Electronic Journal of Linear Algebra
Volume 32 Volume 32 (2017)
Article 3
2017
Generalized left and right Weyl spectra of upper
triangular operator matrices
Guojun Hai
[email protected]
Dragana S. Cvetkovic-Ilic
University of Nis, [email protected]
Follow this and additional works at: http://repository.uwyo.edu/ela
Part of the Algebra Commons
Recommended Citation
Hai, Guojun and Cvetkovic-Ilic, Dragana S.. (2017), "Generalized left and right Weyl spectra of upper triangular operator matrices",
Electronic Journal of Linear Algebra, Volume 32, pp. 41-50.
DOI: http://dx.doi.org/10.13001/1081-3810.3373
This Article is brought to you for free and open access by Wyoming Scholars Repository. It has been accepted for inclusion in Electronic Journal of
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Electronic Journal of Linear Algebra, ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 32, pp. 41-50, February 2017.
http://repository.uwyo.edu/ela
GENERALIZED LEFT AND RIGHT WEYL SPECTRA OF
UPPER TRIANGULAR OPERATOR MATRICES∗
GUOJUN HAI† AND DRAGANA S. CVETKOVIĆ-ILIƇ
A C
is generalized
0 B
Weyl and generalized left (right) Weyl, are completely described. Furthermore, the following intersections and unions of the generalized left Weyl
spectra
Abstract. In this paper, for given operators A ∈ B (H ) and B ∈ B (K ), the sets of all C ∈ B (K , H ) such that MC =
[
g
σlw (MC ) and
C∈B (K ,H )
\
g
σlw (MC )
C∈B (K ,H )
are also described, and necessary and sufficient conditions which two operators A ∈ B (H ) and B ∈ B (K ) have to satisfy in order for MC to be a
generalized left Weyl operator for each C ∈ B (K , H ), are presented.
Key words. Operator matrix, Generalized left(right) Weyl, Spectrum.
AMS subject classifications. 47A10, 47A53, 47A55.
1. Introduction. Let H , K be infinite dimensional complex separable Hilbert spaces, and let B (H , K ) denote
the set of all bounded linear operators from H to K . For simplicity, we also write B (H , H ) as B (H ). By F (H , K ) we
denote the set of all operators from B (H , K ) with a finite dimensional range. For a given A ∈ B (H , K ), the symbols
N (A) and R (A) denote the null space and the range of A, respectively. Let n(A) = dim N (A), β(A) = codimR (A),
and d(A) = dim R (A)⊥ .
If A ∈ B (H , K ) is such that R (A) is closed and n(A) < ∞, then A is said to be a upper semi-Fredholm operator.
If β(A) < ∞, then A is called a lower semi-Fredholm operator. A semi-Fredholm operator is one which is either
upper semi-Fredholm or lower semi-Fredholm. An operator A ∈ B (H , K ) is called Fredholm if it is both lower
semi-Fredholm and upper semi-Fredholm. The subset of B (H , K ) consisting of all Fredholm operators is denoted
by Φ(H , K ). By Φ+ (H , K ) (Φ− (H , K )) we denote the set of all upper (lower) semi-Fredholm operators from
B (H , K ).
If A ∈ B (H , K ) is such that R (A) is closed and n(A) ≤ d(A), then A is a generalized left Weyl operator. If
A ∈ B (H , K ) is such that R (A) is closed and d(A) ≤ n(A), then A is a generalized right Weyl operator. Notice that in
the cases of generalized left (right) Weyl operators, n(A) and d(A) are allowed to be infinity. An operator A ∈ B (H , K )
is a generalized Weyl operator if it is both generalized right Weyl and generalized left Weyl. The set of all generalized
Weyl operators from B (H , K ) is denoted by W g (H , K ).
Let Wgl (H , K ) (Wgr (H , K )) denote the subset of B (H , K ) consisting of all generalized left (right) Weyl operag
g
tors. For an operator C ∈ B (H ), the generalized left (right) Weyl spectrum σlw (C) (σrw (C)) is defined by
g
σlw (C)(σgrw (C)) = {λ ∈ C : C − λI is not generalized left (right) Weyl}.
∗ Received
by the editors on August 10, 2016. Accepted for publication on January 11, 2017. Handling Editor: Torsten Ehrhardt.
of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, PR China ([email protected]).
‡ University of Niš, Department of Mathematics, Faculty of Sciences and Mathematics, 18000 Niš, Serbia ([email protected]). Supported
by grant no. 174007 of the Ministry of Science, Technology and Development, Republic of Serbia.
† School
Electronic Journal of Linear Algebra, ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 32, pp. 41-50, February 2017.
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Guojun Hai and Dragana S. Cvetković-Ilić
42
The generalized Weyl spectrum is defined by
σgw (C) = {λ ∈ C : C − λI is not generalized Weyl}.
