2

Spectrum

Let T : D(T ) → X be a linear transformation, where X is a nonzero normed

space and D(T ), the domain of T , is a linear manifold of X . The general
notion of spectrum, which applies to bounded or unbounded transformations,
goes as follows. Let F denote either the real field R or the complex field C ,
and let I be the identity on X . The resolvent set ρ(T ) of T is the set of all
scalars λ in F for which the linear transformation λI − T : D(T ) → X has a
densely defined continuous inverse. That is,
 
ρ(T ) = λ ∈ F : (λI − T )−1 ∈ B[R(λI − T ), D(T )] and R(λI − T )− = X
(see, e.g., [8, Definition 18.2] — there are different definitions of the resol-
vent set for unbounded linear transformations, but they all coincide for the
bounded case). The spectrum σ(T ) of T is the complement of set ρ(T ) in F .
We shall however restrict the theory to operators on a complex Banach space
(i.e., to bounded linear transformations of a complex Banach space to itself).

2.1 Basic Spectral Properties

Throughout this chapter T : X → X will be a bounded linear transformation
of X into itself (i.e, an operator on X ), so that D(T ) = X , where X = {0} is
a complex Banach space. That is, T ∈ B[X ], where X is a nonzero complex
Banach space. In such a case (i.e., in the unital complex Banach algebra B[X ]),
Theorem 1.2 ensures that the resolvent set ρ(T ) is precisely the set of all
complex numbers λ for which λI − T ∈ B[X ] is invertible (i.e., has a bounded
inverse on X ). Therefore (cf. Theorem 1.1),
 
ρ(T ) = λ ∈ C : λI − T ∈ G[X ]
 
= λ ∈ C : λI − T has an inverse in B[X ]
 
= λ ∈ C : N (λI − T ) = {0} and R(λI − T ) = X ,
and so
 
σ(T ) = C \ ρ(T ) = λ ∈ C : λI − T has no inverse in B[X ]
 
= λ ∈ C : N (λI − T ) = {0} or R(λI − T ) = X .

C.S. Kubrusly, Spectral Theory of Operators on Hilbert Spaces, 27

DOI 10.1007/978-0-8176-8328-3_2, © Springer Science+Business Media, LLC 2012
28 2. Spectrum

Theorem 2.1. The resolvent set ρ(T ) is nonempty and open, and the spec-
trum σ(T ) is compact .
Proof. Take any T ∈ B[X ]. By the Neumann expansion (Theorem 1.3), if
T  < |λ|, then λ ∈ ρ(T ). Equivalently, since σ(T ) = C \ρ(T ),

|λ| ≤ T  for every λ ∈ σ(T ).

Thus σ(T ) is bounded, and therefore ρ(T ) = ∅.

Claim. If λ ∈ ρ(T ), then the open ball Bδ (λ) with center at λ and (positive)
radius δ = (λI − T )−1 −1 is included in ρ(T ).
Proof. If λ ∈ ρ(T ), then λI − T ∈ G[X ] so that (λI − T )−1 is nonzero and
bounded, and hence 0 < (λI − T )−1 −1 < ∞. Set δ = (λI − T )−1 −1 , let
Bδ (0) be the nonempty open ball of radius δ about the origin of the complex
plane C , and take any ν in Bδ (0). Since |ν| < (λI − T )−1 −1 , it follows that
ν(λI − T )−1  < 1. Then [I − ν(λI − T )−1 ] ∈ G[X ] by Theorem 1.3, and so
(λ − ν)I − T = (λI − T )[I − ν(λI − T )−1 ] ∈ G[X ]. Thus λ − ν ∈ ρ(T ) so that
 
Bδ (λ) = Bδ (0) + λ = ν ∈ C : ν = ν + λ for some ν ∈ Bδ (0) ⊆ ρ(T ),

which completes the proof of the claimed result.

Thus ρ(T ) is open (it includes a nonempty open ball centered at each of its
points) and so σ(T ) is closed. Compact in C means closed and bounded. 

Remark. Since Bδ (λ) ⊆ ρ(T ), it follows that the distance of any λ in ρ(T ) to
the spectrum σ(T ) is greater than δ; that is (compare with Proposition 2.E),

λ ∈ ρ(T ) implies (λI − T )−1 −1 ≤ d(λ, σ(T )).

The resolvent function RT : ρ(T ) → G[X ] of an operator T ∈ B[X ] is the

mapping of the resolvent set ρ(T ) of T into the group G[X ] of all invertible
operators from B[X ] defined by

RT (λ) = (λI − T )−1 for every λ ∈ ρ(T ).

Since RT (λ) − RT (ν) = RT (λ)[RT (ν)−1 − RT (λ)−1 ]RT (ν), it follows that

RT (λ) − RT (ν) = (ν − λ)RT (λ)RT (ν)

for every λ, ν ∈ ρ(T ) (because RT (ν)−1 − RT (λ)−1 = (ν − λ)I). This is the

resolvent identity. Swapping λ and ν in the resolvent identity, it follows that
RT (λ) and RT (ν) commute for every λ, ν ∈ ρ(T ). Also, T RT (λ) = RT (λ)T
for every λ ∈ ρ(T ) (since RT (λ)−1 RT (λ) = RT (λ)RT (λ)−1 trivially).
Let Λ be a nonempty open subset of the complex plane C . Take a func-
tion f : Λ → C and a point ν ∈ Λ. Suppose there exists a complex number
 2.1 Basic Spectral Properties 29

f (ν) with the following property. For every ε > 0 there is a δ > 0 such that
 f (λ)−f (ν) 
 − f (ν) < ε for all λ in Λ for which 0 < |λ − ν| < δ. If there exists
λ−ν
such an f (ν) ∈ C , then it is called the derivative of f at ν. If f (ν) exists for
every ν in Λ, then f : Λ → C is analytic (or holomorphic) on Λ. A function
f : C → C is entire if it is analytic on the whole complex plane C . To prove the
next result we need the Liouville Theorem, which says that every bounded
entire function is constant.

Theorem 2.2. If X is nonzero, then the spectrum σ(T ) is nonempty.

Proof. Let T ∈ B[X ] be an operator on a nonzero complex Banach space X ,
∗ ∗
and let B[X ] stand for the dual of B[X ]. That is, B[X ] = B[B[X ], C ] is the
Banach space of all bounded linear functionals on B[X ]. Since X = {0}, it
∗
follows that B[X ] = {O}, and so B[X ] = {0}, which is a consequence of the
Hahn–Banach Theorem (see, e.g., [66, Corollary 4.64]). Take any nonzero η in
∗
B[X ] (i.e., a nonzero bounded linear functional η : B[X ] → C ), and consider
the composition of it with the resolvent function, η ◦ RT : ρ(T ) → C . Recall
that ρ(T ) = C \σ(T ) is nonempty and open in C .
Claim 1. If σ(T ) is empty, then η ◦ RT : ρ(T ) → C is bounded.
Proof. The resolvent function RT : ρ(T ) → G[X ] is continuous (since scalar
multiplication and addition are continuous mappings, and inversion is also a
continuous mapping; see, e.g., [66, Problem 4.48]). Thus RT (·): ρ(T ) → R
is continuous. Then sup|λ|≤T  RT (λ) < ∞ if σ(T ) = ∅. Indeed, if σ(T ) is
empty, then ρ(T ) ∩ BT  [0] = BT  [0] = {λ ∈ C : |λ| ≤ T } is a compact set
in C , so that the continuous function RT (·) attains its maximum on it by
the Weierstrass Theorem: a continuous real-valued function attains its max-
imum and minimum on any compact set in a metric space. On the other hand,
if T  < |λ|, then RT (λ) ≤ (|λ| − T )−1 (cf. Remark that follows Theorem
1.3), so that RT (λ) → 0 as |λ| → ∞. Thus, since RT (·): ρ(T ) → R is con-
tinuous, supT ≤λ RT (λ) < ∞. Hence supλ∈ρ(T ) RT (λ) < ∞, and so
sup (η ◦ RT )(λ) ≤ η sup RT (λ) < ∞,
λ∈ρ(T ) λ∈ρ(T )

which completes the proof of Claim 1.

Claim 2. η ◦ RT : ρ(T ) → C is analytic.
Proof. If λ and ν are distinct points in ρ(T ), then
RT (λ) − RT (ν)  
+ RT (ν)2 = RT (ν) − RT (λ) RT (ν)
λ−ν
by the resolvent identity. Set f = η ◦ RT : ρ(T ) → C . Let f : ρ(T ) → C be
defined by f (λ) = −η(RT (λ)2 ) for each λ ∈ ρ(T ). Therefore,
 f (λ) − f (ν)     
 
 − f (ν) = η RT (ν) − RT (λ) RT (ν) 
λ−ν
≤ ηRT (ν)RT (ν) − RT (λ)
30 2. Spectrum

so that f : ρ(T ) → C is analytic because RT : ρ(T ) → G[X ] is continuous, which

completes the proof of Claim 2.
Thus, by Claims 1 and 2, if σ(T ) = ∅ (i.e., if ρ(T ) = C ), then η ◦ RT : C → C
is a bounded entire function, and so a constant function by the Liouville
Theorem. But (see proof of Claim 1) RT (λ) → 0 as |λ| → ∞, and hence
η(RT (λ)) → 0 as |λ| → ∞ (since η is continuous). Then η ◦ RT = 0 for all η
∗
in B[H] = {0} so that RT = O (by the Hahn–Banach Theorem). That is,
(λI − T )−1 = O for λ ∈ C , which is a contradiction. Thus σ(T ) = ∅. 

Remark. σ(T ) is compact and nonempty, and so is its boundary ∂σ(T ). Hence,
∂σ(T ) = ∂ρ(T ) = ∅.

