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Monatsh Math
DOI 10.1007/s00605-017-1122-2
Pseudo B-Fredholm linear relations and spectral theory
T. Álvarez1
Received: 25 October 2016 / Accepted: 16 October 2017
© Springer-Verlag GmbH Austria 2017
Abstract The notions of pseudo B-Fredholm and pseudo B-Weyl linear relations in
Banach spaces are introduced and studied. We prove that if T is an everywhere defined
closed linear relation with a nonempty resolvent set, then T is Fredholm (resp. Weyl) if
and only if T is pseudo B-Fredholm (resp. pseudo B-Weyl) and 0 is non-isolated in the
Fredholm (resp. Weyl) spectrum of T . We also show that T is pseudo B-Fredholm (resp.
pseudo B-Weyl) if and only if T admits a generalized Kato decomposition and 0 is not
an accumulation point of the Fredholm (resp. Weyl) spectrum of T . As an application,
we calculate the pseudo B-Fredholm, the pseudo B-Weyl and the generalized Kato
spectra of the inverse of the backward unilateral shift operator regarded as a linear
relation.
Keywords Pseudo B-Fredholm linear relation · Pseudo W-Weyl linear relation ·
Generalized Kato decomposition
Mathematics Subject Classification Primary 47A06; Secondary 47A05
1 Introduction
Let T be a bounded operator in a Banach space X . Then T is said to be nilpotent if
T n = 0 for some n ∈ N and T is called quasinilpotent if its spectrum consists only the
number zero. We say that T admits a Kato (resp. a generalized Kato) decomposition if
Communicated by A. Constantin.
B
1
T. Álvarez
seco@uniovi.es
Department of Mathematics, University of Oviedo, 33007 Asturias, Spain
123
T. Álvarez
there exists a pair of T -invariant closed subspaces (M, N ) such that X = M ⊕ N , T =
TM ⊕ TN with TM regular and TN nilpotent (resp. quasinilpotent). Kato [17] showed
that every semiFredholm operator has a Kato decomposition (M, N ) with dim N < ∞.
In [19], Labrousse defined and studied the class of quasi-Fredholm operators in Hilbert
spaces. The main result of [19, Theorem 3.2.1] is that any quasi-Fredholm operator in
a Hilbert space has a Kato decomposition and Müller [25, Theorem 5] proves the same
property for Banach space operators under an additional assumption that the subspaces
that appear in the definition of quasi-Fredholm operators are complemented. Note that
the class of quasi-Fredholm operators contains the class of semiFredholm operators
as a proper subclass.
The generalized Kato decomposition has been studied in several papers by Mbekhta
[21–23], Müller [24,25] and Rakoĉević [27] among others.
B-Fredholm and B-Weyl operators were defined and studied by Berkani et al.
in a series of papers [5–10] and by Cvetković et al. [13], among others. Berkani
[5, Propositions 2.6 and 2.7] established that every B-Fredholm operator is quasiFredholm in the sense of [25] and that T is B-Fredholm (resp. B-Weyl) if and only if
there exist two closed subspaces M, N such that X = M ⊕ N , T = TM ⊕ TN where
TM is Fredhom (resp. Weyl) and TN is nilpotent.
More recently, B-Fredholm and B-Weyl operators have been generalized to pseudo
B-Fredholm and pseudo B-Weyl operators (see [31,32]). Precisely, T is called pseudo
B-Fredholm (resp. pseudo B-Weyl) if there exists a pair of T -invariant closed subspaces (M, N ) such that X = M ⊕ N , T = TM ⊕ TN with TM Fredholm (resp. Weyl)
and TN quasinilpotent.
In 2016 Tajmouati and Karmouni [31, Theorems 2.1 and 2.2] proved that if T is
pseudo B-Fredholm (resp. pseudo B-Weyl) then there is η > 0 such that T − λ is
Fredholm (resp. Weyl) whenever 0 <| λ |< η. As an immediate consequence we get
Theorem 1.1 Let T be a bounded operator in a Banach space. Then T is Fredholm
(resp. Weyl) if and only if T is pseudo B-Fredholm (resp. pseudo B-Weyl) and 0 is
non-isolated in the Fredholm (resp. Weyl) spectrum of T .
On the other hand, in [32, Theorem 2.1], the authors show that every pseudo BFredholm operator admits a generalized Kato decomposition. A combination of this
property and [31, Theorems 2.1 and 2.2] gives
If T is pseudo B-Fredholm (resp. pseudo B-Weyl) then T has a generalized Kato
decomposition and 0 is not an accumulation point of the Fredholm (resp. Weyl) spectrum of T. We remark that the reverse implication follows immediately from the very
known fact that the boundary of the Fredholm spectrum of a bounded operator is contained in the regular spectrum (see, for instance [1, Theorem 1.65]). Hence, we have
the following theorem
Theorem 1.2 Let T be a bounded operator in a Banach space. Then T is pseudo
B-Fredholm (resp. pseudo B-Weyl) if and only if T admits a generalized Kato decomposition and 0 is not an accumulation point of the Fredholm (resp. Weyl) spectrum of
T.
