E extracta mathematicae Vol. 33, Núm. 2, 209 – 218 (2018)

Moving Weyl’s Theorem from f(T ) to T

M. Febronio Rodŕıguez 1, B.P. Duggal2, S.V. Djordjević 1

1 BUAP, Facultad de Ciencias F́ısico-Matemáticas

Rı́o Verde y Av. San Claudio, San Manuel, Puebla, Pue. 72570, Mexico

2 8 Redwood Grove, Northfield Avenue, London W5 4SZ, England, U.K.

mbfebronio222@hotmail.com , bpduggal@yahoo.co.uk , slavdj@fcfm.buap.mx

Presented by Manuel González Received May 25, 2017

Abstract: Schmoeger has shown that if Weyl’s theorem holds for an isoloid Banach space
operator T ∈ B(X) with stable index, then it holds for f(T) whenever f ∈ Holo σ(T) is a
function holomorphic on some neighbourhood of the spectrum of T. In this note we establish
a converse.

Key words: Weyl’s theorem, Browder’s theorem, SVEP.

AMS Subject Class. (2000): 47A10, 47A11.

1. Introduction

Recall that an operator T ∈ B(X) has finite ascent if there is p ∈ N for
which

T −p(0) = T −∞(0) =
∞∪
n=1

T −n(0) ;

if in particular

T −p(0) = T −w(0) =
{
x ∈ X : ||T nx||

1
n → 0

}
(transfinite kernel)

we shall say that T has finite hyperascent. The transfinite range of an operator
is defined by

T w(X) =


x ∈ X :

there exists k > 0 and a sequence {xn} ⊂ X
such that Tx1 = x , Txn+1 = xn

and ∥xn∥ ≤ kn∥x∥ for all positive integers n


 .

If T ∈ B(X) has finite ascent, then, in particular, it has the “single valued
extension property” (SVEP) at zero, which says [6] that the only holomorphic
function f for which (T − z)f(z) = 0 for all z in a neighborhood of zero is

209



210 m.f. rodŕıguez, b.p. duggal, s.v. djordjević

the zero function. Equivalently, [6], 0 is not in the “local point spectrum”

π
left
loc (T). For T ∈ B(X), let N(T)(= T

−1(0)) and R(T)(= T(X)) denote,
respectively, the null space and the range of the mapping T. Let α(T) =
dim N(T) and β(T) = dim X/R(T), if theses spaces are finite dimensional,
otherwise let α(T) = ∞ and β(T) = ∞. If the range R(T) of T ∈ B(X)
is closed and α(T) < ∞ (respectively, β(T) < ∞), then T is said to be
an upper semi-Fredholm (respectively, a lower semi-Fredholm) operator and
we denote T ∈ Φ+(X) (respectively T ∈ Φ−(X)). If T ∈ Φ−(X) ∪ Φ+(X)
then T is called a semi-Fredholm operator (in notation T ∈ Φ±(X)) and
for T ∈ Φ−(X) ∩ Φ+(X) we say that T is a Fredholm operator (in notation
T ∈ Φ(X)). For T ∈ Φ±(X), the index of T is defined by

ind(T) = α(T) − β(T).

T has stable index if ind(T − µI) is either ≥ 0 or ≤ 0 (exclusive or) for all µ
not in the Fredholm spectrum σe(T) of T ; T is isoloid if the isolated points of
the spectrum of T are eigenvalues of T. Denote with π00(T) the set of isolated
eigenvalues of T of finite geometric multiplicity, i.e.

π00(T) = {λ ∈ iso σ(T) : 0 < α(T − λ) < ∞}.

Similarly, with π0(T) we denote the set of all isolated eigenvalues of T of finite
algebraic multiplicity (poles of T). Obviously, π0(T) ⊆ π00(T).