2.2 A Classical Partition of the Spectrum

The spectrum σ(T ) of an operator T in B[X ] is the set of all scalars λ in C
for which the operator λI − T fails to be an invertible element of the algebra
B[X ] (i.e., fails to have a bounded inverse on R(λI − T ) = X ). According to
the nature of such a failure, σ(T ) can be split into many disjoint parts. A
classical partition comprises three parts. The set σP (T ) of those λ for which
λI − T has no inverse (i.e., such that the operator λI − T is not injective) is
the point spectrum of T ,
 
σP (T ) = λ ∈ C : N (λI − T ) = {0} .

A scalar λ ∈ C is an eigenvalue of T if there exists a nonzero vector x in X

such that T x = λx. Equivalently, λ is an eigenvalue of T if N (λI − T ) = {0}.
If λ ∈ C is an eigenvalue of T , then the nonzero vectors in N (λI − T ) are
the eigenvectors of T , and N (λI − T ) is the eigenspace (which is a subspace
of X ), associated with the eigenvalue λ. The multiplicity of an eigenvalue is
the dimension of the respective eigenspace. Thus the point spectrum of T is
precisely the set of all eigenvalues of T . Now consider the set σC (T ) of those
λ for which λI − T has a densely defined but unbounded inverse on its range,
 
σC (T ) = λ ∈ C : N (λI − T ) = {0}, R(λI − T )− = X and R(λI − T ) = X

(see Theorem 1.3), which is referred to as the continuous spectrum of T . The

residual spectrum of T is the set σR (T ) of all scalars λ such that λI − T has
an inverse on its range that is not densely defined:
 
σR (T ) = λ ∈ C : N (λI − T ) = {0} and R(λI − T )− = X .

The collection {σP (T ), σC (T ), σR (T )} is a partition (i.e., it is a disjoint cov-

ering) of σ(T ), which means that they are pairwise disjoint and

σ(T ) = σP (T ) ∪ σC (T ) ∪ σR (T ).
 2.2 A Classical Partition of the Spectrum 31

The diagram below, borrowed from [62], summarizes such a partition of the
spectrum. The residual spectrum is split into two disjoint parts, σR(T ) =
σR1(T ) ∪ σR2(T ), and the point spectrum into four disjoint parts, σP (T ) =
4
i=1 σPi(T ). We adopt the following abbreviated notation: Tλ = λI − T ,
Nλ = N (Tλ ), and Rλ = R(Tλ ). Recall that if N (Tλ ) = {0}, then its linear
inverse Tλ−1 on Rλ is continuous if and only if Rλ is closed (Theorem 1.2).

R−
λ
=X R−
λ
= X

R−
λ
= Rλ R−
λ
= Rλ R−
λ
= Rλ R−
λ
= Rλ

Tλ−1 ∈ B[Rλ ,X ] ρ (T ) ∅ ∅ σR1(T )

Nλ = {0} ⎫
Tλ−1 ∈
/ B[Rλ ,X ] ∅ σC (T ) σR2(T ) ∅ ⎬
σAP (T )
Nλ = {0} σP1(T ) σP2(T ) σP3(T ) σP4(T ) ⎭
  
σCP (T )
Fig. § 2.2. A classical partition of the spectrum

Theorem 2.2 says that σ(T ) = ∅, but any of the above disjoint parts of
the spectrum may be empty (Section 2.7). However, if σP (T ) = ∅, then a set
of eigenvectors associated with distinct eigenvalues is linearly independent.

Theorem 2.3. Let {λγ }γ∈Γ be a family of distinct eigenvalues of T . For

each γ ∈ Γ let xγ be an eigenvector associated with λγ . The set {xγ }γ∈Γ is
linearly independent .
Proof. For each γ ∈ Γ take 0 = xγ ∈ N (λγ I − T ) = {0}, and consider the set
{xγ }γ∈Γ (whose existence is ensured by the Axiom of Choice).
Claim. Every finite subset of {xγ }γ∈Γ is linearly independent.
Proof. Take an arbitrary finite subset of {xγ }γ∈Γ , say {xi }ni=1 . If n = 1, then
linear independence is trivial. Suppose {xi }ni=1 is linearly independent
n for some
n ≥ 1. If {xi }n+1
i=1 is not linearly independent, then xn+1 = i=1 i i , where
α x
the family {αi }ni=1 of complex numbers has at least one nonzero number. Thus
n n
λn+1 xn+1 = T xn+1 = αi T xi = αi λi xi .
i=1 i=1

If λn+1 = 0, then λi = 0 for every i = n + 1 (because the eigenvalues are dis-

n
tinct) and i=1 αi λi xi = 0 so that {xi }ni=1 is not linearly independent, which
is a contradiction. If λn+1 = 0, then xn+1 = i=1 αi λ−1
n
n n+1 λi xi , and therefore
−1
i=1 αi (1 − λ n+1 λi )xi = 0 so that {xi } n
i=1 is not linearly independent (since
λi = λn+1 for every i = n + 1 and αi = 0 for some i), which is again a contra-
diction. This completes the proof by induction:{xi}n+1 i=1 is linearly independent.

However, if every finite subset of {xγ }γ∈Γ is linearly independent, then so is

the set {xγ }γ∈Γ itself (see, e.g., [66, Proposition 2.3]). 
32 2. Spectrum

There are some overlapping parts of the spectrum which are commonly
used. For instance, the compression spectrum σCP (T ) and the approximate
point spectrum (or approximation spectrum) σAP (T ), which are defined by
 
σCP (T ) = λ ∈ C : R(λI − T ) is not dense in X
= σP3(T ) ∪ σP4(T ) ∪ σR (T ),
 
σAP (T ) = λ ∈ C : λI − T is not bounded below
= σP (T ) ∪ σC (T ) ∪ σR2(T ) = σ(T )\σR1(T ).
The points of σAP (T ) are referred to as the approximate eigenvalues of T .

Theorem 2.4. The following assertions are pairwise equivalent .

(a) For every ε > 0 there is a unit vector xε in X such that (λI −T )xε  < ε.
(b) There is a sequence {xn } of unit vectors in X such that (λI −T )xn  → 0.
(c) λ ∈ σAP (T ).

Proof. Clearly (a) implies (b). If (b) holds, then there is no constant α > 0
such that α = αxn  ≤ (λI − T )xn  for all n. Thus λI − T is not bounded
below, and so (b) implies (c). Conversely, if λI − T is not bounded below,
then there is no constant α > 0 such that αx ≤ (λI − T )x for all x ∈ X
or, equivalently, for every ε > 0 there exists a nonzero yε in X such that
(λI − T )yε  < εyε . Set xε = yε −1 yε , and hence (c) implies (a). 

Theorem 2.5. The approximate point spectrum σAP (T ) is a nonempty closed

subset of C that includes the boundary ∂σ(T ) of the spectrum σ(T ).
Proof. Take an arbitrary λ in ∂σ(T ). Recall that ρ(T ) = ∅, σ(T ) is closed
(Theorem 2.1), and ∂σ(T ) = ∂ρ(T ) = ρ(T )− ∩ σ(T ). Thus λ is a point of
adherence of ρ(T ), and so there exists a sequence {λn } with each λn in ρ(T )
such that λn → λ. Since
(λn I − T ) − (λI − T ) = (λn − λ)I
for every n, it follows that λn I − T → λI − T in B[X ].
Claim. supn (λn I − T )−1  = ∞.
Proof. Since each λn lies in ρ(T ) and λ ∈ ∂σ(T ) does not lie in ρ(T ) (because
σ(T ) is closed), it follows that the sequence {λn I − T } of operators in G[X ]
converges in B[X ] to λI − T ∈ B[X ]\G[X ]. If λ ∈ σP (T ), then there exists
x = 0 in X such that, if supn (λn I − T )−1  < ∞,
0 = x = (λn I − T )−1 (λn I − T )x
≤ sup (λn I − T )−1  lim sup (λn I − T )x
n n
−1
≤ sup (λn I − T )  (λI − T )x = 0,
n
 2.2 A Classical Partition of the Spectrum 33

which is a contradiction. If λ ∈ σP (T ), then N (λI − T ) = {0}, and hence there

exists the inverse (λI − T )−1 on R(λI − T ) so that, for each n,

(λn I − T )−1 − (λI − T )−1 = (λn I − T )−1 (λI − T ) − (λn I − T ) (λI − T )−1.

If supn (λn I − T )−1  < ∞, then (λn I − T )−1 → (λI − T )−1 in B[X ], be-
cause (λn I − T ) → (λI − T ) in B[X ], and so (λI − T )−1 ∈ B[X ]. That is,
(λI − T ) ∈ G[X ], which is again a contradiction (since λ ∈ σ(T )). This com-
pletes the proof of the claimed result.
Since (λn I − T )−1  = supy=1 (λn I − T )−1 y, take a unit vector yn in
X for which (λn I − T )−1  − n1 ≤ (λn I − T )−1 yn  ≤ (λn I − T )−1  for
each n. The above claim ensures that supn (λn I − T )−1 yn  = ∞, and hence
inf n (λn I − T )−1 yn −1 = 0, so that there exist subsequences of {λn } and
{yn }, say {λk } and {yk }, for which

(λk I − T )−1 yk −1 → 0.

Set xk = (λk I − T )−1 yk −1 (λk I − T )−1 yk and get a sequence {xk } of unit
vectors in X such that (λk I − T )xk  = (λk I − T )−1 yk −1 . Hence

(λI − T )xk  = (λk I − T )xk − (λk − λ)xk  ≤ (λk I − T )−1yk −1 + |λk − λ|.

Since λk → λ, it then follows that (λI − T )xk  → 0, and so λ ∈ σAP (T ) ac-

cording to Theorem 2.4. Therefore,

∂σ(T ) ⊆ σAP (T ).

