<< стр. 2(всего 2)СОДЕРЖАНИЕ
x в‰¤1

| T x, x |. It is enough to prove that
Proof. Let О± = sup x в‰¤1

| T x, y | в‰¤ О± x y

for all x and y. We can obviously assume that x and y are nonzero. Moreover, we may
multiply y by a complex number of modulus one, so we can assume that T x, y в‰Ґ 0. Then

T (x + y), x + y в€’ T (x в€’ y), x в€’ y = 4 Re T x, y = 4| T x, y |.

so
О± О±
2
+ x в€’ y 2) = 2
+ y 2 ).
| T x, y | в‰¤ ( x+y (x
4 2
Now apply this result with x replaced by y / x x and y replaced by x / y y.

Proof of spectral theorem for compact self-adjoint operators. We п¬Ѓrst show that T has a
nonzero eigenvector. If T = 0, this is obvious, so we assume that T = 0. Choose a sequence
xn в€€ X with xn = 1 so that | T xn , xn | в†’ T . Since T is self-adjoint, T xn , xn в€€ R,
so we may pass to a subsequence (still denoted xn ), for which T xn , xn в†’ О» = В± T .
27

Since T is compact we may pass to a further subsequence and assume that T xn в†’ y в€€ X.
Note that y в‰Ґ |О»| > 0.
Using the fact that T is self-adjoint and О» is real, we get
2 2
в€’ 2О» T xn , xn + О»2 xn 2
T xn в€’ О»xn = T xn
2 2
в€’ 2О»2 = 0.
в‰¤2 T в€’ 2О» T xn , xn в†’ 2 T
Since T xn в†’ y we infer that О»xn в†’ y as well, or xn в†’ y/О» = 0. Applying T we have
T y/О» = y, so О» is indeed a nonzero eigenvalue.
To complete the proof, consider the set of all orthonormal subsets of X consisting of
eigenvectors of T . By ZornвЂ™s lemma, it has a maximal element S. Let W be the closure
of the span of S. Clearly T W вЉ‚ W , and it follows directly (since T is self-adjoint), that
T W вЉҐ вЉ‚ W вЉҐ . Therefore T restricts to a self-adjoint operator on W вЉҐ and thus, unless
W вЉҐ = 0, T has an eigenvector in W вЉҐ . But this clearly contradicts the maximality of S
(since we can adjoin this element to S to get a larger orthonormal set of eigenvectors).
Thus W вЉҐ = 0, and S is an orthonormal basis.
The following structure result on the set of eigenvalues is generally considered part of
the spectral theorem as well.
Theorem. If T is a compact self-adjoint operator on a Hilbert space, then the set of
nonzero eigenvalues of T is either a п¬Ѓnite set or a sequence approaching 0 and the corre-
sponding eigenspaces are all п¬Ѓnite dimensional.
Remark. 0 may or may not be an eigenvalue, and its eigenspace may or may not be п¬Ѓnite.
Proof. Let ei be an orthonormal basis of eigenvectors, with T ei = О»i ei . Here i ranges over
some index set I. It suп¬ѓces to show that S = { i в€€ I | |О»i | в‰Ґ } is п¬Ѓnite for all > 0. Then
if i, j в€€ I
T ei в€’ T ej 2 = О»i ej в€’ О»j ej 2 = |О»i |2 + |О»j |2 ,
so if i, j в€€ S, then T ei в€’ T ej 2 в‰Ґ 2 2 . If S were inп¬Ѓnite, we could then choose a sequence
of unit elements in X whose image under T has no convergent subsequence, which violates
the compactness of T .
Suppose, for concreteness, that X is an inп¬Ѓnite dimensional separable Hilbert space and
that { en }nв€€N is an orthonormal basis adapted to a compact self-adjoint operator T on X.
Then the map U : X в†’ l2 given by
U( cn en ) = (c0 , c1 , . . . ),
n
is an isometric isomorphism. Moreover, when we use this map to transfer the action of T
to l2 , i.e., when we consider the operator U T U в€’1 on l2 , we see that this operator is simply
multiplication by the bounded sequence (О»0 , О»1 , . . . ) в€€ lв€ћ . Thus the spectral theorem says
that every compact self-adjoint T is unitarily equivalent to a multiplication operator on l2 .
(An isometric isomorphism of Hilbert spaces is also called a unitary operator. Note that
it is characterized by the property U в€— = U в€’1 .)
A useful extension is the spectral theorem for commuting self-adjoint compact operators.
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Theorem. If T and S are self-adjoint compact operators in a Hilbert space H and T S =
ST , then there is an orthonormal basis of X whose elements are eigenvectors for both S
and T .