Schmoeger, [8], has shown that “if Weyl’s theorem holds for an isoloid
operator T with stable index”, then it holds for f(T) whenever f ∈ Holo σ(T)
(or f ∈ Holoc σ(T)), the set of all non-trivial holomorphic function on some
neighborhood of the spectrum of T (or all function from Holo σ(T) that are
not constant on connected component). In this note we address the converse
problem. Specifically, we will give conditions under which “Browder’s the-
orem” (respectively “finite hyperascent property”) is transmitted back from
f(T) to T .

2. Browder’s theorem

Recall that T is polaroid if every isolated point λ of the spectrum of T, λ ∈
iso σ(T), is a pole of the resolvent of T. The Browder spectrum σb(T) and the
Weyl spectrum σw(T) of T ∈ B(X) are the sets σb(T) = {λ ∈ σ(T) : T − λ /∈
Φ(X) or asc(T −λ) ̸= des(T −λ)} and σw(T) = {λ ∈ σ(T) : T −λ /∈ Φ0(X)},
where Φ0(X) denotes the set of all Fredholm operators with index zero. The
essential (Fredholm) spectrum is the set σe(T) = {λ ∈ σ(T) : T −λ /∈ Φ(X)}.



moving weyl’s theorem from f(T) to T 211

Here des(T) denotes the descent of T, the smallest positive integer n such that
R(T n) = R(T n+1) (if no such n exists, then des(T) = ∞), asc(T) denotes the
ascent of T, the smallest positive integer n such that T −n(0) = T −(n+1)(0)
(if no such n exists, then asc(T) = ∞). Browder’s theorem holds for T if and
only if

(1) σb(T) ⊆ σw(T),

equivalently, [4, Theorem 8.3.1], if and only if T has SVEP on σ(T) \ σw(T),
equivalently, [6], if and only if

(2) π
left
loc (T) ⊆ σw(T).

Remark 2.1. (i) Let T ∈ B(X) and f ∈ Holo σ(T). Then Browder’s
theorem for f(T) implies the spectral mapping theorem for Weyl spectrum:

σw(f(T)) = σb(f(T)) = f(σb(T)) ⊇ f(σw(T)).

Since opposite inclusion always holds, we have σw(f(T)) = f(σw(T)).
(ii) The SVEP property on σ(T) \ σw(T) guarantees us even more: Brow-

der’s theorem for f(T) for every f ∈ Holoc σ(T). Really, let f ∈ Holoc σ(T)
and f(λ0) ∈ σ(f(T)) \ σw(f(T)). Then there is an r ∈ N, a polynomial h and
g ∈ Holo σ(T) (with no zero in σ(T)) such that

f(z) − f(λ0) ≡ (z − λ0)rh(z)g(z)

with h(λ0) ̸= 0 and h(λ0) ̸∈ g(σ(T)). It follows

f(T) − f(λ0) = (T − λ0)rh(T)g(T) ∈ Φ0(X),

with 0 /∈ σ(h(T)g(T)) and, consequently, λ0 /∈ σw(T). Hence, T has SVEP at
λ0 and, by [1, Theorem 2.39], f(T) has SVEP at f(λ0) that implies Browder’s
theorem for f(T).

(iii) In the case of f ∈ Holo σ(T) we need little more, SVEP at all λ ∈
σ(T) \ σe(T) or injectivity of f.

Hence, for f ∈ Holo σ(T), the passage of Browder’s theorem from T to
f(T), is not a major problem. More interesting question is how to pass Brow-
der’s theorem from f(T) to T. In general, the SVEP and Browder’s theorem
do not move from f(T) to T . To see this, it is enough consider an operator
T without SVEP on σ(T) \ σw(T) (in this case no Browder’s theorem for T)
and f ≡ c ∈ Holo σ(T) (for more details see [5, p. 227]).



212 m.f. rodŕıguez, b.p. duggal, s.v. djordjević

Given T ∈ B(X) and f ∈ Holo σ(T), then we define the set

(3) Af(T) = {λ /∈ σw(T) : f(λ) ∈ σw(f(T))}

and we say that T has the property Sf if

(Sf) T has SVEP at every λ ∈ Af(T).

Theorem 2.2. Let T ∈ B(X) be such that Browder’s theorem holds for
f(T).