This inclusion clearly implies that σAP (T ) = ∅ (for σ(T ) is closed and non-
empty). Finally, take an arbitrary λ ∈ C \σAP (T ) so that λI − T is bounded
below. Therefore, there exists an α > 0 for which

αx ≤ (λI − T )x ≤ (νI − T )x + (λ − ν)x,

and hence (α − |λ − ν|)x ≤ (νI − T )x, for all x ∈ X and ν ∈ C . Then

νI − T is bounded below for every ν such that 0 < α − |λ − ν|. Equivalently,
ν ∈ C \σAP (T )) for every ν sufficiently close to λ (i.e., if |λ − ν| < α). Thus the
nonempty open ball Bα (λ) centered at λ is included in C \σAP (T ). Therefore
C \σAP (T ) is open, and so σAP (T ) is closed. 

Remark. σAP (T ) = σ(T )\σR1(T ) is closed in C and includes ∂σ(T ) = ∂ρ(T ).

So C \σR1(T ) = ρ(T ) ∪ σAP (T ) = ρ(T ) ∪ ∂ρ(T ) ∪ σAP (T ) = ρ(T )− ∪ σAP (T )
is closed in C . Outcome:

σR1(T ) is an open subset of C .

For the remainder of this section we assume that T lies in B[H], where
H is a nonzero complex Hilbert space. This will bring forth some important
34 2. Spectrum

simplifications. A particularly useful instance of such simplifications is the

formula for the residual spectrum in the next theorem. First we need the
following piece of notation. If Λ is any subset of C , then set
 
Λ∗ = λ ∈ C : λ ∈ Λ

so that Λ∗∗ = Λ, (C \Λ)∗ = C \Λ∗, and (Λ1 ∪ Λ2 )∗ = Λ∗1 ∪ Λ∗2 .

Theorem 2.6. If T ∗ ∈ B[H] is the adjoint of T ∈ B[H], then

ρ(T ) = ρ(T ∗ )∗ , σ(T ) = σ(T ∗ )∗ , σC (T ) = σC (T ∗ )∗ ,

and the residual spectrum of T is given by the formula

σR (T ) = σP (T ∗ )∗ \σP (T ).

As for the subparts of the point and residual spectra,

σP1(T ) = σR1(T ∗ )∗ , σP2(T ) = σR2(T ∗ )∗ ,

σP3(T ) = σP3(T ∗ )∗ , σP4(T ) = σP4(T ∗ )∗ .
For the compression and approximate point spectra we get

σCP (T ) = σP (T ∗ )∗ , σAP (T ∗ )∗ = σ(T )\σP1(T ),

∂σ(T ) ⊆ σAP (T ) ∩ σAP (T ∗ )∗ = σ(T )\(σP1(T ) ∪ σR1(T )).

Proof. Since S ∈ G[H] if and only if S ∗ ∈ G[H], we get ρ(T ) = ρ(T ∗ )∗ . Hence
σ(T )∗ = (C \ρ(T ))∗ = C \ρ(T ∗ ) = σ(T ∗ ). Recall that R(S)− = R(S) if and
only if R(S ∗ )− = R(S ∗ ), and N (S) = {0} if and only if R(S ∗ )⊥ = {0} (cf.
Lemmas 1.4 and 1.5), which means that R(S ∗ )− = H. Thus σP1(T ) = σR1(T ∗ )∗,
σP2(T ) = σR2(T ∗ )∗, σP3(T ) = σP3(T ∗ )∗, and σP4(T ) = σP4(T ∗ )∗. Applying the
same argument, σC (T ) = σC (T ∗ )∗ and σCP (T ) = σP (T ∗ )∗ . Hence,

σR (T ) = σCP (T )\σP (T ) implies σR (T ) = σP (T ∗ )∗ \σP (T ).

Moreover, by using the above properties and the definition of σAP (T ∗ ) =

σP (T ∗ ) ∪ σC (T ∗ ) ∪ σR2(T ∗ ) = σCP (T )∗ ∪ σC (T )∗ ∪ σP2(T )∗ , we get

σAP (T ∗ )∗ = σCP (T ) ∪ σC (T ) ∪ σP2(T ) = σ(T )\σP1(T ).

Therefore, σAP (T ∗ )∗ ∩ σAP (T ) = σ(T )\(σP1(T ) ∪ σR1(T )). But σ(T ) is closed
and σR1(T ) is open (and so is σP1(T ) = σR1(T ∗ )∗ ) in C . This implies that
σP1(T ) ∪ σR1(T ) ⊆ σ(T )◦ , where σ(T )◦ denotes the interior of σ(T ), and
∂σ(T ) ⊆ σ(T )\(σP1(T ) ∪ σR1(T )). 

Remark. We have just shown that

σP1(T ) is an open subset of C .

 2.3 Spectral Mapping 35

2.3 Spectral Mapping

Spectral Mapping Theorems are pivotal results in spectral theory. Here we

focus on the important particular case for polynomials. Further versions will
be considered in Chapter 4. Let p: C → C be a polynomial with complex
coefficients, take any subset Λ of C , and consider its image under p, viz.,
 
p(Λ) = p(λ) ∈ C : λ ∈ Λ .

Theorem 2.7. (Spectral Mapping Theorem for Polynomials). Take an

operator T ∈ B[X ] on a complex Banach space X . If p is any polynomial
with complex coefficients, then

σ(p(T )) = p(σ(T )).

Proof. To avoid trivialities, let p: C → C be an arbitrary nonconstant polyno-

mial with complex coefficients,
n
p(λ) = αi λi , with n ≥ 1 and αn = 0,
i=0

for every λ ∈ C . Take an arbitrary ν ∈ C and consider the factorization

n
ν − p(λ) = βn (λi − λ),
i=1

with βn = −1n+1 αn where {λi }ni=1 are the roots of ν − p(λ), so that
n n
νI − p(T ) = νI − αi T i = βn (λi I − T ).
i=0 i=1

If λi ∈ ρ(T ) for every i = 1, . . . , n, then βn ni=1 (λi I − T ) ∈ G[X ] so that
ν ∈ ρ(p(T )). Thus if ν ∈ σ(p(T )), then there exists λj ∈ σ(T ) for some j =
1, . . . , n. However, λj is a root of ν − p(λ), that is,

n
ν − p(λj ) = βn (λi − λj ) = 0,
i=1

and so p(λj ) = ν. Hence, if ν ∈ σ(p(T )), then

 
ν = p(λj ) ∈ p(λ) ∈ C : λ ∈ σ(T ) = p(σ(T ))

because λj ∈ σ(T ), and therefore

σ(p(T )) ⊆ p(σ(T )).

Conversely, if ν ∈ p(σ(T )) = {p(λ) ∈ C : λ ∈ σ(T )}, then ν = p(λ) for some

λ ∈ σ(T ). Thus ν − p(λ) = 0 so that λ = λj for some j = 1, . . . , n, and hence
36 2. Spectrum
n
νI − p(T ) = βn (λi I − T )
i=1
n n
= (λj I − T )βn (λi I − T ) = βn (λi I − T )(λj I − T )
j=i=1 j=i=1

since (λj I − T ) commutes with (λi I − T ) for every i. If ν ∈ ρ(p(T )), then
(νI − p(T )) ∈ G[X ] so that
n  
(λj I − T ) βn (λi I − T ) νI − p(T ) −1
j=i=1
      
= νI − p(T ) νI − p(T ) −1 = I = νI − p(T ) −1 νI − p(T )

 n
= (νI − p(T ) −1 βn (λi I − T ) (λj I − T ).
j=i=1

Then (λj I − T ) has a right and a left inverse (i.e., it is invertible), and so
(λj I − T ) ∈ G[X ] by Theorem 1.1. Hence, λ = λj ∈ ρ(T ), which contradicts
the fact that λ ∈ σ(T ). Conclusion: if ν ∈ p(σ(T )), then ν ∈/ ρ(p(T )), that is,
ν ∈ σ(p(T )). Thus
p(σ(T )) ⊆ σ(p(T )). 

In particular,
σ(T n ) = σ(T )n for every n ≥ 0,
that is, ν ∈ σ(T )n = {λn ∈ C : λ ∈ σ(T )} if and only if ν ∈ σ(T n ), and

σ(αT ) = ασ(T ) for every α ∈ C ,

that is, ν ∈ ασ(T ) = {αλ ∈ C : λ ∈ σ(T )} if and only if ν ∈ σ(αT ). Also notice
(even though this is not a particular case of the Spectral Mapping Theorem
for polynomials) that if T ∈ G[X ], then

σ(T −1 ) = σ(T )−1 ,

which means that ν ∈ σ(T )−1 = {λ−1 ∈ C : 0 = λ ∈ σ(T )} if and only

if ν ∈ σ(T −1 ). Indeed, if T ∈ G[X ] (so that 0 ∈ ρ(T )) and if ν = 0, then
−νT −1 (ν −1 I − T ) = νI − T −1 . Thus ν −1 ∈ ρ(T ) if and only if ν ∈ ρ(T −1 ).
Also note that if T ∈ B[H], then

σ(T ∗ ) = σ(T )∗

by Theorem 2.6, where H is a complex Hilbert space.

The next result is an extension of the Spectral Mapping Theorem for
polynomials which holds for normal operators in a Hilbert space H. If Λ1
and Λ2 are arbitrary subsets of C and p : C ×C → C is any polynomial in two
variables (with complex coefficients), then set
 2.3 Spectral Mapping 37
 
p(Λ1 , Λ2 ) = p(λ1 , λ2 ) ∈ C : λ1 ∈ Λ1 , λ2 ∈ Λ2 ;

in particular, with Λ∗ = {λ ∈ C : λ ∈ Λ},

 
p(Λ, Λ∗ ) = p(λ, λ) ∈ C : λ ∈ Λ .

Theorem 2.8. (Spectral Mapping Theorem for Normal Operators). If

T ∈ B[H] is normal and p(·, ·) is a polynomial in two variables, then
 
σ(p(T, T ∗ )) = p(σ(T ), σ(T ∗ )) = p(λ, λ) ∈ C : λ ∈ σ(T ) .