Proof. For an eigenvalue О» of T , let XО» denote the corresponding eigenspace of T . If
x в€€ XО» , then T Sx = ST x = О»Sx, so Sx в€€ XО» . Thus S restricts to a self-adjoint
operator on XО» , and so there is an orthonormal basis of SвЂ“eigenvectors for XО» . These are
T вЂ“eigenvectors as well. Taking the union over all the eigenvalues О» of T completes the
construction.

Let T1 and T2 be any two self-adjoint operators and set T = T1 + iT2 . Then T1 =
(T + T в€— )/2 and T2 = (T в€’ T в€— )/(2i). Conversely, if T is any element of B(X), then we
can deп¬Ѓne two self-adjoint operators from these formulas and have T = T1 + iT2 . Now
suppose that T is compact and also normal, i.e., that T and T в€— commute. Then T1 and
T2 are compact and commute, and hence we have an orthonormal basis whose elements
are eigenvectors for both T1 and T2 , and hence for T . Since the real and imaginary parts
of the eigenvalues are the eigenvalues of T1 and T2 , we again see that the eigenvalues form
a sequence tending to zero and all have п¬Ѓnite dimensional eigenspaces.
We have thus shown that a compact normal operator admits an orthonormal basis
of eigenvectors. Conversely, if { ei } is an orthonormal basis of eigenvectors of T , then
T в€— ei , ej = 0 if i = j, which implies that each ei is also an eigenvector for T в€— . Thus
T в€— T ei = T T в€— ei for all i, and it follows easily that T is normal. We have thus shown:

Spectral Theorem for compact normal operators. Let T be a compact operator on a
Hilbert space X. Then there exists an orthonormal basis for X consisting of eigenvectors of
T if and only if T is normal. In this case, the set of nonzero eigenvalues form a п¬Ѓnite set or
a sequence tending to zero and the eigenspaces corresponding to the nonzero eigenvalues are
п¬Ѓnite dimensional. The eigenvalues are all real if and only if the operator is self-adjoint.

The spectrum of a general compact operator. In this section we derive the structure
of the spectrum of a compact operator (not necessarily self-adjoint or normal) on a complex
Banach space X.
For any operator T on a complex Banach space, the resolvent set of T , ПЃ(T ) consists
of those О» в€€ C such that T в€’ О»1 is invertible, and the spectrum Пѓ(T ) is the complement.
If О» в€€ Пѓ(T ), then T в€’ О»1 may fail to be invertible in several ways. (1) It may be that
N (T в€’ О»1) = 0, i.e., that О» is an eigenvalue of T . In this case we say that О» belongs to the
point spectrum of T , denoted Пѓp (T ). (2) If T в€’ О»1 is injective, it may be that its range is
dense but not closed in X. In this case we say that О» belongs to the continuous spectrum
of T , Пѓc (T ). Or (3) it may be that T в€’ О»1 is injective but that its range is not even dense
in X. This is the residual spectrum, Пѓr (T ). Clearly we have a decomposition of C into the
disjoint sets ПЃ(T ), Пѓp (T ), Пѓc (T ), and Пѓr (T ). As an example of the continuous spectrum,
consider the operator T en = О»n en where the en form an orthonormal basis of a Hilbert
space and the О»n form a positive sequence tending to 0. Then 0 в€€ Пѓc (T ). If T en = О»n en+1 ,
0 в€€ Пѓr (T ).
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Now if T is compact and X is inп¬Ѓnite dimensional, then 0 в€€ Пѓ(T ) (since if T were
invertible, the image of the unit ball would contain an open set, and so couldnвЂ™t be pre-
compact). From the examples just given, we see that 0 may belong to the point spectrum,
the continuous spectrum, or the residual spectrum. However, we shall show that all other
elements of the spectrum are eigenvalues, i.e., that Пѓ(T ) = Пѓp (T ) в€Є {0}, and that, as in the
normal case, the point spectrum consists of a п¬Ѓnite set or a sequence approaching zero.
The structure of the spectrum of a compact operator will be deduced from two lemmas.
The п¬Ѓrst is purely algebraic. To state it we need some terminology: consider a linear
operator T from a vector space X to itself, and consider the chains of subspaces

0 = N (1) вЉ‚ N (T ) вЉ‚ N (T 2 ) вЉ‚ N (T 3 ) вЉ‚ В· В· В· .