(i) If f ∈ Holo σ(T) and T has the property (Sf), then Browder’s theorem
holds for T.

(ii) If f ∈ Holoc σ(T), then Browder’s theorem holds for g(T), for any g ∈
Holoc σ(T).

Proof. Let λ ∈ σ(T) \ σw(T).
Case I: If f(λ) ∈ f(σ(T)) \ f(σw(T)), then Browder’s theorem for f(T)

and, consequently, the spectral mapping theorem for Weyl spectrum of T,
guarantees us that f(λ) is an isolated point of σ(f(T)) (matter of fact it is a
pole). Then λ is an isolated point of the spectrum of T that implies SVEP
property for T at λ.

Case II: Let f(λ) ∈ f(σw(T))(= σw(f(T))).
(i) Then, by property (Sf), T has SVEP at λ.
(ii) If f(λ) ∈ σw(f(T)), then by injectivity of f ∈ Holoc σ(T), we have that
λ ∈ σw(T), which is a contradiction to our assumption.

Hence, T has SVEP at all λ /∈ σw(T), and Browder’s theorem holds for
T . Moreover, for any g ∈ Holoc σ(T), by Remark 2.1 (ii), Browder’s theorem
holds for g(T), for every g ∈ Holoc σ(T).

The similar behavior we have in the situation of more general versions of
Browder’s theorem: the g-Browder, a-Browder or s-Browder theorems. We
say that T ∈ B(X) obeys

g-Browder’s theorem if σbb(T) ⊆ σbw(T) ,(4)
a-Browder’s theorem if σab(T) ⊆ σaw(T) ,(5)
s-Browder’s theorem if σsb(T) ⊆ σsw(T) ,(6)

where



moving weyl’s theorem from f(T) to T 213

σbb(T) = {λ ∈ σ(T) : T − λ is not B-Fredholm or asc(T − λ) ̸= des(T − λ)},
σbw(T) = {λ ∈ σ(T) : T − λ is not B-Fredholm or ind(T − λ) ̸= 0},
σab(T) = {λ ∈ σa(T) : T − λ /∈ Φ+(T) or asc(T − λ) = ∞},
σaw(T) = {λ ∈ σa(T) : T − λ /∈ Φ+(T) or ind(T − λ) > 0},
σsb(T) = {λ ∈ σs(T) : T − λ /∈ Φ−(T) or des(T − λ) = ∞},
σsw(T) = {λ ∈ σs(T) : T − λ /∈ Φ−(T) or ind(T − λ) < 0}.

Note that T ∈ B(X) is a B-Fredholm operator if for some integer n the range
space R(T n) is closed and Tn = T|R(T n) is a Fredholm operator. In this case
Tm is a Fredholm operator and ind(Tm) = ind(Tn) for each m ≥ n. This
enables us to define the index of a B-Fredholm operator T as the index of the
Fredholm operator Tn where n is any integer such that R(T

n) is closed and
such that Tn is a Fredholm operator.

Let ∗ ∈ {g, a, s}. It is known that ∗-Browder’s theorem holds for T if T has
SVEP at all points λ /∈ σ∗w(T) and that ∗-Browder’s theorem implies Brow-
der’s theorem (matter of fact, g-Browder’s theorem is equivalent to Browder’s
theorem). Moreover, if T has SVEP at all points λ /∈ σ∗w(T), then the spec-
tral mapping theorem holds for σ∗w(T) and the functions from Holoc σ(T).
(For more details see [4]).

Let T ∈ B(X), then for any f ∈ Holo σ(T) and ∗ ∈ {g, a, s}, we define the
sets

A∗f(T) = {λ /∈ σ∗w(T) : f(λ) ∈ σ∗w(f(T))},
and the property S∗f

(S∗f) T has SVEP at every λ ∈ A
∗
f(T),

then we have next theorem:

Theorem 2.3. Let T ∈ B(X) and f ∈ Holo σ(T). If ∗-Browder’s theorem
holds for f(T) and T has the property S∗f, then ∗-Browder’s theorem holds
for T. Moreover, ∗-Browder’s theorem holds for T if and only if it is holds for
g(T), for any g ∈ Holoc σ(T).