Proof. Take any normal operator T ∈ B[H]. If p(λ, λ) = n,m i j
i,j=0 αi,j λ λ , then
∗
 n,m i ∗j ∗ ∗
set p(T, T ) = i,j=0 αi,j T T = p(T , T ). Let P(T, T ) be the collection of
all those polynomials p(T, T ∗ ), which is a commutative subalgebra of B[H]
since T commutes with T ∗. Consider the collection T of all commutative sub-
algebras of B[H] containing T and T ∗, which is partially ordered (in the inclu-
sion ordering) and nonempty (e.g., P(T, T ∗) ∈ T ). Moreover, every chain in
T has an upper bound in T (the union of all subalgebras in a given chain of
subalgebras in T is again a subalgebra in T ). Thus Zorn’s Lemma says that
T has a maximal element, say A(T ). Outcome: If T is normal, then there is
a maximal (thus closed) commutative subalgebra A(T ) of B[H] containing T
and T ∗. Since P(T, T ∗) ⊆ A(T ) ∈ T, and every p(T, T ∗) ∈ P(T, T ∗) is normal,
A(p(T, T ∗ )) = A(T ) for every nonconstant p(T, T ∗). Furthermore,
Φ(p(T, T ∗ )) = p(Φ(T ), Φ(T ∗ ))
for every homomorphism Φ : A(T ) → C . Thus, by Proposition 2.Q(b),
 
 ) .
σ(p(T, T ∗ )) = p(Φ(T ), Φ(T ∗ )) ∈ C : Φ ∈ A(T

Take a surjective homomorphism Φ: A(T ) → C (i.e., take any Φ ∈ A(T  )). Con-
sider the Cartesian decomposition T = A + iB, where A, B ∈ B[H] are self-
adjoint, and so T ∗ = A − iB (Proposition 1.O). Thus Φ(T ) = Φ(A) + iΦ(B)
and Φ(T ∗ ) = Φ(A) − iΦ(B). Since A = 12 (T + T ∗ ) and B = − 2i (T − T ∗ ) lie in
P(T, T ∗), we get A(A) = A(B) = A(T ). Moreover, since they are self-adjoint,
 )} = σ(A) ⊂ R and {Φ(B) ∈ C : Φ ∈ A(T
{Φ(A) ∈ C : Φ ∈ A(T  )} = σ(B) ⊂ R
(Propositions 2.A and 2.Q(b)), and so Φ(A) ∈ R and Φ(B) ∈ R . Hence
Φ(T ∗ ) = Φ(T ).
Therefore, since σ(T ∗ ) = σ(T )∗ for every T ∈ B[H] by Theorem 2.6, and
according to Proposition 2.Q(b),
 
 )
σ(p(T, T ∗ )) = p(Φ(T ), Φ(T )) ∈ C : Φ ∈ A(T
 
 )}
= p(λ, λ) ∈ C : λ ∈ {Φ(T ) ∈ C : Φ ∈ A(T
 
= p(λ, λ) ∈ C : λ ∈ σ(T )
= p(σ(T ), σ(T )∗ ) = p(σ(T ), σ(T ∗ )). 
38 2. Spectrum

2.4 Spectral Radius

The spectral radius of an operator T ∈ B[X ] on a nonzero complex Banach
space X is the nonnegative number
rσ (T ) = sup |λ| = max |λ|.
λ∈σ(T ) λ∈σ(T )

The first identity in the above expression defines the spectral radius rσ (T ),
and the second one is a consequence of the Weierstrass Theorem (cf. proof of
Theorem 2.2) since σ(T ) = ∅ is compact in C and the function | |: C → R is
continuous. A straightforward consequence of the Spectral Mapping Theorem
for polynomials reads as follows.

Corollary 2.9. rσ (T n ) = rσ (T )n for every n ≥ 0.

Proof. Take an arbitrary nonnegative integer n. Theorem 2.7 ensures that
σ(T n ) = σ(T )n . Hence ν ∈ σ(T n ) if and only if ν = λn for some λ ∈ σ(T ),
n
and so supν∈σ(T n ) |ν| = supλ∈σ(T ) |λn | = supλ∈σ(T ) |λ|n = supλ∈σ(T ) |λ| . 

If λ ∈ σ(T ), then |λ| ≤ T . This follows by the Neumann expansion of

Theorem 1.3 (cf. proof of Theorem 2.1). Thus rσ (T ) ≤ T . Therefore, for
every operator T ∈ B[X ], and for each nonnegative integer n,
0 ≤ rσ (T n ) = rσ (T )n ≤ T n  ≤ T n.
Thus rσ (T ) ≤ 1 if T is power bounded (i.e., if supn T n  < ∞). Indeed, in this
1
case, rσ (T )n = rσ (T n ) ≤ supk T k  and limn (supk T k ) n = 1, and so
sup T n < ∞ implies rσ (T ) ≤ 1.
n

Remark. If T is a nilpotent operator (i.e., if T n = O for some n ≥ 1), then

rσ (T ) = 0, and so σ(T ) = σP (T ) = {0} (cf. Proposition 2.J). An operator
T ∈ B[X ] is quasinilpotent if rσ (T ) = 0 (i.e., if σ(T ) = {0}). Thus every
nilpotent is quasinilpotent. Since σP (T ) may be empty for a quasinilpotent
operator (cf. Proposition 2.N), these classes are related by proper inclusion:
Nilpotent ⊂ Quasinilpotent.

The next result is the well-known Gelfand–Beurling formula for the spec-
tral radius. Its proof requires another piece of elementary complex analysis,
viz., every analytic function has a power series representation. That is, if
f : Λ → C is analytic, and if Bα,β (ν) = {λ ∈ C : 0 ≤ α < |λ − ν| < β} lies in
the open set Λ ⊆ C , then f has a unique Laurent expansion about the point
∞
ν, namely, f (λ) = k=−∞ γk (λ − ν)k for every λ ∈ Bα,β (ν).

Theorem 2.10. (Gelfand–Beurling Formula).

1
rσ (T ) = lim T n  n .
n
 2.4 Spectral Radius 39

Proof. Since rσ (T )n ≤ T n  for every positive integer n, and since the limit
1
of the sequence {T n n} exists by Lemma 1.10, we get
1
rσ (T ) ≤ lim T n  n .
n

For the reverse inequality, proceed as follows. Consider the Neumann expan-
sion (Theorem 1.3) for the resolvent function RT : ρ(T ) → G[X ],
∞
RT (λ) = (λI − T )−1 = λ−1 T k λ−k
k=0

for every λ ∈ ρ(T ) such that T  < |λ|, where the above series converges in
the (uniform) topology of B[X ]. Take an arbitrary bounded linear functional
∗
η : B[X ] → C in B[X ] (cf. proof of Theorem 2.2). Since η is continuous,
∞
η(RT (λ)) = λ−1 η(T k )λ−k
k=0

for every λ ∈ ρ(T ) such that T  < |λ|.

Claim. The above displayed identity holds whenever rσ (T ) < |λ|.

Proof. λ−1 ∞ k −k
k=0 η(T )λ is a Laurent expansion of η(RT (λ)) about the ori-
gin for every λ ∈ ρ(T ) such that T  < |λ|. But η ◦ RT is analytic on ρ(T )
(cf. Claim 2 in Theorem 2.2) so that η(RT (λ)) has a unique Laurent ex-
pansion about the origin for every λ ∈ ρ(T ), and hence for every λ ∈ C such
∞
that rσ (T ) < |λ|. Then η(RT (λ)) = λ−1 k=0 η(T k )λ−k, which holds when-
ever rσ (T ) ≤ T  < |λ|, must be the Laurent expansion about the origin for
every λ ∈ C such that rσ (T ) < |λ|, thus proving the claimed result.
∞ k −k
Hence, if rσ (T ) < |λ|, then the series of complex numbers k=0 η(T )λ
−1 k −k
converges, and so η((λ T ) ) = η(T )λ → 0, for every η in the dual space
k
∗
B[X ] of B[X ]. This means that the B[X ]-valued sequence {(λ−1 T )k } con-
verges weakly. Then it is bounded (in the uniform topology of B[X ] as a
consequence of the Banach–Steinhaus Theorem). That is, the operator λ−1 T
is power bounded. Thus |λ|−n T n  ≤ supk (λ−1 T )k  < ∞, so that
1  1
|λ|−1 T n n ≤ sup (λ−1 T )k  n ,
k
1 1
for every n. Therefore, |λ|−1 limn T n n ≤ 1, and so limn T n  n ≤ |λ| for
1
every λ ∈ C such that rσ (T ) < |λ|. That is, limn T n  n ≤ rσ (T ) + ε for every
ε > 0. Outcome:
1
lim T n  n ≤ rσ (T ). 
n

What Theorem 2.10 says is that rσ (T ) = r(T ), where rσ (T ) is the spectral

1
radius of T and r(T ) is the limit of the sequence {T n n } (whose existence
was proved in Lemma 1.10). We shall then adopt one and the same notation
(the simplest, of course) for both of them. Thus, from now on, we write
40 2. Spectrum
1
r(T ) = sup |λ| = max |λ| = lim T n  n .
λ∈σ(T ) λ∈σ(T ) n

A normaloid was defined in Section 1.6 as an operator T for which r(T ) = T .

Since rσ (T ) = r(T ), it follows that a normaloid operator acting on a complex
Banach space is precisely an operator whose norm coincides with the spectral
radius. Moreover, since on a complex Hilbert space H every normal operator
is normaloid, and so is every nonnegative operator, and since T ∗ T is always
nonnegative, it follows that, for every T ∈ B[H],
r(T ∗ T ) = r(T T ∗ ) = T ∗ T  = T T ∗ = T 2 = T ∗ 2 .
Further useful properties of the spectral radius follow from Theorem 2.10:
r(αT ) = |α| r(T ) for every α ∈ C
and, if H is a complex Hilbert space and T ∈ B[H], then
r(T ∗ ) = r(T ).