We emphasize that in the last decade several decompositions of linear relations have
been considered in the context of vector spaces in [28,29] and in the context of Hilbert
123
Pseudo B-Fredholm linear relations and spectral theory
spaces in [15,16]. The motivation of the study of linear relations (sometimes called
multivalued linear operators) is manifold: there is of course interest from a purely
mathematical point of view, see [12], but there are many fields where these objects
appear naturally.
In the last years, the concepts of quasi-Fredholm and B-Fredholm operators are
extended to the more general case of linear relations. So, in [20] quasi-Fredholm linear
relations in Hilbert spaces are defined and studied. They linear relations are completely
characterized in terms of a Kato decomposition with a regular linear relation and a
bounded nilpotent operator [20, Theorem 6.4]. In 2015 [11, Theorems 2.1 and 3.1]
, Chamkha and Mnif consider the classes of quasi-Fredholm and B-Fredholm linear
relations in Banach spaces and give the corresponding Kato decomposition results. But
we remark that in [11] the authors focus on the polynomial spectral mapping theorem
for quasi-Fredholm and B-Fredholm linear relations and the above Theorems 1.1
and 1.2 are not investigated.
The main purpose of the present paper is to show that the above Theorems 1.1
and 1.2 remain valid in the context of linear relations. The obtained results are applied to
describe the pseudo B-Fredholm, the pseudo B-Weyl and the generalized Kato spectra
of the inverse of the backward unilateral shift operator regarded as a multivalued linear
operator.
2 Preliminaries
Let X be a Banach space. A linear relation A in X is a subspace of X × X , it can
interpreted as a multivalued linear operator and it is uniquely determined by its graph
which is defined by
G(A) := {(x, y) ∈ X × X : y ∈ Ax}.
Let A be a linear relation in X . The domain, range, null space and multivalued part of
A are denoted by D(A), R(A), N (A) and A(0) respectively. We say that A is injective
if N (A) = {0} and it is called surjective if R(A) = X . The inverse A−1 of A is defined
by G(A−1 ) := {(y, x) : (x, y) ∈ G(A)}, so that D(A−1 ) = R(A), N (A−1 ) = A(0),
A−1 (0) = N (A) and R(A−1 ) = D(A). Furthermore, A is an operator if and only
if A(0) = {0}. The conjugate A of A is defined by G(A ) := G(−A−1 )⊥ , that is,
(y , x ) ∈ G(A ) if and only if y (y) − x (x) = 0 for all (x, y) ∈ G(A).
Let A, B be linear relations in X and let λ ∈ K. Then the linear relations A + B,
A ⊕ B, λA and AB are given by
G(A + B) := {(x, y + z) : (x, y) ∈ G(A), (x, z) ∈ G(B)},
G(A ⊕ B) := {(x + u, y + v) : (x, y) ∈ G(A),
(u, v) ∈ G(B)} with G(A) ∩ G(B) = {(0, 0)},
G(λA) := {(x, λy) : (x, y) ∈ G(A)}
123
T. Álvarez
while A − λ stands for A − λI , where I is the identity operator in X ,
G(AB) := {(x, z) : (x, y) ∈ G(A), (y, z) ∈ G(B) for some y ∈ X }.
The product of linear relations is clearly associative. Hence An , n ∈ Z, is defined
as usual with Ao = I and A1 = A.
Let A be a linear relation in X . We say that A is continuous if the operator Q A A is
continuous where Q A is the quotient map from X onto X/A(0). In such a case the norm
of A is defined by A := Q A A . Everywhere defined continuous linear relations
are referred to as bounded linear relations. The linear relation A is called closed if its
graph is a closed subspace of X × X , Fredholm, denoted by A ∈ φ(X ), if A is closed
with dim N (A) < ∞ and its range is a closed subspace of finite codimension and so
the index of A is defined by i(A) := dim N (A) − codim R(A) and A is said to be
Weyl, denoted by A ∈ W (X ), if A ∈ φ(X ) with i(A) = 0. Furthermore, we say that
A is regular if A is closed with closed range and N (An ) ⊂ R(A) for all nonnegative
integers n. We note that by virtue of [20, Lemma 2.7] every regular linear relation A
in X is a quasi-Fredholm relation of degree 0 in the sense of [20, Definition 5.1].
Let M, N be closed subspaces of X and let A be a closed linear relation in X . Define
A | M and A M by G(A | M ) := G(A) ∩ (M × X ) and G(A M ) := G(A) ∩ (M × M)
respectively. We say that A is completely reduced by the pair (M, N ), abbreviated as
(M, N ) ∈ Red(A), if X = M ⊕ N and A = A M ⊕ A N .
For a closed linear relation A in X we define the following spectra
the Fredholm spectrum: σφ (A) := {λ ∈ K : A − λ ∈
/ φ(X )},
the Weyl spectrum: σW (A) := {λ ∈ K : A − λ ∈
/ W (X )}.
The resolvent set of A is defined by ρ(A) := {λ ∈ K : A − λ is injective and
surjective }. The complementary of ρ(A) is called the spectrum of A which is denoted
by σ (A).