In this paper, we address the question for which operators A ∈ B (H ) and B ∈ B (K ), there exists an operator C ∈
B (K , H ) such that an upper-triangular operator matrix
MC =
A C
0 B
H
H
,
→
:
K
K
is generalized left (right) Weyl. There are many papers which consider some types of invertibility, regularity and some
other properties of an upper-triangular operator matrix MC (see [1]–[17] and references therein) as well as various
S
g
types of spectra of MC . This paper is a continuation of the work presented in [10], where the sets C∈B (K ,H ) σw (MC )
T
g
and C∈B (K ,H ) σw (MC ) are described and some necessary and sufficient conditions for the existence of C ∈ B (K , H )
such that MC is generalized Weyl are given, but the set of all such operators C is not described. As a corollary of our
main results we obtain a description of all C ∈ B (K , H ) such that MC is generalized Weyl, and we denote this set by
S
T
g
g
SGW (A, B). The sets C∈B (K ,H ) σlw (MC ) and C∈B (K ,H ) σlw (MC ) are described for given A ∈ B (H ) and B ∈ B (K )
as well as the set of all C ∈ B (K , H ) such that MC is generalized left Weyl which is denoted by SGLW (A, B). In an
S
T
g
g
analogous way, similar results can be provided for C∈B (K ,H ) σrw (MC ) and C∈B (K ,H ) σrw (MC ).
2. Results. In this section, by H , K we denote complex separable Hilbert spaces. For given operators A ∈ B (H )
and B ∈ B (K ), by MC we denote
A C
H
H
MC =
:
→
,
0 B
K
K
where C ∈ B (K , H ). Evidently, for given A ∈ B (H ) and B ∈ B (K ), arbitrary C ∈ B (K , H ) can be represented by
!
C1 C2
N (B)
R (A)
C=
:
→
.
(2.1)
C3 C4
N (B)⊥
R (A)⊥
First, we will state some auxiliary lemmas which will be used in the proof of the main result.
L EMMA 2.1. If A ∈ B (H ) and D ∈ F (H ), then R (A + D) is closed if and only if R (A) is closed.
L EMMA 2.2. Let S ∈ B (H ), T ∈ B (K , H ) and R ∈ B (H , K ) be given operators.
(i) If R (S) is non-closed and R ( S T ) is closed, then n( S T ) = ∞.
S
S
(ii) If R (S) is non-closed and R (
) is closed, then d(
) = ∞.
R
R
Proof. (i) Suppose that R (S) is non-closed, R ( S T ) is closed and n( S T ) < ∞. Then
X
H
:H →
such that
S T is a left Fredholm operator which implies that there exists an operator
Y
K
X
Y
S
T
= I + K,
for some compact operator K ∈ B (H ⊕ K ). Hence, XS = IH + K1 , for some compact operator K1 ∈ B (H ) which
implies that S is left Fredholm and so R (S) is closed, which is a contradiction.
Electronic Journal of Linear Algebra, ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 32, pp. 41-50, February 2017.
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43
Generalized Left and Right Weyl Spectra of Upper Triangular Operator Matrices
(ii) The proof follows by taking adjoints in (i).
In the following theorem, for given operators A ∈ B (H ) and B ∈ B (K ), we present necessary and sufficient
conditions for the existence of C ∈ B (K , H ) such that MC is a generalized left Weyl operator, and we completely
describe the set of all such C ∈ B (K , H ).
T HEOREM 2.3. Let A ∈ B (H ) and B ∈ B (K ). There exists C ∈ B (K , H ) such that MC is generalized left Weyl
if and only if one of the following conditions is satisfied:
(i) R (A) and R (B) are closed and n(A) + n(B) ≤ d(A) + d(B). In this case,
SGLW (A, B) = C ∈ B (K , H ) : C is given by (2.1),C3 has closed range,
o
n(A) + n(C3 ) ≤ d(C3 ) + d(B) .
(ii) R (A) is closed, R (B) is non-closed and d(A) = ∞. In this case,
n
o
SGLW (A, B) = C ∈ B (K , H ) : R (B∗ ) + R(C∗ PR (A)⊥ ) is closed .
(iii) R (A) is non-closed, R (B) is closed and n(B) = d(A) + d(B) = ∞. In this case,
n
SGLW (A, B) = C ∈ B (K , H ) : R (A) + R (CPN (B) ) is closed,
o
d(B) + codim(R (A) + R (CPN (B) )) = ∞ .
(iv) R (A) and R (B) are non-closed and n(B) = d(A) = ∞. In this case,
SGLW (A, B) = {C ∈ B (K , H ) : R (MC ) is closed}.
For simplicity, we will divide the statement of this theorem into four propositions and prove each of them separately.
P ROPOSITION 2.4. Let A ∈ B (H ) and B ∈ B (K ) be such that R (A) and R (B) are closed. There exists C ∈
B (K , H ) such that MC is generalized left Weyl if and only if n(A) + n(B) ≤ d(A) + d(B). In this case,
n
SGLW (A, B) = C ∈ B (K , H ) : C is given by (2.1),C3 has closed range,
o
n(A) + n(C3 ) ≤ d(C3 ) + d(B) .
Proof. If n(A) + n(B) ≤ d(A) + d(B), then M0 is a generalized left Weyl operator. Conversely, suppose that there
exists C ∈ B (K , H ) such that MC is a generalized left Weyl operator and that C is given by (2.1). Then MC has a
matrix representation