Either this chain is strictly increasing forever, or there is a least n в‰Ґ 0 such that N (T n ) =
N (T n+1 ), in which case only the п¬Ѓrst n spaces are distinct and all the others equal the
nth one. In the latter case we say that the kernel chain for T stabilizes at n. In particular,
the kernel chain stabilizes at 0 iп¬Ђ T is injective. Similarly we may consider the chain

X = R(1) вЉѓ R(T ) вЉѓ R(T 2 ) вЉѓ R(T 3 ) вЉѓ В· В· В· ,

and deп¬Ѓne what it means for the range chain to stabilize at n > 0. (So the range stabilizes
at 0 iп¬Ђ T is surjective.) It could happen that neither or only one of these chains stabilizes.
However:

Lemma. Let T be a linear operator from a vector space X to itself. If the kernel chain
stabilizes at m and the range chain stabilizes at n, then m = n and X decomposes as the
direct sum of N (T n ) and R(T n ).

Proof. Suppose m were less than n. Since the range chain stabilizes at n, there exists x with
T nв€’1 x в€€ R(T n ), and then there exists y such that T n+1 y = T n x. Thus x в€’ T y в€€ N (T n ),
/
and, since kernel chain stabilizes at m < n, N (T n ) = N (T nв€’1 ). Thus T nв€’1 x = T n y, a
contradiction. Thus m в‰Ґ n. A similar argument, left to the reader, establishes the reverse
inequality.
Now if T n x в€€ N (T n ), then T 2n x = 0, whence T n x = 0. Thus N (T n ) в€© R(T n ) = 0.
Given x, let T 2n y = T n x, so x decomposes as T n y в€€ R(T n ) and x в€’ T n y в€€ N (T n ).

The second lemma brings in the topology of compact operators.

Lemma. Let T : X в†’ X be a compact operator on a Banach space and О»1 , О»2 , . . . a
sequence of complex numbers with inf |О»n | > 0. Then the following is impossible: There
exists a strictly increasing chain of closed subspaces S1 вЉ‚ S2 вЉ‚ В· В· В· with (О»n 1 в€’ T )Sn вЉ‚
Snв€’1 for all n.

Proof. Suppose such a chain exists. Note that each T Sn вЉ‚ Sn for each n. Since Sn /Snв€’1
contains an element of norm 1, we may choose yn в€€ Sn with yn в‰¤ 2, dist(yn , Snв€’1 ) = 1.
If m < n, then
T ym в€’ (О»n 1 в€’ T )yn
в€€ Snв€’1 ,
z :=
О»n
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and
T ym в€’ T yn = |О»n | yn в€’ zn в‰Ґ |О»n |.
This implies that the sequence (T yn ) has no Cauchy subsequence, which contradicts the
compactness of T .

We are now ready to prove the result quoted at the beginning of the subsection.

Theorem. Let T be a compact operator on a Banach space X. Then any nonzero ele-
ment of the spectrum of T is an eigenvalue. Moreover Пѓ(T ) is either п¬Ѓnite or a sequence
approaching zero.

Proof. Consider the subspace chains N [(О»1 в€’ T )n ] and R[(О»1 в€’ T )n ] (these are closed
subspaces by a previous result). Clearly О»1 в€’ T maps N [(О»1 в€’ T )n ] into N [(О»1 в€’ T )nв€’1 ],
so the previous lemma implies that the kernel chain stabilizes, say at n. Now R[(О»1в€’T )n ] =
N [(О»1 в€’ T в€— )n ] (since the range is closed), and since these last stabilize, the range chain
a

stabilizes as well.
Thus we have X = N [(О»1 в€’ T )n ] вЉ• R[(О»1 в€’ T )n ]. Thus

R(О»1 в€’ T ) = X =в‡’ R(О»1 в€’ T )n = X =в‡’ N (О»1 в€’ T )n = 0 =в‡’ N (О»1 в€’ T ) = 0.