Note that an everywhere defined closed linear relation in a complex Banach space
X is bounded [12, Corollary III.5.4] and it may have a spectrum that coincides with the
whole complex plane; in contrast, if A is a bounded operator in X then σ (A) ⊂ {λ ∈
C :| λ |≤ A } (see [12, Example VI.2.6] for more details). Furthermore, it follows
from [2, Proposition 3] that if A is an everywhere defined closed linear relation in
X with ρ(A) = ∅, then σ (A), σφ (A) and σW (A) are nonempty proper subsets of C.
However, it is easy to see that, in general, these properties are not true if ρ(A) = ∅.
Indeed, let A be the linear relation whose graph is X × X . Then it is clear that ρ(A) = ∅
and since N (λ − T ) = X for all λ ∈ C we have that σφ (A) = σW (A) = σ (A) = C.
By the above remarks, in the sequel X will denote a complex Banach space and T
will always denote an everywhere defined closed linear relation having a nonempty
resolvent set except stated otherwise.
The notion of Kato decomposition for T was introduced in [4, Definition 2.3] as
follows
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Pseudo B-Fredholm linear relations and spectral theory
Definition 2.1 We say that T admits (or has) a Kato decomposition, denoted as T ∈
K D(X ), if there is (M, N ) ∈ Red(T ) such that TM is a regular linear relation and TN
is a bounded nilpotent operator.
Our next objective is to show that if T is Fredholm then it admits a Kato decomposition. To this end, we start with a purely algebraic lemma which gives information
about the linear relations TM and TN .
Lemma 2.2 Let (M, N ) ∈ Red(T ). Then
(i) For all n ∈ N, T n = TMn ⊕ TNn , N (T n ) = N (TMn ) ⊕ N (TNn ), T n (0) = TMn (0) ⊕
TNn (0) and R(T n ) = R(TMn ) ⊕ R(TNn ).
(ii) If PM and PN are the projections of X onto M and N respectively, then for all
n ∈ N, T n = TMn PM + TNn PN . Furthermore PM T , T PM , PN T and T PN are
bounded linear relations with G(PM T ) ⊂ G(T PM ) and G(PN T ) ⊂ G(T PN ).
Proof (i) A proof can be found in [20, (2.8) and (2.9)] .
(ii) The equality T n = TMn PM + TNn PN was obtained in [20, Lemma 2.1]. On the
other hand, it is clear that PM T , T PM , PN T and T PN are everywhere defined and
the continuity follows immediately from [12, Corollary II.3.13]. Let now (x, y) ∈
G(PM T ). Then y ∈ PM T x = PM (TM PM + TN PN )x = TM PM x = T PM x, so
that (x, y) ∈ G(T PM ). Hence G(PM T ) ⊂ G(T PM ) and similarly we obtain that
G(PN T ) ⊂ G(T PN ).
Lemma 2.3 Let (M, N ) ∈ Red(T ). Then T ∈ φ(X ) (resp. T ∈ W (X )) if and only if
TM ∈ φ(M) and TN ∈ φ(N ) (resp. TM ∈ W (M) and TN ∈ W (N )).
Proof From Lemma 2.2 we have that
dim N (T ) < ∞ if and only if dim N (TM ) < ∞ and dim N (TN ) < ∞. In this case
dim N (T ) = dim N (TM ) + dim N (TN ).
codim R(T ) < ∞ if and only if codim R(TM ) < ∞ and codim R(TN ) < ∞. In
this case codim R(T ) = codim R(TM ) + codim R(TN ).
Hence, it only remains to verify that R(T ) is closed if and only if R(TM ) and R(TN )
are both closed. Let PM and PN be the projections of X onto M and N respectively.
First assume that R(T ) is closed and let (yn ) ⊂ R(TM ) such that yn → y for some
y ∈ X . Since R(T ) is closed and R(TM ) ⊂ M, there exist (m n ) ⊂ M, m ∈ M and
u ∈ N such that yn ∈ TM m n , y = PM y ∈ PM T (m + u) and thus we infer from
Lemma 2.2 (i) that y ∈ PM (TM PM + TN PN )(m + u) = TM m, so R(TM ) is closed.
The closeness of R(TN ) can be proved in a similar way.
On the other hand, if both R(TM ) and R(TN ) are closed, suppose (yn ) ⊂ R(T )
such that yn → y for some y ∈ X . Then there are (m n ) ⊂ M and (u n ) ⊂ N such that
yn ∈ T (m n + u n ) and applying Lemma 2.2 we infer that PM yn ∈ PM T (m n + u n ) =
T PM (m n + u n ) = (TM PM + TN PN )PM m n = TM m n , so that PM yn ∈ TM m n .
Similarly, we get that PN yn ∈ TN u n and since PM yn → PM y and PN yn → PN y one
has that PM y ∈ R(TM ) and PN y ∈ R(TN ). Now the use of Lemma 2.2 (i) allowed
us to conclude that y = PM y + PN y ∈ R(TM ) ⊕ R(TN ) = R(T ). The proof is
completed.
123
T. Álvarez
Now, we are in the position to prove that if T is Fredholm then it has a Kato
decomposition in the sense of Definition 2.1.