An important application of the Gelfand–Beurling formula is the charac-

terization of uniform stability in terms of the spectral radius. An operator T
in B[X ] is uniformly stable if the power sequence {T n } converges uniformly
u
to the null operator (i.e., if T n  → 0). Notation: T n −→ O.

Corollary 2.11. If T ∈ B[X ] is an operator on a complex Banach space X ,

then the following assertions are pairwise equivalent .
u
(a) T n −→ O.
(b) r(T ) < 1.
(c) T n  ≤ β αn for every n ≥ 0, for some β ≥ 1 and some α ∈ (0, 1).

Proof. Since r(T )n = r(T n ) ≤ T n for each n ≥ 1, it follows that T n → 0

implies r(T ) < 1. Now suppose r(T ) < 1 and take any α in (r(T ), 1). Since
1
r(T ) = limn T n  n (Gelfand–Beurling formula), there is an integer nα ≥ 1
such that T  ≤ αn for every n ≥ nα . Thus T n  ≤ β αn for every n ≥ 0 with
n

β = max 0≤n≤nα T n α−nα , which clearly implies T n → 0. 

An operator T ∈ B[H] on a complex Hilbert space H is strongly stable or

weakly stable if the power sequence {T n } converges strongly or weakly to the
null operator (i.e., if T n x → 0 for every x in H, or T n x ; y → 0 for every
x and y in H — equivalently, T n x ; x → 0 for every x in H — cf. Section
s w
1.1). These are denoted by T n −→ O and T n −→ O, respectively. Therefore,
from what we have considered so far,
u
r(T ) < 1 ⇐⇒ T n −→ O =⇒
s w
T n −→ O =⇒ T n −→ O =⇒ sup T n < ∞ =⇒ r(T ) ≤ 1.
n
 2.5 Numerical Radius 41

The converses to the above one-way implications fail in general. The next
result applies the preceding characterization of uniform stability to extend
the Neumann expansion of Theorem 1.3.

Corollary 2.12. Let T ∈ B[X ] be an operator on a complex Banach space,

and let λ ∈ C be any nonzero complex number .
n  T k
(a) r(T ) < |λ| if and only if converges uniformly. In this case we
k=0 λ ∞   ∞  
get λ ∈ ρ(T ) and RT (λ) = (λI − T )−1 = λ1 k=0 Tλ k where k=0 Tλ k
n  T k
denotes the uniform limit of k=0 λ .
 n  T k 
(b) If r(T ) = |λ| and λ  converges strongly, then λ ∈ ρ(T ) and
∞
k=0  ∞  
RT (λ) = (λI − T )−1 = λ1 k=0 Tλ k where k=0 Tλ k denotes the strong
n  T k
limit of k=0 λ .
n  T k
(c) If |λ| < r(T ), then k=0 λ does not converge strongly.
n  T k  n u
Proof. If converges uniformly, then Tλ −→ O, and therefore

k=0 λ
|λ|−1 r(T ) = r Tλ < 1 by Corollary 2.11. On the other hand, if r(T ) < |λ|,
 
then
 T λ∈ ρ(T ) so that λI − T ∈ G[X ], and also r Tλ = |λ|−1 r(T ) < 1. Hence,
 n
≤ β αn for every n ≥ 0, for some β ≥ 1 and α ∈ (0, 1), according to
λ ∞      n
Corollary 2.11, and so k=0  Tλ k  < ∞, which means that Tλ is an
absolutely summable sequence in B[X ]. Now followthe steps
 in the proof of
n T k
Theorem 1.3 to conclude the results in (a). If k=0  converges strongly,
 n n 
λ 
then Tλ x → 0 in X for every x ∈ X so that supn  Tλ x < ∞ for every

x ∈ X , and hence supn  Tλ n  < ∞ (by the Banach–Steinhaus Theorem).
Thus |λ|−1 r(T ) = r( Tλ ) ≤ 1, which proves (c). Moreover,
n  T k n  T k  T n+1
(λI − T ) λ1 = 1
(λI − T ) = I − s
−→ I.
k=0 λ λ k=0 λ λ

  T k   T k
Therefore, (λI − T )−1 = λ1 ∞ , where ∞ ∈ B[X ] is the strong
n  T k k=0 λ k=0 λ
limit of k=0 λ , which concludes the proof of (b). 

2.5 Numerical Radius

The numerical range W (T ) of an operator T ∈ B[H] acting on a nonzero
complex Hilbert space H is the (nonempty) set consisting of the inner products
T x ; x for unit vectors x ∈ H; that is,
 
W (T ) = λ ∈ C : λ = T x ; x for some x = 1 .

It can be shown that W (T ) is a convex set in C (see, e.g., [50, Problem 210]),
and it is clear that
W (T ∗ ) = W (T )∗ .
42 2. Spectrum

Theorem 2.13. σP (T ) ∪ σR (T ) ⊆ W (T ) and σ(T ) ⊆ W (T )− .

Proof. Take any operator T ∈ B[H] on a nonzero complex Hilbert space H.
If λ ∈ σP (T ), then there is a unit vector x ∈ H such that T x = λx. Hence
T x ; x = λx2 = λ so that λ ∈ W (T ). If λ ∈ σR (T ), then λ ∈ σP (T ∗) by
Theorem 2.6, and so λ ∈ W (T ∗ ). Thus λ ∈ W (T ). Therefore,

σP (T ) ∪ σR (T ) ⊆ W (T ).

If λ ∈ σAP (T ), then there is a sequence {xn } of unit vectors in H such that

(λI − T )xn  → 0 by Theorem 2.4. Hence

0 ≤ |λ − T xn ; xn | = |(λI − T )xn ; xn | ≤ (λI − T )xn  → 0

so that T xn xn  → λ. Since each T xn ; xn  lies in W (T ), the classical Closed

Set Theorem says that λ ∈ W (T )−. Thus σAP (T ) ⊆ W (T )−, and so

σ(T ) = σR (T ) ∪ σAP (T ) ⊆ W (T )− . 

The numerical radius of an operator T ∈ B[H] on a nonzero complex

Hilbert space H is the nonnegative number

w(T ) = sup |λ| = sup |T x ; x|.

λ∈W (T ) x=1

It is readily verified that

w(T ∗ ) = w(T ) and w(T ∗ T ) = T 2.

Unlike the spectral radius, the numerical radius is a norm on B[H]. That is,
0 ≤ w(T ) for every T ∈ B[H] and 0 < w(T ) if T = O, w(αT ) = |α|w(T ), and
w(T + S) ≤ w(T ) + w(S) for every α ∈ C and every S, T ∈ B[H]. However, the
numerical radius does not have the operator norm property in the sense that
the inequality w(S T ) ≤ w(S)w(T ) is not true for all operators S, T ∈ B[H].
Nevertheless, the power inequality holds: w(T n ) ≤ w(T )n for all T ∈ B[H] and
every positive integer n (see, e.g., [50, p. 118 and Problem 221]). Moreover,
the numerical radius is a norm equivalent to the (induced uniform) operator
norm of B[H] and dominates the spectral radius, as in the next theorem.

Theorem 2.14. 0 ≤ r(T ) ≤ w(T ) ≤ T  ≤ 2w(T ).

Proof. Take any T ∈ B[H]. Since σ(T ) ⊆ W (T )−, we get r(T ) ≤ w(T ). More-
over, w(T ) = supx=1 |T x ; x| ≤ supx=1 T x = T . Now recall that, by
the polarization identity (cf. Proposition 1.A),

T x ; y = 14 T (x + y) ; (x + y) − T (x − y) ; (x − y)

+ iT (x + iy) ; (x + iy) − iT (x − iy) ; (x − iy)
 2.5 Numerical Radius 43

for every x, y in H. Therefore, since |T z ; z| ≤ supu=1 |T u ; u|z2 =

w(T )z2 for every z ∈ H, it follows that, for every x, y ∈ H,

1

|T x ; y| ≤ 4 |T (x + y) ; (x + y)| + |T (x − y) ; (x − y)|

+ |T (x + iy) ; (x + iy)| + |T (x − iy) ; (x − iy)|
 
≤ 14 w(T ) x + y2 + x − y2 + x + iy2 + x − iy2 .

So it follows by the parallelogram law (cf. Proposition 1.A) that

 
|T x ; y| ≤ w(T ) x2 + y2 ≤ 2w(T )

whenever x = y = 1. Thus, since T  = supx=y=1 |T x ; y| (see, e.g.,
[66, Corollary 5.71]), it follows that T  ≤ 2w(T ). 

An operator T ∈ B[H] is spectraloid if r(T ) = w(T ). Recall that an oper-

ator is normaloid if r(T ) = T  or, equivalently, if T n  = T n for every
n ≥ 1 (see Theorem 1.11). The next result is a straightforward application of
the previous theorem.

Corollary 2.15. Every normaloid operator is spectraloid .

Indeed, r(T ) = T  implies r(T ) = w(T ), but r(T ) = T  also implies

w(T ) = T  (according to Theorem 2.14). Thus w(T ) = T  is a property
of every normaloid operator on H. Actually, this can be viewed as a third
definition of a normaloid operator on a complex Hilbert space.

Theorem 2.16. T ∈ B[H] is normaloid if and only if w(T ) = T .

Proof. Half of the proof was presented above. It remains to prove that

w(T ) = T  implies r(T ) = T .

