dz = p(a)q(a)−1
q(z)
(note that q(a)−1 exists by Theorem 34).
Proof. We first prove that for k = 0, 1, . . . and for λ ∈ C not surrounded by Γ we
have
1
1
(z − a)−1 dz = (a − λ)−k .
(1)
2πi Γ (z − λ)k
For k = 0 this was proved in Proposition 35. Suppose that (1) is true for some
k ≥ 0. We have
(z − a)−1 − (λ − a)−1 = (z − a)−1 (λ − a) − (z − a) (λ − a)−1
= (λ − z)(z − a)−1 (λ − a)−1 .
Thus (z − a)−1 = (λ − a)−1 + (a − λ)−1 (z − λ)(z − a)−1 and
1
2πi
1
(z − a)−1 dz
k+1
Γ (z − λ)
1
1
(a − λ)−1
−1
=
(λ
−
a)
dz
+
2πi Γ (z − λ)k+1
2πi
Γ
1
(z − a)−1 dz
(z − λ)k
−(k+1)
= (a − λ)
by the induction assumption (the first integral is equal to 0 since the function
1
z → (z−λ)
k+1 is analytic inside Γ).
Let now
p(z)
q(z)
be an arbitrary rational function, let λ1 , . . . , λn be the roots
of q of multiplicities k1 , . . . , kn . Then
p(z)
q(z)
can be expressed as
n
kj
cj,s
p(z)
= p1 (z) +
q(z)
(z
−
λj )s
j=1 s=1
for some polynomial p1 and complex numbers cj,s . It is easy to verify that
n
−1
p(a)q(a)
kj
cj,s (a − λj )−s ,
= p1 (a) +
j=1 s=1
and, by (1), we have
1
2πi
n
Γ
kj
p(z)
cj,s (a − λj )−s = p(a)q(a)−1 .
(z − a)−1 dz = p1 (a) +
q(z)
j=1 s=1
www.pdfgrip.com
1. Basic Concepts
15
For a non-empty compact set K ⊂ C denote by HK the set of all functions
analytic on a neighbourhood of K. We identify two such functions if they coincide
on a neighbourhood of K. Thus, more precisely, HK is the algebra of all germs of
functions analytic on a neighbourhood of K.
Theorem 37 (functional calculus). Let a be an element of a Banach algebra A.
Then there exists a homomorphism f → f (a) from the algebra Hσ(a) into A with
the following properties:
(i) if f (z) =
n
i
i=0 αi a ;
n
i=0
αi z i is a polynomial with complex coefficients αi , then f (a) =
(ii) f (a) ∈ {a} for each f ;
(iii) if U is a neighbourhood of σ(a), f, fk are analytic on U and fk → f uniformly
on U , then fk (a) → f (a);
(iv) σ f (a) = f σ(a) .
Properties (i) and (iii) determine this homomorphism uniquely.
Proof. Define
f (a) =
1
2πi
f (z)(z − a)−1 dz,
Γ
where Γ is a contour surrounding σ(a) in the domain of definition of f . The
linearity of the mapping f → f (a) is clear and (i) was proved in Proposition 35.
(ii) follows directly from the definition since {a} is a closed algebra containing
(z − a)−1 for every z ∈ Γ.
To prove (iii), we can replace Γ by a contour Γ surrounding σ(a) in U . Then
fk (z) → f (z) uniformly on Γ and (iii) is clear.
By Proposition 36, (f1 f2 )(a) = f1 (a)f2 (a) if f1 , f2 are rational functions with
poles outside σ(a). By the Runge theorem, any f ∈ Hσ(a) can be approximated
uniformly on some neighbourhood of σ(a) by rational functions, so we conclude
that the mapping f → f (a) is multiplicative. Since property (i) determines f (a)
uniquely for rational functions f , we see that properties (i) and (iii) determine the
functional calculus uniquely.
It remains to prove (iv). If λ ∈
/ f (σ(a)), then g(z) = (f (z) − λ)−1 is a function
analytic on a neighbourhood of σ(a). Thus (f (a) − λ)g(a) = 1 and λ ∈
/ σ(f (a)).
Conversely, if λ ∈ f (σ(a)), then there exists z0 ∈ σ(a) with f (z0 ) = λ and
f (z) − λ = (z − z0 )g(z) for some function g ∈ Hσ(a) . Then f (a) − λ = (a − z0 )g(a)
and, since a − z0 ∈
/ Inv(A), we have f (a) − λ ∈
/ Inv(A). Hence λ ∈ σ(f (a)).
We mention at least one important corollary of the functional calculus for
the algebra of operators:
Corollary 38. Let T be an operator on a Banach space X, dim X ≥ 1. Suppose
that U1 , U2 are disjoint open subsets of C such that σ(T ) ⊂ U1 ∪ U2 . Then there
www.pdfgrip.com