Proposition 2.4 If T ∈ φ(X ) then there are two closed subspaces M and N such that
X = M ⊕ N , dim N < ∞, T = TM ⊕ TN , TM is a regular linear relation in M and
TN is a bounded nilpotent operator in N .
Proof First we note that by virtue of [14, Lemma 3.1], T n is everywhere defined and
closed for all nonnegative integer n. Since N (T ) is finite dimensional, it follows that
dim(N (T ) ∩ R(T n )) is a nonincreasing sequence and has therefore a limit. Hence,
there is some smallest q ∈ N ∪ {0} for which N (T ) ∩ R(T q ) = N (T ) ∩ R(T n+q )
for all n ∈ N. Moreover, N (T ) ∩ R(T q ) is closed and complemented in X because it
is finite dimensional. Since dim N (T q ) ≤ q.dim N (T ) [29, Lemma 5.4] and R(T ) is
closed of finite codimension one deduces that the subspace R(T ) + N (T q ) is closed
of finite codimension and hence it is complemented in X . In this situation, proceeding
exactly as in the proof of [20, Theorem 5.2] we obtain the desired conclusion.
As it is usual, for a subset K of C, ∂ K denotes the boundary of K , isoK , accK and
int K denotes the set of all isolated, accumulation and interior points of K respectively.
The last part of this section is devoted to prove that if T is regular and 0 ∈
accC\σφ (T ) (resp. 0 ∈ accC\σW (T ) ) then T is Fredholm (resp. T is Weyl). This
result will be crucial for the proof of the second main theorem of this paper.
We begin recalling the notions of minimum modulus of T and the gap between
two closed subspaces of X . Following [12, Definition II.2.1 and Proposition II.2.2]
the minimum modulus of T is defined by
γ (T ) := ∞ if N (T ) = X and
Tx :x∈
/ N (T )}.
γ (T ) := in f {
d(x, N (T ))
Let M, N be two closed subspaces of X . The gap between M and N is defined by
δ (M, N ) := max{δ(M, N ), δ(N , M)}
where δ(M, N ) := sup{d(m, N ) : m ∈ M, m = 1} if M = {0} and δ({0}, N ) :=
0.
Then δ(M, N ) = 0 if and only if M ⊂ N , δ (M, N ) = 0 if and only if M = N and
δ (M, N ) = δ (M ⊥ , N ⊥ ). (For more details about the gap between closed subspaces
see, for instance [18]).
The following lemma is a technical result from which information concerning the
regular linear relations will follow.
Lemma 2.5 let T be a regular linear relation in X . Then
(i) T is regular.
(ii) γ (T − λ) ≥ γ (T ) − 3 | λ | for all λ ∈ C.
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Pseudo B-Fredholm linear relations and spectral theory
(iii)
(iv)
(v)
There exists η > 0 such that T − λ is regular if 0 <| λ |< η.
δ (N (T ), N (T − λ)) → 0 as λ → 0.
δ (R(T ), R(T − λ)) → 0 as λ → 0.
Proof The statements (i), (ii) and (iii) are established in [3, Theorems 13 and 23].
(iv) As T is regular it is closed with closed range, so that, its minimum modulus is
a positive number (see [12, Theorem III.4.2]). Let us consider two cases for γ (T ):
Case 1 γ (T ) = ∞. In this case it follows from the definition of minumum modulus
combined with (ii) that N (T ) = D(T ) = D(T − λ) = N (T − λ) for all scalar λ, so
that, δ (N (T ), N (T − λ)) = 0 → 0 as λ → 0
Case 2 γ (T ) < ∞. By (iii) there is η > 0 such that T − λ is regular whenever
0 <| λ |< η. Then 0 < γ (T − λ) and according to [3, Lemma 2.10 (ii)] we have that
|λ|
.
min{γ (T ), γ (T − λ)}
If γ (T − λ) = ∞ or min{γ (T ), γ (T − λ)} = γ (T ) then
|λ|
δ (N (T ), N (T − λ)) ≤
→ 0 as λ → 0.
γ (T )
δ (N (T ), N (T − λ)) ≤
Assume now that 0 < γ (T − λ) and min{γ (T ), γ (T − λ)} = γ (T − λ). Since
T − λ is regular and T = T − λ + λ, one deduces from (ii) applied to T − λ that
γ (T ) ≥ γ (T − λ) − 3 | λ | which implies that | γ (T − λ) − γ (T ) |≤ 3 | λ |. If
)
moreover we suppose that | λ |< γ (T
6 we deduce that
γ (T )
3γ (T )
< −3 | λ | +γ (T ) < γ (T − λ) < γ (T ) + 3 | λ |<
2
2
)
which imply that 0 < γ (T
2 < γ (T − λ) and hence
γ (T )
δ (N (T ), N (T − λ)) < γ2|λ|
(T ) if 0 <| λ |< min{η, 6 }. So,
δ (N (T ), N (T − λ)) → 0 as λ → 0, as desired.
(v) Let η be a positive number such that T − λ is regular whenever 0 <| λ |< η.