In other words О» в€€ Пѓ(T ) =в‡’ О» в€€ Пѓp (T ).
Finally we prove the last statement. If it were false we could п¬Ѓnd a sequence of
eigenvalues О»n with inf |О»n | > 0. Let x1 , x2 , . . . be corresponding nonzero eigenvectors
and set Sn = span[x1 , . . . , xn ]. These form a strictly increasing chain of subspaces (re-
call that eigenvectors corresponding to distinct eigenvalues are linearly independent) and
(О»n 1 в€’ T )Sn вЉ‚ Snв€’1 , which contradicts the lemma.

The above reasoning also gives us the Fredholm alternative:

Theorem. Let T be a compact operator on a Banach space X and О» a nonzero complex
number. Then either (1) О»1 в€’ T is an isomorphism, or (2) it is neither injective nor
surjective.

Proof. Since the kernel chain and range chain for S = О»1 в€’ T stabilize, either they both
stabilize at 0, in which case S is injective and surjective, or neither does, in which case it
is neither.

We close this section with a result which is fundamental to the study of Fredholm
operators.

Theorem. Let T be a compact operator on a Banach space X and О» a nonzero complex
number. Then

dim N (О»1 в€’ T ) = dim N (О»1 в€’ T в€— ) = codim R(О»1 в€’ T ) = codim R(О»1 в€’ T в€— ).
31

Proof. Let S = О»1 в€’ T . Since R(S) is closed

[X/ R(S)]в€— в€ј R(S)a = N (S в€— ).
=

Thus [X/ R(S)]в€— is п¬Ѓnite dimensional, so X/ R(S) is п¬Ѓnite dimensional, and these two
spaces are of the same dimension. Thus codim R(S) = dim N (S в€— ).
For a general operator S we only have R(S в€— ) вЉ‚ N (S)a , but, as we now show, when
R(S) is closed, R(S в€— ) = N (S)a . Indeed, S induces an isomorphism of X/ N (S) onto
R(S), and for any f в€€ N (S)a , f induces a map X/ N (S) to R. It follows that f = gS
for some bounded linear operator g on R(S), which can be extended to an element of X в€—
by Hahn-Banach. But f = gS simply means that f = S в€— g, showing that N (S)a вЉ‚ R(S в€— )
(and so equality holds) as claimed.
Thus
N (S)в€— в€ј X в€— / N (S)a = X в€— / R(S в€— ),
=
so codim R(S в€— ) = dim N (S)в€— = dim N (S).
We complete the theorem by showing that dim N (S) в‰¤ codim R(S) and dim N (S в€— ) в‰¤
codim R(S в€— ). Indeed, since R(S) is closed with п¬Ѓnite codimension, it is complemented by a
п¬Ѓnite dimensional space M (with dim M = codim R(S). Since N (S) is п¬Ѓnite dimensional,
it is complemented by a space N . Let P denote the projection of X onto N (S) which is
a bounded map which to the identity on N (S) and to zero on N . Now if codim R(S) <
dim N (S), then there is a linear map of N (S) onto M which is not injective. But then T в€’
f P is a compact operator and О»1 в€’ T + f P is easily seen to be surjective. By the Fredholm
alternative, it is injective as well. This implies that f is injective, a contradiction. We
have thus shown that dim N (S) в‰¤ codim R(S). Since T в€— is compact, the same argument
shows that dim N (S в€— ) в‰¤ codim R(S в€— ). This completes the proof.

VI. Introduction to General Spectral Theory

In this section we skim the surface of the spectral theory for a general (not necessarily
compact) operator on a Banach space, before encountering a version of the Spectral The-
orem for a bounded self-adjoint operator in Hilbert space. Our п¬Ѓrst results donвЂ™t require
the full structure of in the algebra of operators on a Banach space, but just an arbitrary
Banach algebra structure, and so we start there.

The spectrum and resolvent in a Banach algebra. Let X be a Banach algebra with
an identity element denoted 1. We assume that the norm in X has been normalized so that
1 = 1. The two main examples to bear in mind are (1) B(X), where X is some Banach
space; and (2) C(G) endowed with the sup norm, where G is some compact topological
space, the multiplication is just pointwise multiplication of functions, and 1 is the constant
function 1.
In this set up the resolvent set and spectrum may be deп¬Ѓned as before: ПЃ(x) = { О» в€€
C | x в€’ О»1 is invertible }, Пѓ(x) = C \ ПЃ(x). The spectral radius is deп¬Ѓned to be r(x) =
sup |Пѓ(x)|. For О» в€€ ПЃ(x), the resolvent is deп¬Ѓned as Rx (О») = (x в€’ О»1)в€’1 .
32

Lemma. If x, y в€€ X with x invertible and xв€’1 y < 1, then x в€’ y is invertible,
в€ћ
в€’1
(xв€’1 y)n xв€’1 ,
(x в€’ y) =
n=0

and (x в€’ y)в€’1 в‰¤ xв€’1 /(1 в€’ xв€’1 y ).