 



A1 C1 C2
H
R (A)
MC =  0 C3 C4  :  N (B)  −→  R (A)⊥  ,
0
0 B1
N (B)⊥
K
where A1 : H −→ R (A) is right invertible and B1 : N (B)⊥ −→ K is left invertible. Evidently, there exists invertible
U,V ∈ B (H ⊕ K ) such that

 



A1 0
0
H
R (A)
UMCV =  0 C3 0  :  N (B)  −→  R (A)⊥  .
(2.2)
⊥
0
0 B1
N (B)
K
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Guojun Hai and Dragana S. Cvetković-Ilić
44
Hence, UMCV is a generalized left Weyl which implies that
n(A1 ) + n(C3 ) ≤ d(C3 ) + d(B1 ).
(2.3)
Since,
n(A1 ) = n(A), n(B) = n(C3 ) + dim N (C3 )⊥ ,
d(B1 ) = d(B) and d(A) = d(C3 ) + dim R (C3 ),
having in mind that dim N (C3 )⊥ = dim R (C3 ) and (2.3), we get
n(A) + n(B) ≤ d(A) + d(B).
To describe the set of all C ∈ B (K , H ) such that MC is a generalized left Weyl, notice that for arbitrary C given by
(2.1), there exists invertible U,V ∈ B (H ⊕ K ) such that UMCV is given by (2.10). Hence, MC is a generalized left
Weyl if and only if UMCV is a generalized left Weyl which is equivalent with the fact that R (C3 ) is closed and that
(2.3) holds.
P ROPOSITION 2.5. Let A ∈ B (H ) and B ∈ B (K ) be such that R (A) is closed and R (B) is non-closed. There
exists C ∈ B (K , H ) such that MC is generalized left Weyl if and only if d(A) = ∞. In this case,
n
o
SGLW (A, B) = C ∈ B (K , H ) : R (B∗ ) + R (C∗ PR (A)⊥ ) is closed .
Proof. Suppose that d(A) = ∞. Then MC0 is a generalized left Weyl operator for C0 given by
0
C0 =
: K −→ R (A) ⊕ R (A)⊥ ,
J
where J : K −→ R (A)⊥ is unitary. Evidently, MC0 is represented by

MC0
A1
= 0
0

0
J  : H ⊕ K −→ R (A) ⊕ R (A)⊥ ⊕ K ,
B
where A1 : H −→ R (A) is right invertible. Since J is invertible, there exists an invertible operator U ∈ B (H ⊕ K )
such that


A1 0
UMC0 =  0 J  : H ⊕ K −→ R (A) ⊕ R (A)⊥ ⊕ K .
0 0
Now, it is clear that UMC0 is a generalized left Weyl operator, and so MC0 is a generalized left Weyl operator.
Conversely, suppose that there exists C ∈ B (K , H ) such that MC is generalized left Weyl. Then MC has a matrix
representation