Then T and (T − λ) = T − λ are regular by virtue of (i), in particular R(T ) and
R(T − λ) are both closed subspaces of X and since N (T − λ) = R(T − λ)⊥ ([12,
Proposition III.1.4]) we deduce that
δ (R(T ), R(T − λ)) = δ (R(T )⊥ , R(T − λ)⊥ )
=
δ (N (T ), N (T − λ)) if 0 <| λ |< η
and thus the use of (iv) applied to T makes us to conclude that
δ (R(T ), R(T − λ)) → 0 as λ → 0.
The proof is completed.
123
T. Álvarez
Proposition 2.6 Let T be a closed linear relation in X . If T is regular and 0 ∈
accC\σφ (T ) (resp. 0 ∈ accC\σW (T )) then T ∈ φ(X ) (resp. T ∈ W (X )).
Proof Assume that T is regular. By Lemma 2.5 it follows
(†) There exists η > 0 such that T − λ is regular, δ (N (T ), N (T − λ)) < 1 and
δ (R(T ), R(T − λ)) < 1 for all | λ |< η.
A combination of (†) and [26, Corollary 10.10] gives
(‡) There exists η > 0 such that dim N (T ) = dim N (T − λ), R(T − λ) is closed
and codim R(T ) = codim R(T − λ) for all | λ |< η.
Assume now that 0 ∈ accC\σφ (T ) (resp. 0 ∈ accC\σW (T )). Then there is μ ∈ K
such that 0 <| λ |< μ and T − μ ∈ φ(X ) (resp. T − μ ∈ W (X )).This fact together
with (‡) ensures that T ∈ φ(X ) (resp. T ∈ W (X )), as desired.
3 Pseudo B-Fredholm and Pseudo B-Weyl linear relations
Following [11, Definition 3.1] we say that T is B-Fredholm, denoted by T ∈ Bφ(X ),
if there exists n ∈ N such that R(T n ) is closed and TR(T n ) is a Fredholm linear relation.
Chamkha and Mnif [11, Theorem 3.1] have shown that T is B-Fredholm if and only
if there exists (M, N ) ∈ Red(T ) such that TM is a Fredholm linear relation in M and
TN is a bounded nilpotent operator in N . Such characterization result combined with
Definition 2.1 and the notions of pseudo B-Fredholm and pseudo B-Weyl operators
suggest to introduce the following concepts
Definition 3.1 The linear relation T is called pseudo B-Fredholm (resp. pseudo BWeyl), abbreviated as T ∈ p Bφ(X ) (resp. T ∈ p BW (X )), if there is (M, N ) ∈
Red(T ) such that TM is a Fredholm linear relation (resp. TM is a Weyl linear relation)
and TN is a bounded quasinilpotent operator. We say that T admits a generalized Kato
decomposition, denoted by T ∈ G K D(X ), if there exists (M, N ) ∈ Red(T ) such
that TM is a regular linear relation and TN is a bounded quasinilpotent operator.
The corresponding spectra are defined as follows
/ p Bφ(X )},
the pseudo B-Fredholm spectrum: σ p Bφ (T ) := {λ ∈ C : T − λ ∈
the pseudo B-Weyl spectrum: σ p BW (T ) := {λ ∈ C : T − λ ∈
/ p BW (X )},
the generalized Kato spectrum: σgK (T ) := {λ ∈ C : T − λ ∈
/ G K D(X )}.
The above spectra can be empty sets even for bounded operators since in [32,
Proposition 2.2] it is proved that if S is a bounded operator in X then σ (S) is a finite
set if and only if σgK (S) = ∅ if and only if σ p Bφ (S) = ∅ if and only if σ p BW (S) = ∅.
Clearly Bφ(X ) ⊂ p Bφ(X ) and the following example shows that the above inclusion can be proper.
Example 3.2 Let Q be the operator in l2 defined by
Q(x1 , x2 , x3 , . . .) := (0, x1 ,
x2 x3
, , . . .), (x1 , x2 , x3 , . . .) ∈ l2 .
2 3
Then Q is a bounded pseudo B-Fredholm operator but it is not B-Fredholm.
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Pseudo B-Fredholm linear relations and spectral theory
1 1/n
Indeed, it is easy to see that lim n→∞ Q n 1/n = lim n→∞ ( n!
)
= 0, so that, Q
is a bounded quasinilpotent operator and hence it is pseudo B-Fredholm. Obviously,
Q is the limit of finite rank operators Fn , n ∈ N, given by
Fn (x1 , x2 , x3 , . . .) := (0, x1 ,
xn
x2 x3
, , . . . , , 0, 0, . . .), n ∈ N
2 3
n
and hence Q is compact. Since Q n is compact and R(Q n ) is infinite dimensional we
have that R(Q n ) is not closed for all positive integer n. This last property together
with [5, Proposition 2.1 and Theorem 2.7] ensures that Q is not B-Fredholm. Recall that a linear relation A in a vector space is of finite descent if there is a
nonnegative integer p such that R(A p ) = R(A p+1 ) (see [29, (3.4)])
Proposition 3.3 Assume that T has finite descent. Then T is pseudo B-Fredholm
(resp. pseudo B-Weyl) if and only if T is B-Fredholm (resp. B-Weyl).