Proof.
(xв€’1 y)n xв€’1 в‰¤ xв€’1 xв€’1 y в‰¤ xв€’1 /(1 в€’ xв€’1 y ),
n

so the sum converges absolutely and the norm bound holds. Also
в€ћ в€ћ в€ћ
в€’1 n в€’1 в€’1
(xв€’1 y)n+1 = 1,
n
(x в€’ y) = y) в€’
(x y) x (x
n=0 n=0 n=0

and similarly for the product in the reverse order.

As a corollary, we see that if |О»| > x , then О»1 в€’ x is invertible, i.e., О» в€€ ПЃ(x). In other
words:

Proposition. r(x) в‰¤ x .

We also see from the lemma that limО»в†’в€ћ Rx (О») = 0. Another corollary is that if
О» в€€ ПЃ(x) and |Вµ| < Rx (О») в€’1 , then О» в€’ Вµ в€€ ПЃ(x) and
в€ћ
Rx (О»)n+1 Вµn .
Rx (О» в€’ Вµ) =
n=0

Theorem. The resolvent ПЃ(x) is always open and contains a neighborhood of в€ћ in C and
the spectrum is always non-empty and compact.

Proof. The above considerations show that the resolvent is open, and so the spectrum is
closed. It is also bounded, so it is compact.
To see that the spectrum is non-empty, let f в€€ X в€— be arbitrary and deп¬Ѓne П†(О») =
f [Rx (О»)]. Then П† maps ПЃ(x) into C, and it is easy to see that it is holomorphic (since we
have the power series expansion
в€ћ
f [R(О»)n+1 ]Вµn
П†(О» в€’ Вµ) =
n=0

if Вµ is suп¬ѓciently small). If Пѓ(x) = в€…, then П† is entire. It is also bounded (since it tends to 0
at inп¬Ѓnity), so LiouvilleвЂ™s theorem implies that it is identically zero. Thus for any f в€€ X в€— ,
f [(О»1 в€’ x)в€’1 ] = 0 . This implies that (О»1 в€’ x)в€’1 = 0, which is clearly impossible.
33

Corollary (GelfandвЂ“Mazur). If X is a complex Banach division algebra, then X is
isometrically isomorphic to C.

Proof. For each 0 = x в€€ X, let О» в€€ Пѓ(x). Then x в€’ О»1 is not invertible, and since X is a
division algebra, this means that x = О»1. Thus X = C1.

Now we turn to a bit of вЂњfunctional calculus.вЂќ Let x в€€ X and let f be a complex
function of a complex variable which is holomorphic on the closed disk of radius x about
the origin. Then we make two claims: (1) plugging x into the power series expansion of f
deп¬Ѓnes an element f (x) в€€ X; and (2) the complex function f maps the spectrum of x into
the spectrum of f (x). (In fact onto, as we shall show later in the case f is polynomial.)
To prove these claims, note that, by assumption, the radius of convergence of the power
в€ћ n
series for f about the origin exceeds x , so we can expand f (z) = n=0 an z where
|an | x n < в€ћ. Thus the series an xn is absolutely convergent in the Banach space
X; we call its limit f (x). (This is the deп¬Ѓnition of f (x). It is a suggestive abuse of
notation to use f to denote the this function, which maps a subset of X into X, as well as
the original complex-valued function of a complex variable.) Now suppose that О» в€€ Пѓ(x).
Then
в€ћ в€ћ в€ћ
n n
f (О»)1 в€’ f (x) = an (О» 1 в€’ x ) = (О»1 в€’ x) an Pn (О»1 в€’ x),
an Pn =
n=1 n=1 n=1

where
nв€’1
О»k xnв€’kв€’1 .
Pn =
k=0
в€ћ
nв€’1
Note that Pn в‰¤ n x an Pn converges to some y в€€ X. Thus
, so n=1

f (О»)1 в€’ f (x) = (О»1 в€’ x)y = y(О»1 в€’ x).