A1 C1
MC =  0 C2  : H ⊕ K −→ R (A) ⊕ R (A)⊥ ⊕ K ,
(2.4)
0 B
Electronic Journal of Linear Algebra, ISSN 1081-3810
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45
Generalized Left and Right Weyl Spectra of Upper Triangular Operator Matrices
where A1 : H −→ R (A) is right invertible. Thus, there exists an invertible operator V ∈ B (H ⊕ K ) such that


A1 0
MCV =  0 C2  : H ⊕ K −→ R (A) ⊕ R (A)⊥ ⊕ K .
0 B
(2.5)
Now we will show that d(A) = ∞: Indeed, if d(A) < ∞, then R (C2∗ ) is finite dimensional. Since R (MCV ) is closed, we
C2
have that R
is closed, which implies that R (B∗ ) + R (C2∗ ) is closed. This, together with dim R (C2∗ ) < ∞,
B
implies that R (B) is closed. This is a contradiction. Hence, d(A) = ∞.
In order to describe the set SGLW (A, B), notice that for arbitrary C ∈ B (K , H ), MC has a form (2.4) and that there
exists an invertible operator V ∈ B (H ⊕ K ) such that MCV is given by (2.5). Hence, MC is generalized left Weyl if
C2
and only if C is such that R
is closed and that
B
C2
C2
n(A1 ) + n (
) ≤ d(A1 ) + d (
) .
(2.6)
B
B
C2
⊥
Notice that by Lemma 2.2, we have that for each C2 ∈ B (K , R (A) ) such that R
is closed, it follows that
B
C2
d
= ∞. Thus,
B
n
o
SGLW (A, B) = C ∈ B (K , H ) : R (B∗ ) + R (C∗ PR (A)⊥ ) is closed .
P ROPOSITION 2.6. Let A ∈ B (H ) and B ∈ B (K ) be such that R (A) is non-closed and R (B) is closed. There
exists C ∈ B (K , H ) such that MC is generalized left Weyl if and only if n(B) = d(A) + d(B) = ∞. In this case,
n
SGLW (A, B) = C ∈ B (K , H ) : R (A) + R(CPN (B) ) is closed,
o
d(B) + codim(R (A) + R (CPN (B) )) = ∞ .
Proof. Suppose that n(B) = d(A) + d(B) = ∞. Then there exists a left invertible operator C1 : N (B) −→ H such
that R (C1 ) = R (A). We will prove that MC is a generalized left Weyl operator for C given by
C = C1 0 : N (B) ⊕ N (B)⊥ −→ H .
Evidently, MC is represented by
MC =
0
B1
A C1
0 0
: H ⊕ N (B) ⊕ N (B)⊥ −→ H ⊕ K ,
where B1 : N (B)⊥ −→ K is left invertible and
R (MC ) = (R (A) + R (C1 )) ⊕ R (B1 ) = R (A) ⊕ R (B).
Thus, R (MC ) is closed and
d(MC ) = d(A) + d(B) = ∞,
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Guojun Hai and Dragana S. Cvetković-Ilić
46
i.e., MC is a generalized left Weyl operator.
Conversely, suppose that there exists C ∈ B (K , H ) such that MC is generalized left Weyl. It follows that MC has
a matrix representation
A C1 C2
MC =
: H ⊕ N (B) ⊕ N (B)⊥ −→ H ⊕ K ,
(2.7)
0 0 B1
where B1 : N (B)⊥ −→ K is left invertible and there exists an invertible operator U ∈ B (H ⊕ K ) such that
A C1 0
UMC =
: H ⊕ N (B) ⊕ N (B)⊥ −→ H ⊕ K .
0 0 B1
(2.8)
Since UMC has a closed range, by Lemma 2.1 and the fact that R (A) is non-closed, we have that n(B) = ∞. Also,
applying Lemma 2.2, we get that n( A C1 ) = ∞ which implies that d(UMC ) = d(B) + d( A C1 ) = ∞. Since
d( A C1 ) ≤ d(A), it follows that d(A) + d(B) = ∞.
In order to describe the set SGLW (A, B), notice that for arbitrary C ∈ B (K , H ), MC has a form (2.7) and that there
exists an invertible operator V ∈ B (H ⊕ K ) such that UMC is given by (2.8). Hence, MC is generalized left Weyl if
and only if C is such that R
is closed and that
A C1
n
+ n(B1 ) ≤ d
+ d(B1 ).
(2.9)
A C1
A C1
Notice that if R
is closed, then by Lemma 2.2, we have that n
= ∞. Hence, MC is a
A C1
A C1
generalized left Weyl operator for C ∈ B (K , H ) if and only if R
is closed and d
+d(B1 ) =
A C1
A C1
∞. Obviously, d(B1 ) = d(B).
P ROPOSITION 2.7. Let A ∈ B (H ) and B ∈ B (K ) be such that R (A) and R (B) are non-closed. There exists
C ∈ B (K , H ) such that MC is generalized left Weyl if and only if n(B) = d(A) = ∞. In this case,
SGLW (A, B) = C ∈ B (K , H ) : R(MC ) is closed .
Proof. Since R (A) and R (B) are non-closed, by Lemma 2.2, we conclude that if C ∈ B (K , H ) is such that
R (MC ) is closed, then n(MC ) = d(MC ) = ∞. Hence, MC is generalized left Weyl if and only if R(MC ) is closed. Now,
the proof directly follows by Theorem 2.6 of [4].
R EMARK 1. It is interesting to notice that the condition d(B) + codim(R (A) + R (CPN (B) )) = ∞ from Proposition
2.6, appearing also in item (iii) of Theorem 2.3, can be replaced by the condition d(C3 ) + d(B) = ∞, where C3 is the
block-operator defined by (2.1). So, if A ∈ B (H ) and B ∈ B (K ) are such that R (A) is non-closed and R (B) is closed,
then