Proof It is evident that T ∈ p Bφ(X ) if T ∈ Bφ(X ). Suppose now that T is pseudo
B-Fredholm and let M, N be two closed subspaces of X such that X = M ⊕ N ,
TM ⊕ TN where TM is Fredholm (resp. Weyl) and TN is a bounded quasinilpotent
operator. If T has a finite descent, then TM and TN have finite descent by virtue of
[29, Lemma 8.2] and since any bounded quasinilpotent operator having finite descent
is nilpotent (see, for instance [33, Corollary 10.6]) we conclude that T is B-Fredholm
(resp. B-Weyl)).
For bounded operators this Proposition was proved in [34, Proposition 2.4].
Note that the operator Q considered in the example 3.2 is injective with R(Q) = l2 ,
so that we infer from [29, Corollary 6.7] that Q has infinite descent.
Next we investigate the interrelations between the classes φ(X ), p Bφ(X ),
p BW (X ) and G K D(X ). For this end, we first give a punctured neighbourhood theorem for pseudo B-Fredholm (resp. pseudo B-Weyl) linear relations.
Proposition 3.4 If T ∈ p Bφ(X ) (resp. T ∈ p BW (X )) then there exists η > 0 such
that T − λ ∈ φ(X ) (resp. T − λ ∈ W (X )) whenever 0 <| λ |< η.
Proof Let M and N be two closed subspaces of X such that X = M⊕N , T = TM ⊕TN ,
TM ∈ φ(M) (resp. TM ∈ W (M)) and TN is a bounded quasinilpotent operator in N .
Since σφ (TM ) and σφ (TN ) are closed subsets of C [12, Corollary V.15.7], there is η > 0
for which TM −λ ∈ φ(M) (resp. TM −λ ∈ W (M)) for all λ ∈ C with 0 <| λ |< η. On
the other hand, TN is quasinilpotent, so that σ (TN ) = {0} which implies that TN − λ
is an isomorphism for all nonzero scalar λ. Then, applying Lemma 2.3 we obtain that
T − λ = (T − λ) M ⊕ (T − λ) N = (TM − λ) ⊕ (TN − λ) is a Fredholm (resp. Weyl)
linear relation whenever 0 <| λ |< η.
Proposition 3.4 represents an improvement of Theorems 2.1 and 2.2 in [31] to linear
relations.
We now state the first main result of the present paper which generalizes the above
Theorem 1.1 to linear relations.
Theorem 3.5 We have:
123
T. Álvarez
(i) σφ (T ) = σ p Bφ (T ) ∪ isoσφ (T ).
(ii) σW (T ) = σ p BW (T ) ∪ isoσW (T ).
Proof (i) Let λ ∈ σφ (T )\σ p Bφ (T ). Then T − λ is pseudo B-Fredholm and according
to Proposition 3.4, there is η > 0 such that T −λ−μ ∈ φ(X ) if 0 <| μ |< η. Then
for every δ ∈ C with 0 <| δ − λ |< η, we have that T − δ = (T − λ) + (λ − δ)
is a Fredholm linear relation. So λ is not an accumulation point of the Fredholm
spectrum of T and since λ ∈ σφ (T ) we obtain that λ is an isolated point of the
Fredholm spectrum of T . Therefore σφ (T ) ⊂ σ p Bφ (T )∪isoσφ (T ) and the reverse
inclusion is always true.
(ii) The same arguments used in the proof of (i) prove the assertion (ii).
Proposition 3.6 If T ∈ p Bφ(X ) then T ∈ G K D(X ).
Proof Assume that there is (M, N ) ∈ Red(T ) such that TM is a Fredholm linear
relation and TN is a bounded quasinilpotent operator. Since D(T ) = X we have that
D(TM ) = M and thus the condition ρ(T ) = ∅ combined with Lemma 2.2 (i) ensures
that TM has a nonempty resolvent set. Now, from Proposition 2.4 it follows that there
exist two TM - invariant closed subspaces M1 and M2 such that M = M1 ⊕ M2 ,
dim M2 < ∞, TM1 is regular and TM2 is a bounded nilpotent operator. So that,
X = M1 ⊕ (M2 ⊕ N ), M2 ⊕ N is closed, TM2 ⊕N = TM2 ⊕ TN is a bounded
quasinilpotent operator and T = TM1 ⊕ TM2 ⊕N . Therefore T admits a generalized
Kato decomposition.
The following examples show that the inclusions p BW (X ) ⊂ p Bφ(X ) ⊂
G K D(X ) can be proper.
Example 3.7 Let Q, S and V be the operators in l2 defined by
x2 x3
Q(x1 , x2 , x3 , . . .) := 0, x1 , , , . . .
2 3
S(x1 , x2 , x3 , . . .) := (x1 , 0, x2 , 0, x3 , 0 . . .)
V (x1 , x2 , x3 , . . .) := (x2 , x3 , . . .), (x1 , x2 , x3 , . . .) ∈ l2 .
Then
(i) S is a bounded regular operator in l2 , S is not Fredholm and S ⊕ Q ∈ G K D(l2 ⊕
l2 )\ p Bφ(l2 ⊕ l2 ).