Now f (О»)1 в€’ f (x) canвЂ™t be invertible, because these formulas would then imply that О»1 в€’ x
would be invertible as well, but О» в€€ Пѓ(x). Thus we have veriп¬Ѓed that f (О») в€€ Пѓ f (x) for
all О» в€€ Пѓ(x).

Theorem (Spectral Radius Formula). r(x) = limnв†’в€ћ xn 1/n
= inf n xn 1/n
.

Proof. If О» в€€ Пѓ(x), then О»n в€€ Пѓ(xn ) (which is also evident algebraically), so |О»n | в‰¤ xn .
This shows that r(x) в‰¤ inf n xn 1/n .
Now take f в€€ X в€— , and consider
в€ћ
в€’1
О»в€’nв€’1 f (xn ).
П†(О») = f [(О»I в€’ x) ]=
n=0

Then П† is clearly holomorphic for О» > x , but we know it extends holomorphically to
О» > r(x) and tends to 0 as О» tends to inп¬Ѓnity. Let П€(О») = П†(1/О»). Then П€ extends
34

analytically to zero with value zero and deп¬Ѓnes an analytic function on the open ball of

в€ћ
f (О»n xn ).
П€(О»)/О» =
n=0

This shows that for each |О»| < 1/r(x) and each f в€€ X в€— , f (О»n xn ) is bounded. By the
uniform boundedness principle, the set of elements О»n xn are bounded in X, say by K.
Thus xn 1/n в‰¤ K 1/n /|О»| в†’ 1/|О»|. This is true for all |О»| < 1/r(x), so lim sup xn 1/n в‰¤
r(x).

Corollary. If H is a Hilbert space and T в€€ B(H) a normal operator, then r(T ) = T .

Proof.
= sup T x, T x = sup T в€— T x, x = T в€— T ,
2
T
x в‰¤1 x в‰¤1

since T в€— T is self-adjoint. Using the normality of T we also get

T в€—T = sup T в€— T x, T в€— T x = sup T T в€— T x, T x = sup T в€— T 2 x, T x
2
x в‰¤1 x в‰¤1 x в‰¤1

= sup T 2 x, T 2 x = T 2 2 .
x в‰¤1

Thus T 2 = T 2 . Replacing T with T 2 gives, T 4 = T 4 , and similarly for all powers
of 2. The result thus follows from the spectral radius formula.

As mentioned, we can now show that p maps Пѓ(x) onto Пѓ p(x) if p is a polynomial.

Spectral Mapping Theorem. Let X be a complex Banach algebra with identity, x в€€ X,
and let p a polynomial in one variable with complex coeп¬ѓcients. Then p Пѓ(x) = Пѓ p(x) .

Proof. We have already shown that p Пѓ(x) вЉ‚ Пѓ p(x) . Now suppose that О» в€€ Пѓ p(x) .
By the Fundamental Theorem of Algebra we can factor p в€’ О», so

p(x) в€’ О»1 = aО n (x в€’ О»i 1),
i=1

for some nonzero a в€€ C and some roots О»i в€€ C. Since p(x) в€’ О»1 is not invertible, it
follows that x в€’ О»i 1 is not invertible for at least one i. In orther words, О»i в€€ Пѓ(x), so
О» = p(О»i ) в€€ p Пѓ(x) .
35

Spectral Theorem for bounded self-adjoint operators in Hilbert space. We now
restrict to self-adjoint operators on Hilbert space and close with a version of the Spectral
Theorem for this class of operator. We follow HalmosвЂ™s article вЂњWhat does the Spec-
tral Theorem Say?вЂќ (American Mathematical Monthly 70, 1963) both in the relatively
elementary statement of the theorem and the outline of the proof.
First we note that self-adjoint operators have real spectra (not just real eigenvalues).

Proposition. If H is a Hilbert space and T в€€ B(H) is self-adjoint, then Пѓ(T ) вЉ‚ R.

Proof.
| (О»1 в€’ T )x, x | в‰Ґ | Im (О»1 в€’ T )x, x | = | Im О»| x 2 ,
so if Im О» = 0, О»1 в€’ T is injective with closed range. The same reasoning shows that
ВЇ
(О»1 в€’ T )в€— = О»1 в€’ T is injective, so R(О»1 в€’ T ) is dense. Thus О» в€€ ПЃ(T ).