Suppose w(T ) = T  (and T = O to avoid trivialities). Recall that W (T )−

is compact in C (for W (T ) is clearly bounded). Thus maxλ∈W (T )− |λ| =
supλ∈W (T )− |λ| = supλ∈W (T ) |λ| = w(T ) = T , and therefore there exists
λ ∈ W (T )− such that λ = T . Since W (T ) is nonempty, λ is a point of ad-
herence of W (T ), and so there is a sequence {λn } with each λn in W (T ) such
that λn → λ. This means that there is a sequence {xn } of unit vectors in H
(i.e., xn  = 1) such that λn = T xn ; xn  → λ, where |λ| = T  = 0. Hence,
if S = λ−1 T ∈ B[H], then
Sxn ; xn  → 1.

Claim. Sxn  → 1 and ReSxn ; xn  → 1.

Proof. |Sxn ; xn | ≤ Sxn  ≤ S = 1 for each n. But Sxn ; xn  → 1 implies
|Sxn ; xn | → 1 (and so Sxn  → 1) and also ReSxn ; xn  → 1 (since | · | and
Re(·) are continuous functions), which concludes the proof.
44 2. Spectrum

Then (I − S)xn 2 = Sxn − xn 2 = Sxn 2 − 2ReSxn ; xn  + xn 2 → 0 so

that 1 ∈ σAP (S) ⊆ σ(S) (cf. Theorem 2.4). Hence r(S) ≥ 1, and so r(T ) =
r(λ S) = |λ| r(S) ≥ |λ| = T , which implies that r(T ) = T  (because
r(T ) ≤ T  for every operator T ). 

Remark. If T ∈ B[H] is spectraloid and quasinilpotent, then T = O. In fact,

if w(T ) = 0, then T = O (since the numerical radius is a norm — also see
Theorem 2.14); in particular, if w(T ) = r(T ) = 0, then T = O. Therefore, the
unique normal (or hyponormal, or normaloid, or spectraloid ) quasinilpotent
operator is the null operator .

Corollary 2.17. If there exists λ ∈ W (T ) such that |λ| = T , then T is

normaloid and λ ∈ σP (T ). In other words, if there exists a unit vector x such
that T  = |T x ; x|, then r(T ) = w(T ) = T  and T x ; x ∈ σP (T ).
Proof. If λ ∈ W (T ) is such that |λ| = T , then w(T ) = T  (Theorem 2.14)
so that T is normaloid (Theorem 2.16). Moreover, since λ = T x ; x for some
unit vector x, it follows that T  = |λ| = |T x ; x| ≤ T xx ≤ T ,
and hence |T x ; x| = T xx. That is, the Schwarz inequality becomes an
identity, which implies that T x = αx for some α ∈ C (see, e.g., [66, Problem
5.2]). Thus α ∈ σP (T ). But α = αx2 = αx ; x = T x ; x = λ. 

2.6 Spectrum of Compact Operators

The spectral theory of compact operators plays a central role in the Spectral
Theorem for compact normal operators of the next chapter. Normal operators
were defined on Hilbert spaces; thus we keep on working with compact opera-
tors on Hilbert spaces, as we did in Section 1.8, although the spectral theory
of compact operators can be equally developed on nonzero complex Banach
spaces. So we assume that all operators in this section act on a nonzero com-
plex Hilbert space H. The main result for characterizing the spectrum of
compact operators is the Fredholm Alternative of Corollary 1.20, which can
be restated as follows.

Theorem 2.18. (Fredholm Alternative). Take T ∈ B∞[H]. If λ ∈ C \{0},

then λ ∈ ρ(T ) ∪ σP (T ). Equivalently,

σ(T )\{0} = σP (T )\{0}.

Moreover, if λ ∈ C \{0}, then dim N (λI − T ) = dim N (λI − T ∗ ) < ∞ so that

λ ∈ ρ(T ) ∪ σP4(T ). Equivalently,

σ(T )\{0} = σP4(T )\{0}.

Proof. Take a compact operator T on a Hilbert space H and a nonzero scalar

λ in C . Corollary 1.20 and the diagram of Section 2.2 ensure that
 2.6 Spectrum of Compact Operators 45

λ ∈ ρ(T ) ∪ σP1(T ) ∪ σR1(T ) ∪ σP4(T ).

Also by Corollary 1.20, N (λI − T ) = {0} if and only if N (λI − T ∗ ) = {0}, so
that λ ∈ σP (T ) if and only if λ ∈ σP (T ∗ ). Thus, λ ∈ σP1(T ) ∪ σR1(T ) by The-
orem 2.6, so that λ ∈ ρ(T ) ∪ σP4(T ) or, equivalently, λ ∈ ρ(T ) ∪ σP (T ) (since
λ ∈ σP1(T ) ∪ σP2(T ) ∪ σP3(T )). Therefore,
σ(T )\{0} = σP (T )\{0} = σP4(T )\{0}. 

The scalar 0 may be anywhere. That is, if T ∈ B∞[H], then λ = 0 may lie
in σP (T ), σR (T ), σC (T ), or ρ(T ). However, if T is a compact operator on a
nonzero space H and 0 ∈ ρ(T ), then H must be finite dimensional. Indeed,
if 0 ∈ ρ(T ), then T −1 ∈ B[H] so that I = T −1 T is compact (since B∞[H] is
an ideal of B[H]), which implies that H is finite dimensional (cf. Proposition
1.Y). The preceding theorem in fact is a rewriting of the Fredholm Alternative
(and it is also referred to as the Fredholm Alternative). It will be applied
often from now on. Here is a first application. Let B0 [H] denote the class of
all finite-rank operators on H (i.e., the class of all operators from B[H] with
a finite-dimensional range). Recall that B0 [H] ⊆ B∞[H] (finite-rank operators
are compact — cf. Proposition 1.X). Let #A denote the cardinality of a set
A, so that #A < ∞ means “A is a finite set”.

Corollary 2.19. If T ∈ B0 [H], then

σ(T ) = σP (T ) = σP4(T ) and #σ(T ) < ∞.

Proof. If dim H < ∞, then an injective operator is surjective, and linear mani-
folds are closed (see, e.g., [66, Problem 2.18 and Corollary 4.29]), and so the di-
agram of Section 2.2 says that σ(T ) = σP (T ) = σP4(T ) (for σP1(T ) = σR1(T ∗ )∗
according to Theorem 2.6). On the other hand, suppose dim H = ∞. Since
B0 [H] ⊆ B∞[H], Theorem 2.18 says that
σ(T )\{0} = σP (T )\{0} = σP4(T )\{0}.
Since dim R(T ) < ∞ and dim H = ∞, it follows that R(T )− = R(T ) = H and
N (T ) = {0} (because dim N (T ) + dim R(T ) = dim H; see, e.g., [66, Problem
2.17]). Then 0 ∈ σP4(T ) (cf. diagram of Section 2.2), and therefore
σ(T ) = σP (T ) = σP4(T ).
If σP (T ) is infinite, then there exists an infinite set of linearly independent
eigenvectors of T (Theorem 2.3). Since every eigenvector of T lies in R(T ),
this implies that dim R(T ) = ∞ (because every linearly independent subset
of a linear space is included in some Hamel bases — see, e.g., [66, Theorem
2.5]), which is a contradiction. Conclusion: σP (T ) must be finite. 

In particular, the above result clearly holds if H is finite dimensional since,

as we saw above, dim H < ∞ implies B[H] = B0 [H].
46 2. Spectrum

Corollary 2.20. Take an arbitrary compact operator T ∈ B∞[H].

(a) 0 is the only possible accumulation point of σ(T ).
(b) If λ ∈ σ(T )\{0}, then λ is an isolated point of σ(T ).
(c) σ(T )\{0} is a discrete subset of C .
(d) σ(T ) is countable.

Proof. Let T be a compact operator on H.

Claim. An infinite sequence of distinct points of σ(T ) converges to zero.
Proof. Let {λn }∞n=1 be an infinite sequence of distinct points of σ(T ). Without
loss of generality, suppose that every λn is nonzero. Since T is compact and
0 = λn ∈ σ(T ), it follows by Theorem 2.18 that λn ∈ σP (T ). Let {xn }∞ n=1 be
a sequence of eigenvectors associated with {λn }∞ n=1 (i.e., T xn = λn xn with
each xn = 0), which is a sequence of linearly independent vectors by Theorem
2.3. For each n ≥ 1, set
Mn = span{xi }ni=1 ,
which is a subspace of H with dim Mn = n, and

Mn ⊂ Mn+1

for every n ≥ 1 (because {xi }n+1

i=1 is linearly independent and so xn+1 lies
in Mn+1 \Mn ). From now on the argument is similar to that in the proof
of Theorem 1.18. Since each Mn is a proper subspace of the Hilbert space
Mn+1 , it follows that there exists yn+1 in Mn+1 with yn+1  = 1 for which
1 n+1
2 < inf u∈Mn yn+1 − u. Write yn+1 = i=1 αi xi in Mn+1 so that

n+1 n
(λn+1 I − T )yn+1 = αi (λn+1 − λi )xi = αi (λn+1 − λi )xi ∈ Mn .
i=1 i=1

Recall that λn = 0 for all n, take any pair of integers 1 ≤ m < n, and set

y = ym − λ−1 −1
m (λm I − T )ym + λn (λn I − T )yn

so that T (λ−1 −1
m ym ) − T (λn yn ) = y − yn . Since y lies in Mn−1 ,

1
2 < y − yn  = T (λ−1 −1
m ym ) − T (λn yn ),

which implies that the sequence {T (λ−1 n yn )} has no convergent subsequence.