3. Weyl’s theorem

If Browder’s theorem holds for some T ∈ B(X) together with π0(T) =
π00(T), then we say that T satisfies Weyl’s theorem. SVEP alone is not enough



214 m.f. rodŕıguez, b.p. duggal, s.v. djordjević

for T to satisfy Weyl’s theorem: consider, for example, the quasinilpotent op-
erator Q ∈ B(ℓ2), Q(x1, x2, x3, . . . ) = (x22 ,

x3
3
, . . . ). A necessary and sufficient

condition for T to satisfy Weyl’s theorem is that T satisfies Browder’s theorem
and, for every λ ∈ π00(T), T − λ has finite hyperascent. Furthermore, if T is
polaroid and has SVEP, then both f(T) and f(T ∗) satisfy Weyl’s theorem for
every f ∈ Holoc σ(T) [3].

For moving Weyl’s theorem from f(T) to T we need a variant of (3). Let
T ∈ B(X) and f ∈ Holo σ(T), then we define the set

(7) Πf(T) = {λ ∈ π00(T) : f(λ) ∈ σw(f(T))}

and we say that T has the property Πf if

(Πf) T − λ has a finite hyperascent, for every λ ∈ Πf(T).

Remark 3.1. (i) Let T ∈ B(X) and {λ1, λ2, . . . , λn} ⊂ C be a finite set of
distinct complex numbers. Then, for any polynomial p(λ) =

∏n
i=1(λi − λ)

mi

we have

p(T)−1(0) =

n⊕
i=1

(T − λi)−mi(0).

Moreover, if p(λ0) ̸= 0, for some complex number λ0, then

(T − λ0)−w(0) ∩ p(T)−1(0) = {0}.

(ii) If T, S ∈ B(X) is a pair of commuting operators, then T −w(0)
⊆ (TS)−w(0). Moreover, if S is an invertible operator, then T −w(0) =
(TS)−w(0).

Theorem 3.2. Let T ∈ B(X) and f ∈ Holoc σ(T). If Weyl’s theorem
holds for f(T) and T has the property Πf, then Weyl’s theorem holds for T.

Proof. By Theorem 2.2, Browder’s theorem holds for T, hence we have
to show that T − λ has a finite hyperascent, for every λ ∈ π00(T) (see [4,
Theorem 8.4.5 (vi)]). Let λ0 ∈ π00(T); then f(λ0) ∈ f(σ(T)) = σ(f(T)) =
σw(f(T)) ∪ π00(f(T)).

Case I: f(λ0) ∈ π0(f(T)). Since Weyl’s theorem holds for f(T), f(T) −
f(λ0) has a finite hyperascent, i.e., there exists a positive integer p ∈ N such
that (f(T) − f(λ0))−w(0) = (f(T) − f(λ0))−p(0).

Since λ ∈ iso σ(T), X splits into the direct sum of the transfinite kernel
(T − λI)−w(0) and the transfinite range (T − λI)wX, both hyperinvariant



moving weyl’s theorem from f(T) to T 215

under T . If we write S0 and S1 for the restriction of S ∈ comm(T) to the
kernel and the range respectively, then

σ(S0) = {λ0} ̸⊆ σ(S1).

Let r ∈ N, the polynomial h and g ∈ Holoc σ(T) (with no zero in σ(T)) be
such that

f(z) − f(λ0) ≡ (z − λ0)rh(z)g(z)

with h(λ0) ̸= 0 and h(λ0) ̸∈ g(σ(T)). It follows

f(T) − f(λ0) = (T − λ0)rh(T)g(T)

with 0 /∈ σ(g(T)).
Then

(8) (T − λ0)−1(0) ⊆ (T − λ0)−r(0) ⊆ (f(T) − f(λ0))−1(0)

and, by Remark 3.1 (i),

(f(T) − f(λ0))−1(0) = ((T − λ0)rh(T))−1(0)

= (T − λ0)−r(0)
⊕

h(T)−1(0).
(9)

Since f(T0) − f(λ0) has hyperascent ≤ p, by Remark 3.1 (ii), we have

(T − λ0)−w(0) ⊆ (f(T) − f(λ0))−w(0)
= (f(T) − f(λ0))−p(0) = (T − λ0)−pr(0) ⊕ h(T)−p(0).