Spectral Theorem for self-adjoint operators in Hilbert space. If H is a complex
Hilbert space and T в€€ B(H) is self-adjoint, then there exists a measure space в„¦ with
measure Вµ, a bounded measurable function П† : в„¦ в†’ R, and an isometric isomorphism
U : L2 в†’ H such that
U в€’1 T U = MП†
where MП† : L2 в†’ L2 is the operation of multiplication by П†. (Here L2 means L2 (в„¦, Вµ; C),
the space of complex-valued functions on в„¦ which are square integrable with respect to the
measure Вµ.)

Sketch of proof. Let x be a nonzero element of H, and consider the smallest closed sub-
space M of H containing T n x for n = 0, 1, . . . , i.e., M = { p(T )x | p в€€ PC }. Here PC is the
space of polynomials in one variable with complex coeп¬ѓcients. Both M and its orthogonal
complement are invariant under T (this uses the self-adjointness of T ). By a straightfor-
ward application of ZornвЂ™s lemma we see that H can be written as a Hilbert space direct
sum of T invariant spaces of the form of M . If we can prove the theorem for each of these
subspaces, we can take direct products to get the result for all of H. Therefore we may
assume from the start that H = { p(T )x | p в€€ PC } for some x. (In other terminology, that
T has a cyclic vector x.)
Now set в„¦ = Пѓ(T ), which is a compact subset of the real line, and consider the space
C = C(в„¦, R), the space of all continuous real-valued functions on в„¦. The subspace of
real-valued polynomial functions is dense in C (since any continuous function on в„¦ can
be extended to the interval [в€’r(T ), r(T )] thanks to TietzeвЂ™s extension theorem and then
approximated arbitrarily closely by a polynomial thanks to the Weierstrass approximation
theorem). For such a polynomial function, p, deп¬Ѓne Lp = p(T )x, x в€€ R. Clearly L is
linear and
|Lp| в‰¤ p(T ) x 2 = r p(T ) x 2 ,
by the special form of the spectral radius formula for self-adjoint operators in Hilbert space.
Since Пѓ p(T ) = p Пѓ(T ) , we have r p(T ) = p Lв€ћ (в„¦) = p C , and thus,
2
|Lp| в‰¤ x p C.
36

This shows that L is a bounded linear functional on a dense subspace of C and so extends
uniquely to deп¬Ѓne a bounded linear functional on C.
Next we show that L is positive in the sense that Lf в‰Ґ 0 for all non-negative functions
f в€€ C. Indeed, if f = p2 for some polynomial, then

Lf = p(T )2 x, x = p(T )x, p(T )x в‰Ґ 0.
в€љ
For an arbitrary non-negative f , we can approximate f uniformly by polynomials pn , so
f = lim p2 and Lf = lim Lp2 в‰Ґ 0.
n n

We now apply the Riesz Representation Theorem for the representation of the linear
functional L on C. It state that there exists a п¬Ѓnite measure on в„¦ such that Lf = f dВµ
for f в€€ C (it is a positive measure since L is positive). In particular, p(T )x, x = p dВµ
for all p в€€ PR .
We now turn to the space L2 of complex-valued functions on в„¦ which are square inte-
grable with respect to the measure Вµ. The subspace of complex-valued polynomial func-
tions is dense in L2 (since the measure is п¬Ѓnite, the L2 norm is dominated by the supremum
norm). For such a polynomial function, q, deп¬Ѓne U q = q(T )x. Then

2 2
|q|2 dВµ = q 2
Uq = q(T )x = q(T )x, q(T )x = q (T )q(T )x, x =
ВЇ L2 .

Thus U is an isometry of a dense subspace of L2 into H and so extends to an isometry of
L2 onto a closed subspace of H. In fact, U is onto H itself, since, by the assumption that
x is a cyclic vector for T , the range of U is dense.
Finally, deп¬Ѓne П† : в„¦ в†’ R by П†(О») = О». If q is a complex polynomial, then (MП† q)(О») =
О»q(О»), which is also a polynomial. Thus

U в€’1 T U q = U в€’1 T q(T )x = U в€’1 (MП† q)(T ) x = MП† q.

Thus the bounded operators U в€’1 T U and MП† coincide on a dense subset of L2 , and hence
they are equal.

For a more precise description of the measure space and the extension to normal oper-
ators, see Zimmer.

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