Thus, since T is compact, Proposition 1.S ensures that {λ−1 n yn } has no
bounded subsequence. That is, supk |λk |−1 = supk λ−1 k y k  = ∞, and so
inf k |λk | = 0 for every subsequence {λk }∞
k=1 of {λn } ∞
n=1 . Thus λn → 0, which
concludes the proof of the claimed result.
(a) Thus, if λ = 0, then there is no sequence of distinct points in σ(T ) that
converges to λ; that is, λ = 0 is not an accumulation point of σ(T ).
 2.7 Additional Propositions 47

(b) Therefore, every λ in σ(T )\{0} is not an accumulation point of σ(T );

equivalently, every λ in σ(T )\{0} is an isolated point of σ(T ).
(c) Hence σ(T )\{0} consists entirely of isolated points, which means that
σ(T )\{0} is a discrete subset of C .
(d) Since a discrete subset of a separable metric space is countable (see, e.g.,
[66, Example 3.Q]), and since C is separable, σ(T )\{0} is countable. 

Corollary 2.21. If an operator T ∈ B[H] is compact and normaloid, then

σP (T ) = ∅ and there exists λ ∈ σP (T ) such that |λ| = T .
Proof. Suppose T is normaloid (i.e., r(T ) = T ). Thus σ(T ) = {0} only if
T = O. If T = O and H = {0}, then 0 ∈ σP (T ) and T  = 0. If T = O, then
σ(T ) = {0} and T  = r(T ) = maxλ∈σ(T ) |λ|, so that there exists λ = 0 in
σ(T ) such that |λ| = T . Moreover, if T is compact and σ(T ) = {0}, then
∅ = σ(T )\{0} ⊆ σP (T ) by Theorem 2.18. Hence r(T ) = maxλ∈σ(T ) |λ| =
maxλ∈σP (T ) |λ| = T . Thus there exists λ ∈ σP (T ) such that |λ| = T . 

Corollary 2.22. Every compact hyponormal operator is normal .

Proof. Suppose T ∈ B[H] is a compact hyponormal operator on a nonzero
 H. Corollary 2.21
complex Hilbert space − says that σP (T ) = ∅. Consider the
subspace M = λ∈σP (T ) N (λI − T ) of Theorem 1.16 with {λγ }γ∈Γ =
σP (T ). Observe that σP (T |M⊥ ) = ∅. Indeed, if there is a λ ∈ σP (T |M⊥ ),
then there exists 0 = x ∈ M⊥ such that λx = T |M⊥ x = T x, and so
x ∈ N (λI − T ) ⊆ M, which is a contradiction. Moreover, recall that T |M⊥
is compact and hyponormal (Propositions 1.O and 1.U). Thus, if M⊥ = {0},
then Corollary 2.21 says that σP (T |M⊥ ) = ∅, which is another contradic-
tion. Therefore, M⊥ = {0} so that M = H (see Section 1.3), and hence
T = T |H = T |M is normal according to Theorem 1.16. 

2.7 Additional Propositions

Proposition 2.A. Let H = {0} be a complex Hilbert space and let T denote
the unit circle about the origin of the complex plane.
(a) If H ∈ B[H] is hyponormal, then σP (H)∗ ⊆ σP (H ∗ ) and σR (H ∗ ) = ∅.
(b) If N ∈ B[H] is normal, then σP (N ∗ ) = σP (N )∗ and σR (N ) = ∅.
(c) If U ∈ B[H] is unitary, then σ(U ) ⊆ T .
(d) If A ∈ B[H] is self-adjoint, then σ(A) ⊂ R .
(e) If Q ∈ B[H] is nonnegative, then σ(Q) ⊂ [0, ∞).
(f) If R ∈ B[H] is strictly positive, then σ(R) ⊂ [α, ∞) for some α > 0.
(g) If E ∈ B[H] is a nontrivial projection, then σ(E) = σP (E) = {0, 1}.
48 2. Spectrum

Proposition 2.B. Similarity preserves the spectrum and its parts, and so it
preserves the spectral radius. That is, let H and K be nonzero complex Hilbert
spaces. For every T ∈ B[H] and W ∈ G[H, K],
(a) σP (T ) = σP (W T W −1 ),
(b) σR (T ) = σR (W T W −1 ),
(c) σC (T ) = σC (W T W −1 ).
Hence σ(T ) = σ(W T W −1 ), ρ(T ) = ρ(W T W −1 ), and r(T ) = r(W T W −1 ).
Unitary equivalence also preserves the norm: if W is a unitary transforma-
tion, then, in addition, T  = W T W −1 .

Proposition 2.C. σ(S T )\{0} = σ(T S)\{0} for every S, T ∈ B[H].

1 1
Proposition 2.D. If Q ∈ B[H] is nonnegative, then σ(Q 2 ) = σ(Q) 2 .

Proposition 2.E. Take any operator T ∈ B[H]. Let d denote the usual dis-
tance in C . If λ ∈ ρ(T ), then

r((λI − T )−1 ) = [d(λ, σ(T ))]−1 .

If T is hyponormal and λ ∈ ρ(T ), then

(λI − T )−1  = [d(λ, σ(T ))]−1 .

Proposition 2.F. Let {Hk } be a collection of Hilbert spaces, let {Tk } be a

(similarly indexed ) collection
of operators with each Tk in B[Hk ], and consider
the (orthogonal ) direct sum k T k in B[ k Hk ]. Then

(a) σP ( k Tk ) = k σP (Tk ),

(b) σ( k Tk ) = k σ(Tk ) if the collection {Tk } is finite.
In general (if the collection {Tk } is not finite), then
 − 
(c) k σ(Tk ) ⊆ σ( k Tk ) and the inclusion may be proper.
However, if (λI − Tk )−1  = [d(λ, σ(Tk ))]−1 for each k and every λ ∈ ρ(Tk ),
 − 
(d) k σ(Tk ) = σ( k Tk ), which happens whenever each Tk is hyponormal.

Proposition 2.G. An operator T ∈ B[X ] on a complex Banach space is nor-

maloid if and only if there is a λ ∈ σ(T ) such that |λ| = T .

Proposition 2.H. For every operator T ∈ B[H] on a complex Hilbert space,

 
σR (T ) ⊆ λ ∈ C : |λ| < T  .

Proposition 2.I. If H and K are complex Hilbert spaces and T ∈ B[H], then
 2.7 Additional Propositions 49

r(T ) = inf W T W −1 .
W ∈G[H,K]

The spectral radius expression in the preceding proposition ensures that an

operator is uniformly stable if and only if it is similar to a strict contraction.

Proposition 2.J. If T ∈ B[X ] is a nilpotent operator on a complex Banach

space, then σ(T ) = σP (T ) = {0}.

Proposition 2.K. An operator T ∈ B[H] on a complex Hilbert space is spec-

traloid if and only if w(T n ) = w(T )n for every n ≥ 0.
An operator T ∈ B[H] on a separable infinite-dimensional Hilbert space H
is diagonalizable if T x = ∞k=0 αk x ; ek ek for every x ∈ H, for some orthonor-
mal basis {ek }∞k=0 for H and some bounded sequence {αk }∞ k=0 of scalars.

Proposition 2.L. If T ∈ B[H] is diagonalizable and H is complex, then

 
σP (T ) = λ ∈ C : λ = αk for some k ≥ 1 , σR (T ) = ∅, and
 
σC (T ) = λ ∈ C : inf k |λ − αk | = 0 and λ = αk for every k ≥ 1 .

An operator S+ ∈ B[K+ ] on a Hilbert space K+ is a unilateral shift , and

an operator S ∈ B[K] on a Hilbert space K is a bilateral shift , if there exists
an infinite sequence {Hk }∞ k=0 and an infinite family {Hk }k=−∞
∞
∞of nonzero
pairwise∞ orthogonal subspaces of K+ and K such that K+ = k=0 Hk and
K = k=−∞ Hk (cf. Section 1.3), and both S+ and S map each Hk isometri-
cally onto Hk+1 , so that each transformation U+ (k+1) = S+ |Hk : Hk → Hk+1 ,
and each transformation Uk+1 = S|Hk : Hk → Hk+1 , is unitary, and therefore
dim Hk+1 = dim Hk . Such a common dimension is the multiplicity of S+ and
2 ∞ 2
S.
∞ If H k = H for all k, then K + = + (H) = k=0 H and K =  (H) =
k=−∞ H are the direct orthogonal sums of countably infinite copies of a
single nonzero Hilbert space H, indexed either by the nonnegative integers or
by all integers, which are precisely the Hilbert spaces consisting of all square-
summable H-valued sequences {xk }∞ k=0 and of all square-summable H-valued
families {xk }∞ k=−∞ . In this case (if Hk = H for all k), U+ (k+1) = S+ |H = U+
and Uk+1 = S|H = U for all k, where U+ and U are any unitary operators on
H. In particular, if U+ = U = I, the identity on H, then S+ and S are referred
2
to as the canonical unilateral and bilateral shifts on + (H) and on  2 (H). The
∗ ∗
adjoint S+ ∈ B[K+ ] of S+ ∈ B[K+ ] and the adjoint of S ∈ B[K] of S ∈ B[K] are
referred  to as a backward unilateral ∞shift and as a∞ backward bilateral shift .
∞ ∞ ∞
Writing
∞ k=0 xk for {xk } k=0 in k=0 Hk , and k=−∞ x k for {xk } k=−∞ in
∗
k=0 Hk , it follows that S+ : K+ → K+ and S+ : K+ → K+ , and S : K → K and
S ∗ : K → K, are given by the formulas
∞ ∞
S+ x = 0 ⊕ k=1 U+ (k) xk−1 and S+∗ x = k=0 U+∗(k+1) xk+1
∞ ∞
for all x = k=0 xk in K+ = k=0 Hk , with 0 being the origin of H0 , where
U+ (k+1) is any unitary transformation of Hk onto Hk+1 for each k ≥ 0, and
50 2. Spectrum
∞ ∞
Sx = k=−∞ Uk xk−1 and S∗x = ∗
k=∞ Uk+1 xk+1
∞ ∞
for all x = k=−∞ xk in K = k=−∞ Hk , where, for each integer k, Uk+1
is any unitary transformation of Hk onto Hk+1 . The spectrum of a bilateral
shift is simpler than that of a unilateral shift, for bilateral shifts are unitary
operators (i.e., besides being isometries as unilateral shifts are, bilateral shifts
are normal too). For a full treatment on shifts on Hilbert spaces see [49].