 



A1 C1 C2
H
R (A)
MC =  0 C3 C4  :  N (B)  −→  R (A)⊥  ,
0
0 B1
N (B)⊥
K
where A1 : H −→ R (A) is with a dense range and B1 : N (B)⊥ −→ K is left invertible. There exists an invertible
U ∈ B (H ⊕ K ) such that

 



A1 C1 0
H
R (A)
UMC =  0 C3 0  :  N (B)  −→  R (A)⊥  .
(2.10)
0
0 B1
N (B)⊥
K
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47
Generalized Left and Right Weyl Spectra of Upper Triangular Operator Matrices
Now, it is evident that R (MC ) is closed if and only if
A1
0
C1
C3
is closed which is equivalent with the fact that
R (A) + R(CPN (B) ) is closed. Also,
d
A1
0
C1
C3
=n
A∗1
C1∗
0
C3∗
= n(C3∗ ) = d(C3 ),
since A∗1 is injective (R (A1 ) = R (A)). Hence, in this case, the set SGLW can also be described by
n
SGLW (A, B) = C ∈ B (K , H ) : C is given by (2.1), R (A) + R(CPN (B) )
is closed, d(C3 ) + d(B) = ∞ .
As a corollary of the previous theorem, we get the description of the set
g
C∈B (K ,H ) σlw (MC ):
T
C OROLLARY 2.8. Let A ∈ B (H ) and B ∈ B (K ) be given operators. Then
\
g
σlw (MC ) = λ ∈ C : R (A − λI) is not closed, n(B − λI) < ∞
C∈B (K ,H )
∪ λ ∈ C : R (B − λI) is not closed, d(A − λI) < ∞
∪ λ ∈ C : R (A − λI) is not closed, R (B − λI) is closed,
d(A − λI) + d(B − λI) < ∞
∪ λ ∈ C : R (A − λI), R (B − λI) are closed,
n(A − λI) + n(B − λI) > d(A − λI) + d(B − λI) .
Using Theorem 2.3, Remark 1 and the fact that A is generalized left Weyl if and only if A∗ is generalized right Weyl,
we can give the description of the set SGW (A, B) which consists of all C ∈ B (K , H ) such that MC is generalized Weyl.
Notice that necessary and sufficient conditions for the existence of C ∈ B (K , H ) such that MC is generalized Weyl are
given in [10].
T HEOREM 2.9. Let A ∈ B (H ) and B ∈ B (K ) be given operators. There exists C ∈ B (K , H ) such that MC is
generalized Weyl if and only if one of the following conditions is satisfied:
(i) R (A) and R (B) are closed and n(A) + n(B) = d(A) + d(B). In this case,
n
SGW (A, B) = C ∈ B (K , H ) : C is given by (2.1),C3 has closed range,
o
n(A) + n(C3 ) = d(C3 ) + d(B) .
(ii) R (A) is closed, R (B) is non-closed and d(A) = n(A) + n(B) = ∞. In this case,
SGW (A, B) = {C ∈ B (K , H ) : C is given by (2.1), n(A) + n(C3 ) = ∞,
R (B∗ ) + R(C∗ PR (A)⊥ ) is closed}.
(iii) R (A) is non-closed, R (B) is closed and n(B) = d(A) + d(B) = ∞. In this case,
SGW (A, B) = C ∈ B (K , H ) : C is given by (2.1), d(B) + d(C3 ) = ∞,
R (A) + R (CPN (B) ) is closed .
Electronic Journal of Linear Algebra, ISSN 1081-3810
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Guojun Hai and Dragana S. Cvetković-Ilić
48
(iv) R (A) and R (B) are non-closed and n(B) = d(A) = ∞. In this case,
SGW (A, B) = {C ∈ B (K , H ) : R (MC ) is closed}.
Proof. Since necessary and sufficient conditions for the existence of C ∈ B (K , H ) such that MC is generalized
Weyl are given in [10], we need only prove that the set SGW (A, B) is given as claimed in each of the four possible cases