(ii) V is a bounded Fredholm operator in l2 , V is not Weyl and V ⊕ Q ∈ p Bφ(l2 ⊕
l2 )\ p BW (l2 ⊕ l2 ).
First we recall that it was proved in Example 3.2 that Q is a bounded quasinilpotent
operator but it is not nilpotent.
(i) It is obvious that S is a bounded injective operator with closed range of infinite
codimension, so that S is regular but it is not Fredholm. This combined with Proposition 2.6 leads to 0 ∈
/ accC\σφ (S). Since S is regular and Q is quasinilpotent we have
that S ⊕ Q ∈ G K D(l2 ⊕ l2 ). Assume that S ⊕ Q ∈ p Bφ(l2 ⊕ l2 ), then we infer from
Theorem 1.2 that 0 is not an accumulation point of the Fredholm spectrum of S ⊕ Q,
123
Pseudo B-Fredholm linear relations and spectral theory
so that 0 ∈
/ (accσφ (S) ∪ accσφ (Q)) by virtue of Lemma 2.2 (i). Consequently, 0 is not
an accumulation point of the Fredholm spectrum of S which contradicts the condition
0∈
/ accC\σφ (S). Therefore (i) holds.
(ii) Evidently V is a bounded surjective operator in l2 with dim N (V ) = 1, so that
V is Fredholm but it is not Weyl and V ⊕ Q ∈ p Bφ(l2 ⊕ l2 ). Now, reasoning in
the same way as in the assertion (i) we deduce that V ⊕ Q is not a pseudo B-Weyl
operator.
We are ready to give the second main result of this paper which is an extension of
Theorem 1.2 to the case of linear relations.
Theorem 3.8 The following statements are equivalent:
(i) T ∈ p Bφ(X ) (resp. T ∈ p BW (X )).
/ accσW (T )).
(ii) T ∈ G K D(X ) and 0 ∈
/ accσφ (T ) (resp. 0 ∈
/ intσW (T )).
(iii) T ∈ G K D(X ) and 0 ∈
/ intσφ (T ) (resp. 0 ∈
Proof (i) ⇒ (ii) Combine Propositions 3.4 and 3.5.
(ii) ⇒ (iii) It is evident.
(iii) ⇒ (i) Suppose that T has a generalized Kato decomposition and let M, N be
two closed subspaces of X such that (M, N ) ∈ Red(T ) where TM is an everywhere
defined regular linear relation and TN is a bounded quasinilpotent operator. Assume
that 0 is not an interior point of the Fredholm (resp. Weyl) spectrum of T , that is,
0 ∈ accC\σφ (T ) (resp. 0 ∈ accC\σW (T )) and so, according to Lemma 2.2, one has
that 0 ∈ accC\σφ (TM ) (resp. 0 ∈ accC\σW (TM )). Now, the use of Proposition 2.6
allowed us to conclude that TM ∈ φ(M) (resp. TM ∈ W (M)) and hence T ∈ p Bφ(X )
(resp. T ∈ p BW (X )), as required.
.
As an immediate consequence we get
Corollary 3.9 The following statements hold:
(i) σ p Bφ (T ) = σgK (T ) ∪ accσφ (T ) = σgK (T ) ∪ intσφ (T ).
(ii) σ p BW (T ) = σgK (T ) ∪ accσW (T ) = σgK (T ) ∪ intσW (T ).
The second part of this section is concentrated on the description of the pseudo
B-Fredholm, the pseudo B-Weyl and the generalized Kato spectra of the inverse of the
backward unilateral shif operator regarded as a multivalued linear operator. For this,
we shall use the main results obtained in this paper and the following two auxiliary
lemmas
Lemma 3.10 [4, Proposition 4.1] Let S be an everywhere defined closed linear relation in X with 0 ∈ ρ(S) and let λ ∈ C\{0}. Then
(i) S − λ = −λS(S −1 − λ−1 ) = −λ(S −1 − λ−1 )S.
(ii) N ((S − λ)n ) = N ((S −1 − λ−1 )n ) for all n ∈ N.
(iii) R((S − λ)n ) = R((S −1 − λ−1 )n ) for all n ∈ N.
Lemma 3.11 Let A, B be closed linear relations in X . The following implications
hold:
(i) If A and B are Fredholm then AB and B A are both Fredholm linear relations.
123
T. Álvarez
(ii) If D(A) = D(B) = X AB = B A and AB ∈ φ(X ), then A and B are both
Fredholm linear relations.
Proof (i) By virtue of [12, Proposition II.5.1], AB is a closed linear relation in
X and according to [30, Lemma 5.1], dim N (AB) ≤ dim N (A) + dim N (B) and
codim R(AB) ≤ codim R(A) + codim R(B). Hence it only remains to verify that
R(AB) is closed. For this, we note that since the closed linear relations A and
A | R(B)+N (A) have the same null space we obtain that γ (A) ≤ γ (A | R(B)+N (A) )
and thus by applying [12, Theorem III.4.2] it follows that A | R(B)+N (A) has closed
range.