Proposition 2.M. Let D and T = ∂ D denote the open unit disk and the unit
circle about the origin of the complex plane, respectively. If S+ ∈ B[K+] is a
unilateral shift and S ∈ B[K] is a bilateral shift on complex spaces, then
(a) σP (S+ ) = σR (S+∗ ) = ∅, σR (S+ ) = σP (S+∗ ) = D , σC (S+ ) = σC (S+∗ ) = T .
(b) σ(S) = σ(S ∗ ) = σC (S ∗ ) = σC (S) = T .
2
A unilateral weighted shift T+ = S+ D+ in B[+ (H)] isthe product of a
∞
canonical unilateral shift S+ and a diagonal operator D+ = k=0 αk I, both in
2
B[+ (H)], where {αk }∞
k=0 is a bounded sequence of scalars. A bilateral weighted
shift T = SD in B[ 2 (H)] is the product of a canonical bilateral shift S and
∞
a diagonal operator D = k=−∞ αk I, both in B[ 2 (H)], where {αk }∞ k=−∞ is
a bounded family of scalars.
2
Proposition 2.N. Let T+ ∈ B[+ (H)] be a unilateral weighted shift, and let
2
T ∈ B[ (H)] be a bilateral weighted shift, where H is complex .
(a) If αk → 0 as |k| → ∞, then T+ and T are compact and quasinilpotent .
If , in addition, αk = 0 for all k, then
(b) σ(T+) = σR (T+) = σR2(T+) = {0} and σ(T+∗ ) = σP (T+∗ ) = σP2(T+∗ ) = {0},
(c) σ(T ) = σC (T ) = σC (T ∗ ) = σ(T ∗ ) = {0}.

Proposition 2.O. Let T ∈ B[H] be an operator on a complex Hilbert space

and let D be the open unit disk about the origin of the complex plane.
w
(a) If T n −→ O, then σP (T ) ⊆ D .
u
(b) If T is compact and σP (T ) ⊆ D , then T n −→ O.
(c) The concepts of weak, strong, and uniform stabilities coincide for a com-
pact operator on a complex Hilbert space.

Proposition 2.P. Take an operator T ∈ B[H] on a complex Hilbert space and

let T = ∂ D be the unit circle about the origin of the complex plane.
u w
(a) T n −→ O if and only if T n −→ O and σC (T ) ∩ T = ∅.
(b) If the continuous spectrum does not intersect the unit circle, then the
concepts of weak, strong, and uniform stabilities coincide.
 2.7 Additional Propositions 51

The concepts of resolvent set ρ(T ) and spectrum σ(T ) of an operator T

in the unital complex Banach algebra B[X ] as in Section 2.1, namely, ρ(T ) =
{λ ∈ C : λI − T has an inverse in B[X ]} and σ(T ) = C \ρ(T ), of course, hold
in any unital complex Banach algebra A, and so does the concept of spectral
radius r(T ) = supλ∈σ(T ) |λ|, where the Gelfand–Beurling formula of Theorem
1
2.10, namely, r(T ) = limn T n  n , holds in any unital complex Banach algebra
(whose proof is essentially the same as the proof of Theorem 2.10). Recall that
a component of a set in a topological space is any maximal (in the inclusion
ordering) connected subset of it. A hole of a compact set is any bounded
component of its complement. Thus the holes of the spectrum σ(T ) are the
bounded components of the resolvent set ρ(T ) = C \σ(T ).
If A is a closed unital subalgebra of a unital complex Banach algebra A
(for instance, A = B[X ] where X is a Banach space), and if T ∈ A , then let
ρ (T ) be the resolvent set of T with respect to A , let σ (T ) = C \ρ (T ) be
the spectrum of T with respect to A , and set r (T ) = supλ∈σ (T ) |λ|. Recall
that a homomorphism (or an algebra homomorphism) between two algebras
is a linear transformation between them that also preserves product. Let A
be a maximal (in the inclusion ordering) commutative subalgebra of a unital
complex Banach algebra A (i.e., a commutative subalgebra of A that is not
included in any other commutative subalgebra of A). Note that A is trivially
unital, and closed in A because the closure of a commutative subalgebra of a
Banach algebra is again commutative since multiplication is continuous in A.
Consider the (unital complex commutative) Banach algebra C (of all complex
numbers). Let A = {Φ: A → C : Φ is an homomorphism} stand for the col-
lection of all algebra homomorphisms of A onto C .

Proposition 2.Q. Let A be any unital complex Banach algebra (for instance,
A = B[X ]). If T ∈ A , where A is any closed unital subalgebra of A, then
(a) ρ (T ) ⊆ ρ(T ) and r (T ) = r(T ) (invariance of the spectral radius).
Hence ∂σ (T ) ⊆ ∂σ(T ) and σ(T ) ⊆ σ (T ).
Thus σ (T ) is obtained by adding to σ(T ) some holes of σ(T ).
(b) If A is a maximal commutative subalgebra of A, then
 
σ(T ) = Φ(T ) ∈ C : Φ ∈ A for each T ∈A.

Moreover, in this case,

σ(T ) = σ (T ).

Proposition 2.R. If A is a unital complex Banach algebra, and if S, T in

A commute (i.e., if S, T ∈ A and S T − T S), then

σ(S + T ) ⊆ σ(S) + σ(T ) and σ(ST ) ⊆ σ(S) · σ(T ).

52 2. Spectrum

If M is an invariant subspace for T , then it may happen that σ(T |M ) ⊆

σ(T ). Sample: every unilateral shift is the restriction of a bilateral shift to
an invariant subspace (see, e.g., [62, Lemma 2.14]). However, if M reduces
T , then σ(T |M ) ⊆ σ(T ) by Proposition 2.F(b). The full spectrum of T ∈ B[H]
(notation: σ(T )# ) is the union of σ(T ) and all bounded components of ρ(T )
(i.e., σ(T )# is the union of σ(T ) and all holes of σ(T )).

Proposition 2.S. If M is T -invariant, then σ(T |M ) ⊆ σ(T )# .

Proposition 2.T. Let T ∈ B[H] and S ∈ B[K] be operators on Hilbert spaces

H and K. If σ(T ) ∩ σ(S) = ∅, then for every bounded linear transformation
Y ∈ B[H, K] there exists a unique bounded linear transformation X ∈ B[H, K]
such that X T − SX = Y . In particular ,

σ(T ) ∩ σ(S) = ∅ and X T = SX =⇒ X = O.

This is the Rosenblum Corollary, which will be used to prove the Fuglede
Theorems in Chapter 3.

Notes: Again, as in the previous chapter, these are basic results that will be
needed throughout the text. We will not point out here the original sources
but, instead, well-known secondary sources where the reader can find the
proofs for those propositions, as well as some deeper discussions on them.
Proposition 2.A holds independently of the forthcoming Spectral Theorem
(Theorem 3.11) — see, e.g., [66, Corollary 6.18]. A partial converse, however,
needs the Spectral Theorem (see Proposition 3.D in Chapter 3). Proposition
2.B is a standard result (see, e.g., [50, Problem 75] and [66, Problem 6.10]), as
is Proposition 2.C (see, e.g., [50, Problem 76]). Proposition 2.D also dismisses
the Spectral Theorem of the next chapter; it is obtained by the Square Root
Theorem (Proposition 1.M) and by the Spectral Mapping Theorem (Theorem
2.7). Proposition 2.E is a rather useful technical result (see, e.g., [63, Problem
6.14]. Proposition 2.F is a synthesis of some sparse results (cf. [28, Proposition
I.5.1], [50, Solution 98], [55, Theorem 5.42], [66, Problem 6.37], and Propo-
sition 2.E). For Propositions 2.G and 2.H see [66, Sections 6.3 and 6.4]. The
spectral radius formula in Proposition 2.I (see, e.g., [42, p. 22]) ensures that an
operator is uniformly stable if and only if it is similar to a strict contraction
(hint: use the equivalence between (a) and (b) in Corollary 2.11). For Propo-
sitions 2.J and 2.K see [66, Sections 6.3 and 6.4]. The examples of spectra
in Propositions 2.L, 2.M, and 2.N are widely known (see, e.g., [66, Examples
6.C, 6.D, 6.E, 6.F, and 6.G]). Proposition 2.O deals with the equivalence be-
tween uniform and weak stabilities for compact operators, and it is extended
to a wider class of operators in Proposition 2.P (see [63, Problems 8.8. and
8.9]). Proposition 2.Q(a) is readily verified, where the invariance of the spec-
tral radius follows by the Gelfand–Beurling formula. Proposition 2.Q(b) is a
key result for proving both Theorem 2.8 (the Spectral Mapping Theorem for
normal operators) and also Lemma 5.43 of Chapter 5 (for the characterization
 2.7 Additional Propositions 53

of the Browder spectrum in Theorem 5.44) — see, e.g., [76, Theorems 0.3 and
0.4]. For Propositions 2.R, 2.S, and 2.T see [80, Theorem 11.22], [76, Theorem
0.8], and [76, Corollary 0.13], respectively. Proposition 2.T will play a central
role in the proof of the Fuglede Theorems of the next chapter (precisely, in
Corollary 3.5 and Theorem 3.17).

Suggested Reading

Bachman and Narici [8] Halmos [50]

Berberian [17] Herrero [55]
Conway [27, 28] Istrǎţescu [58]
Douglas [34] Kato [60]
Dowson [35] Kubrusly [62, 63, 66]
Dunford and Schwartz [39] Radjavi and Rosenthal [76]
Fillmore [42] Rudin [80]
Gustafson and Rao [45] Taylor and Lay [87]
http://www.springer.com/978-0-8176-8327-6