appearing above.
(i) Suppose that R (A) and R (B) are closed and n(A) + n(B) = d(A) + d(B). Using Theorem 2.3, we have that
C ∈ B (K , H ) is such that MC is generalized left Weyl if and only if C is given by (2.1), where C3 has closed range
and
n(A) + n(C3 ) ≤ d(C3 ) + d(B).
Since we are looking for C ∈ B (K , H ) such that MC is generalized Weyl, we are asking for which C ∈ B (K , H )
satisfying the previously mentioned condition, MC is generalized right Weyl i.e (MC )∗ is generalized left Weyl. Since,
∗
(MC ) =
B∗
0
C∗
A∗
K
K
:
→
H
H
and for C given by (2.1), C∗ is given by
∗
C =
C4∗
C3∗
C2∗
C1∗
N (A∗ )
:
→
N (A∗ )⊥
R (B∗ )
R (B∗ )⊥
!
,
(2.11)
applying Theorem 2.3 we get that (MC )∗ is a generalized left Weyl operator if and only if R (C3∗ ) is closed and
n(B∗ ) + n(C3∗ ) ≤ d(C3∗ ) + d(A∗ )
which is equivalent with R (C3 ) being closed and the inequality d(C3 ) + d(B) ≤ n(A) + n(C3 ). Hence, MC is a generalized Weyl operator if and only if C is given by (2.1), where C3 has closed range and n(A) + n(C3 ) = d(C3 ) + d(B).
(ii) Suppose that R (A) is closed, R (B) is non-closed and d(A) = n(A) + n(B) = ∞. Using Theorem 2.3, we have
that C ∈ B (K , H ) is such that MC is generalized left Weyl if and only if R (B∗ ) + R (C∗ PR (A)⊥ ) is closed. By item
(iii) of Theorem 2.3, using the representations of (MC )∗ given above, we get that (MC )∗ is a generalized left Weyl
operator if and only if R (B∗ ) + R (C∗ PR (A)⊥ ) is closed and d(A∗ ) + codim(R (B∗ ) + R (C∗ PN (A∗ ) )) = ∞. By Remark
1, we have that the last condition is equivalent with d(C3∗ ) + d(A∗ ) = ∞, i.e., n(A) + n(C3 ) = ∞, where C3 is the block
operator in the representation (2.1) of C.
Hence, MC is a generalized Weyl operator if and only if C is given by (2.1), where R (B∗ ) + R (C∗ PR (A)⊥ ) is
closed and n(A) + n(C3 ) = ∞.
Items (iii) and (iv) can be proved in a similar manner.
In the next theorem, we present necessary and sufficient conditions which two operators A ∈ B (H ) and B ∈ B (K )
have to satisfy in order for MC to be a generalized left Weyl operator for each C ∈ B (K , H ).
T HEOREM 2.10. Let A ∈ B (H ) and B ∈ B (K ). Then MC is a generalized left Weyl operator for each C ∈
B (K , H ) if and only if R (A) and R (B) are closed and one of the following conditions is satisfied:
(1) d(A) < ∞, n(B) = ∞, d(B) = ∞,
Electronic Journal of Linear Algebra, ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 32, pp. 41-50, February 2017.
http://repository.uwyo.edu/ela
49
Generalized Left and Right Weyl Spectra of Upper Triangular Operator Matrices
(2) d(A) = ∞, n(B) < ∞,
(3) d(A), n(B) < ∞, n(A) + n(B) ≤ d(A) + d(B).
Proof. Suppose that MC is a generalized left Weyl operator for each C ∈ B (K , H ). If at least one of R (A) and
R (B) is not closed, we have that M0 is not a generalized left Weyl operator since its range is not closed. So, it follows
that R (A) and R (B) are closed subspaces.
Notice that for arbitrary C ∈ B (K , H ), MC is given by