But R(A | R(B)+N (A) ) = A(R(B)+N (A)) = R(AB)+ A A−1 (0) = R(AB)+ A(0)
([12, Corollary I.2.10])= R(AB). Hence R(AB) is closed, as desired. That B A is
Fredholm can be proved similarly.
(ii) dim N (A) ≤ dim N (B A) < ∞. Furthermore, since R(AB) ⊂ R(A) and
codim R(AB) < ∞ we have that
dim
R(A)
X
X
+ dim
= dim
< ∞,
R(A)
R(AB)
R(AB)
X
so that, dim R(A)
< ∞ and thus one deduces from [14, Corollary 3.1] that R(A) is
closed. Hence A ∈ φ(X ) and a similar reasoning shows that B ∈ φ(X ).
Proposition 3.12 Let X ∈ {co , c, l∞ , l p }, p ≥ 1 and let V be the backward unilateral
shift operator in X defined by
V (x1 , x2 , x3 , . . .) := (x2 , x3 , . . .), (x1 , x2 , x3 , . . .) ∈ X.
Then
(i) σφ (V ) = σ p Bφ (V ) = σgK (V ) = ∂D
and
σW (V ) = σ p BW (V ) = σ (V ) = D,
where D := {λ ∈ C :| λ |≤ 1}.
(ii) V −1 is an everywhere defined closed linear relation such that 0 ∈ ρ(V −1 ). Furthermore,
σφ (V −1 ) = σ p Bφ (V −1 ) = σgK (V −1 ) = ∂D
and
σW (V −1 ) = σ p BW (V −1 ) = {λ ∈ C :| λ |≥ 1}.
Proof (i) Clearly V = 1 and it is proved in [35, Theorems 4.1 and 4.2] that σ (V ) =
D and σφ (V ) = ∂D, so that accσφ (V ) = ∂D and hence intσφ (V ) = isoσφ (V ) = ∅.
123
Pseudo B-Fredholm linear relations and spectral theory
Now, we infer from Theorem 3.5 (i) and Corollary 3.9 that
σ p Bφ (V ) = σgK (V ) = σφ (V ).
Since the null space of V coincides with the subspace generated by e1 :=
(1, 0, 0, . . .) and V is surjective we have that 0 ∈ σW (V ).
Let U be the forward unilateral shift operator in X defined by
U (x1 , x2 , x3 , . . .) := (0, x1 , x2 , x3 , . . .), (x1 , x2 , x3 , . . .) ∈ X.
Then U = 1, V U = I and σ (U ) = D (see [12, Theorem 4.1]).
Let now 0 <| λ |< 1, then V − λ = λV (λ−1 − U ) which implies that R(λ −
V ) = R(V ) and N (V − λ) = N (V ) because λ−1 − U is an isomorphism. Hence
intD ⊂ σW (V ) ⊂ D and since the Weyl spectrum of a bounded operator is closed
(see, for instance [1, Corollary 3.41]) we conclude that σW (V ) = D, in particular
isoσW (V ) = ∅. In this situation we infer from Theorem 3.5 (ii) that σ p BW (V ) = D.
Therefore (i) holds.
(ii) Since D(V −1 ) = R(V ) = X , N (V −1 ) = V (0) = {0}, V −1 (0) = N (V ) = {0},
R(V −1 ) = D(V ) = X and V is bounded we have that V −1 is an everywhere defined
/ σW (V −1 ) (hence 0 ∈
/ σφ (V −1 )).
closed linear relation with 0 ∈ ρ(V −1 ), so that 0 ∈
−1
we have that for every nonzero
According to Lemma 3.10 (i) taking S := V
scalar λ, S − λ = −λS(S −1 − λ−1 ) = −λ(S −1 − λ−1 )S and thus it follows from
Lemma 3.11 that S −λ is Fredholm if and only if S −1 −λ−1 is Fredholm. Furthermore
in such case one deduces from [12, Theorem I.6.11] that i(S − λ) = i(S) + i(S −1 −
λ−1 ) − dim{(S −1 − λ−1 )(0) ∩ N (S)} and since i(S) = i(V −1 ) = 0 and (S −1 −
λ−1 )(0) = S −1 (0) = N (S) = V (0) = {0} we obtain that i(S − λ) = i(S −1 − λ−1 ).
In consequence
For λ ∈ C\{0}, V −λ is Fredholm (resp. Weyl) if and only if V −1 −λ−1 is Fredholm
(resp. Weyl).
These properties together with the equalities σφ (V ) = ∂D and σW (V ) = D established in (i) lead to
σφ (V −1 ) = ∂D and σW (V −1 ) = {λ ∈ C :| λ |≥ 1}.
Now, since intσφ (V −1 ) = isoσφ (V −1 ) = isoσW (V −1 ) = ∅ we infer from Theorem 3.5 and Corollary 3.9 that
σ p Bφ (V −1 ) = σgK (V −1 ) = ∂D and σ p BW (V −1 ) = {λ ∈ C :| λ |≥ 1}.
The proof is completed.
Acknowledgements This work was supported by MICINN (Spain) Grant MTM2013-45643.
123
T. Álvarez
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