Again, by (T − λ0)−w(0) ∩ h(T)−p(0) = {0} and Remark 3.1, we have

(T − λ0)−w(0) ⊆ (T − λ0)−pr(0).

Since the opposite inclusion is always valid, we have that T − λ0 has finite
hyperascent.

Case II: If f(λ0) ∈ f(σw(f(T)), since T has a property Πf, follows that
T − λ0 has finite hyperascent.

Remark 3.3. A slight modification of the proofs of Theorem 2.2 and The-
orem 3.2 give us the conditions for moving Weyl’s theorem form f(T), f ∈
Holo σ(T), to T. For this, beside the property Πf, we need to suppose that f
is an injective function.



216 m.f. rodŕıguez, b.p. duggal, s.v. djordjević

If we replace the condition Πf with stronger condition

(10) λ ∈ iso σ(T) =⇒ f(T) − f(λ) has finite hyperascent,

then slight modification of part of the proof of Theorem 3.2 give us that
T is polaroid. In this case we can extend Weyl’s theorem on g(T), for all
g ∈ Holoc σ(T).

Theorem 3.4. Let T ∈ B(X) and f ∈ Holoc σ(T) such that Weyl’s theo-
rem holds for f(T) and T has property (10). Then Weyl’s theorem holds for
g(T) and g(T ∗) for all g ∈ Holoc σ(T).

Proof. In view of the hypothesis Theorem 2.2 implies that T (so also,
T ∗) satisfies Browder’s theorem. Recall now from Theorem 3.2 that if f(T)
satisfies condition (10), then T is polaroid (which, in turn, implies that T ∗ is
polaroid); hence T and T ∗ satisfy Weyl’s theorem. Browder’s theorem for T
implies that T has SVEP at points in σ(T) \ σw(T) and by [3, Theorem 2.4]
g(T) and g(T ∗) satisfy Weyl’s theorem for every g ∈ Holoc σ(T).

4. Applications

A Banach space operator T ∈ B(X) is hereditarily polaroid, T ∈ HP,
if every part of T (i.e., its restriction to an invariant subspace) is polaroid.
The class of HP operators is large. It contains amongst others the following
classes of operators. (We refer the interested reader to [2] for further, but by
no means exhaustive, list of HP operators.)

(a) H(p) operators (operators T ∈ B(X) such that H0(T − λ) = (T −
λ)−p(0) for some integer p = p(λ) ≥ 0 and all complex λ). This class of
operators contains next well known classes:

(a-i) Hilbert space operators T ∈ B(H) which are either hyponormal (|T ∗|2 ≤
|T |2), or p-hyponormal (|T ∗|2p ≤ |T|2p) for some 0 < p ≤ 1 or (p, k)-
quasihyponormal (T ∗k(|T|2p − |T ∗|2p)T k ≥ 0) for some integer k ≥ 1
and 0 < p ≤ 1.

(a-ii) w-hyponormal (|T̃ ∗| ≤ |T | ≤ ˜|T|, where, for the polar decomposition
T = U|T| of T, T̃ = |T|

1
2 U|T |

1
2 ).

(a-iii) M-hyponormal (||(T − λ)∗||2 ≤ M||T − λ||2 for some M ≥ 1 and all
complex λ) or class A (|T|2 ≤ |T 2|).



moving weyl’s theorem from f(T) to T 217

(b) Paranormal operators T ∈ B(X) (||Tx||2 ≤ ||T 2x|| for all unit vectors
x ∈ X).

(c) Totally paranormal operators T ∈ B(X) (||(T − λ)x||2 ≤ ||(T − λ)2x||
for all unit vectors x ∈ X and complex λ).