 
0 A1 C1 C2
N (A)
 0 0 C3 C4   N (A)⊥
:
MC = 
 0 0
0 B1   N (B)
0 0
0
0
N (B)⊥
R (A)
  R (A)⊥ 
→

  R (B)  ,
R (B)⊥



where A1 , B1 are invertible operators and that there exist invertible U,V ∈ B (H ⊕ K ) such that

 

 
R (A)
N (A)
0 A1 0
0
 0 0 C3 0   N (A)⊥   R (A)⊥ 
.
→
:
UMCV = 
 0 0
0 B1   N (B)   R (B) 
R (B)⊥
N (B)⊥
0 0
0
0
(2.12)
(2.13)
So, for any C3 ∈ B (N (B), R (A)⊥ ), we have that R (C3 ) is closed and
n(A) + n(C3 ) ≤ d(B) + d(C3 ).
Hence, at least one of d(A) and n(B) is finite. So, we will consider all possible cases (there are 3 in total) when at least
one of d(A) and n(B) is finite.
Suppose first that d(A) < ∞ and n(B) = ∞. Since for any C3 ∈ B (N (B), R (A)⊥ ), it follows that n(C3 ) = ∞,
and since there exists C3 ∈ B (N (B), R (A)⊥ ) such that d(C3 ) = 0, we conclude that n(MC ) ≤ d(MC ), for each C ∈
B (K , H ) if and only if d(B) = ∞.
If d(A) = ∞ and n(B) < ∞ then for any C3 ∈ B (N (B), R (A)⊥ ), we have that d(C3 ) = ∞, so n(MC ) ≤ d(MC ) is
satisfied for any C ∈ B (K , H ).
If d(A), n(B) < ∞ then for any C3 ∈ B (N (B), R (A)⊥ ), we have that n(B) − n(C3 ) = d(A) − d(C3 ), so n(MC ) ≤
d(MC ) if and only if n(A) + n(B) ≤ d(A) + d(B).
The converse implication can be proved in the same manner.
As a corollary of the previous theorem, we also get the description of the set
g
C∈B (K ,H ) σlw (MC ):
S
C OROLLARY 2.11. For given operators A ∈ B (H ) and B ∈ B (K ) we have
[
g
σlw (MC ) = λ ∈ C : R (A − λI) is not closed
C∈B (K ,H )
∪ λ ∈ C : R (B − λI) is not closed
∪ λ ∈ C : d(A − λI) = n(B − λI) = ∞
∪ λ ∈ C : d(A − λI), n(B − λI) < ∞,
n(A − λI) + n(B − λI) > d(A − λI) + d(B − λI)
∪ λ ∈ C : d(B − λI) < n(B − λI) = ∞ .
Electronic Journal of Linear Algebra, ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 32, pp. 41-50, February 2017.
http://repository.uwyo.edu/ela
Guojun Hai and Dragana S. Cvetković-Ilić
50
R EMARK 2. Throughout the paper, we have used the following fact: For given operators A ∈ B (H ) and B ∈ B (K )
in the each of following three cases:
(i) R (A) and R (B) are closed,
(ii) R (A) is closed, R (B) is non-closed,
(iii) R (A) is non-closed, R (B) is closed,
we have that R (MC ) is closed if and only if the respective condition below is satisfied:
(1) R (C3 ) is closed,
(2) R (B∗ ) + R (C∗ PR (A)⊥ ) is closed,
(3) R (A) + R(CPN (B) ) is closed.
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