The classes consisting of paranormal operators and H(p) operators are
substantial. Thus, the classes consisting of hyponormal or p-hyponormal or
(p, 1)-quasihyponormal or (1, 1)-quasihyponormal and class A Hilbert space
operators are proper subclasses of the class of paranormal operators; the class
H(p) contains in particular the classes consisting of operators which are ei-
ther totally paranormal or generalized scalar or subscalar or multipliers of
commutative semi-simple Banach algebras [1, p. 175].

Moving Weyl’s theorem from f(T) to T, for f ∈ Holoc σ(T) and T ∈ H(p),
is possible applying Theorem 3.4. This fact is known and we can find more
details in [7]. We have:

Theorem 4.1. If f(T) ∈ H(p) for some T ∈ B(X) and f ∈ Holoc σ(T),
then T satisfies Weyl’s theorem. Moreover, g(T) and g(T ∗) satisfy Weyl’s
theorem for every g ∈ Holoc σ(T).

More is true. Recall, [1, p. 177], that a Banach space operator T satisfies
a-Weyl’s theorem if σa(T)\σaw(T) = πa00(T), where σa(A) is the approximate
point spectrum of T, πa00(T) = {λ ∈ iso σa(T) : 0 < dim(T − λI)

−1(0) < ∞}
and σaw(T) = {λ ∈ σa(T) : T −λ is not lower semi–Fredholm or ind(A−λ) ̸≤
0} is the Weyl essential approximate point spectrum of A. If T has SVEP, then
σ(T) = σa(T

∗), σw(T) = σaw(T
∗) and π00(T) = π

a
00(T

∗). Since T satisfies
Weyl’s theorem if and only if σ(T) \ σw(T) = π00(T) [1, p. 166], we have the
following:

Corollary 4.2. If f(T) ∈ H(p) for some T ∈ B(X) and f ∈ Holoc σ(T),
then g(T ∗) satisfies a-Weyl’s theorem for every g ∈ Holoc σ(T).

Proof. Since f(T) has SVEP implies T has SVEP implies g(T) has SVEP
[1], σa(g(T

∗)) \ σaw(g(T ∗)) = σ(g(T)) \ σw(g(T)). This, since g(T) sat-
isfies Weyl’s theorem (see Theorem 4.1), implies σa(g(T

∗) \ σaw(g(T ∗)) =
π0(g(T)) = π

a
0(g(T

∗)).

HP operators have SVEP [2, Theorem 2.8], so that if f(T) ∈ HP, for
some f ∈ Holoc σ(T), then g(T) and g(T ∗) satisfy Browder’s theorem for
every g ∈ Holo σ(T). However, since isolated points of σ(T) may not survive



218 m.f. rodŕıguez, b.p. duggal, s.v. djordjević

passage from σ(T) to σ(f(T)), f ∈ Holoc σ(T), HP operators do not in general
satisfy condition (5). (There is no such problem with H(p) operators.) Now,
using that λ ∈ iso σ(T) if and only if f(λ) ∈ iso σ(f(T)), the condition (10)
and, that, f(T) has SVEP implies T has SVEP, way we have new version of
[2, Theorem 3.6].

Theorem 4.3. Suppose that f(T) ∈ HP for some T ∈ B(X) and f ∈
Holoc σ(T). If f preserves isolated points of σ(T), then T satisfies Weyl’s
theorem. Moreover, g(T) satisfies Weyl’s theorem and g(T ∗) satisfies a-Weyl’s
theorem for every g ∈ Holoc σ(T).

References

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[3] B.P. Duggal, Polaroid operators satisfying Weyl’s theorem, Linear Algebra
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[4] B.P. Duggal, SVEP, Browder and Weyl theorems, in “ Topisc in Approxi-
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[5] K.B. Laursen, M.M. Neumann, “ An Introduction to Local Spectral The-
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[6] R. Harte, On local spectral theory, in “ Recent Advances in Operator Theory
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[7] M. Oudghiri, Weyl’s and Browder’s theorems for operators satisfying the
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