Recent result for generalized
inverses of a linear operator
Slavǐsa Djordjević
Benemérita Universidad Autónoma de Puebla
August 2014
MP-inverse
Recent result for generalized inverses – 2 / 20
■ Let R be a ring with zero 0 and unit 1. Suppose thatthere is an involution x 7→ x∗ in R.
(x∗)∗ = x, (x+y)∗ = x∗+y∗, (xy)∗ = y∗x∗, 1∗ = 1, 0∗ = 0.
■ x ∈ R is self-adjoint if x = x∗.
■ a† is the Moore-Penrose inverse of a ∈ R if the followinghold:
aa†a = a, a†aa† = a†, (aa†)∗ = aa†, (a†a)∗ = a†a.
■ Moore 1920− 1935■ Penrose 1955
MP-inverse
Recent result for generalized inverses – 2 / 20
■ Let R be a ring with zero 0 and unit 1. Suppose thatthere is an involution x 7→ x∗ in R.
(x∗)∗ = x, (x+y)∗ = x∗+y∗, (xy)∗ = y∗x∗, 1∗ = 1, 0∗ = 0.
■ x ∈ R is self-adjoint if x = x∗.
■ a† is the Moore-Penrose inverse of a ∈ R if the followinghold:
aa†a = a, a†aa† = a†, (aa†)∗ = aa†, (a†a)∗ = a†a.
■ Moore 1920− 1935■ Penrose 1955
MP-inverse
Recent result for generalized inverses – 2 / 20
■ Let R be a ring with zero 0 and unit 1. Suppose thatthere is an involution x 7→ x∗ in R.
(x∗)∗ = x, (x+y)∗ = x∗+y∗, (xy)∗ = y∗x∗, 1∗ = 1, 0∗ = 0.
■ x ∈ R is self-adjoint if x = x∗.
■ a† is the Moore-Penrose inverse of a ∈ R if the followinghold:
aa†a = a, a†aa† = a†, (aa†)∗ = aa†, (a†a)∗ = a†a.
■ Moore 1920− 1935■ Penrose 1955
MP-inverse
Recent result for generalized inverses – 2 / 20
■ Let R be a ring with zero 0 and unit 1. Suppose thatthere is an involution x 7→ x∗ in R.
(x∗)∗ = x, (x+y)∗ = x∗+y∗, (xy)∗ = y∗x∗, 1∗ = 1, 0∗ = 0.
■ x ∈ R is self-adjoint if x = x∗.
■ a† is the Moore-Penrose inverse of a ∈ R if the followinghold:
aa†a = a, a†aa† = a†, (aa†)∗ = aa†, (a†a)∗ = a†a.
■ Moore 1920− 1935■ Penrose 1955
Generalized inverse
Recent result for generalized inverses – 3 / 20
■ Let A be a Banach algebra with unit e (with or withoutinvolution).
■ b is inner inverse of a ∈ A, in notation b ∈ a(1), if:
aba = a.
■ b is outer inverse of a ∈ A, b ∈ a(2), if:
bab = b
■ b ∈ a(1) ⇒ bab ∈ a(1,2).
Generalized inverse
Recent result for generalized inverses – 3 / 20
■ Let A be a Banach algebra with unit e (with or withoutinvolution).
■ b is inner inverse of a ∈ A, in notation b ∈ a(1), if:
aba = a.
■ b is outer inverse of a ∈ A, b ∈ a(2), if:
bab = b
■ b ∈ a(1) ⇒ bab ∈ a(1,2).
Generalized inverse
Recent result for generalized inverses – 3 / 20
■ Let A be a Banach algebra with unit e (with or withoutinvolution).
■ b is inner inverse of a ∈ A, in notation b ∈ a(1), if:
aba = a.
■ b is outer inverse of a ∈ A, b ∈ a(2), if:
bab = b
■ b ∈ a(1) ⇒ bab ∈ a(1,2).
Drazin type generalized inverse
Recent result for generalized inverses – 4 / 20
■ Let a ∈ A■ b is Drazin inverse of a ∈ A, in notation b = aD, if:
bab = b, ba = ab, an+1b = an,
for some non-negative integer n (called index of a).■ b is Koliha-Drazin inverse of a, in notation b = ad, if:
bab = b, ba = ab, a(1− ab) is quasinilpotent.
■ ind (a) = 0 ⇒ a is invertible;■ ind (a) = 1 ⇒ a is group invertible, i.e.
aba = a, bab = b, ba = ab
and we write b = a♯
Drazin type generalized inverse
Recent result for generalized inverses – 4 / 20
■ Let a ∈ A■ b is Drazin inverse of a ∈ A, in notation b = aD, if:
bab = b, ba = ab, an+1b = an,
for some non-negative integer n (called index of a).■ b is Koliha-Drazin inverse of a, in notation b = ad, if:
bab = b, ba = ab, a(1− ab) is quasinilpotent.
■ ind (a) = 0 ⇒ a is invertible;■ ind (a) = 1 ⇒ a is group invertible, i.e.
aba = a, bab = b, ba = ab
and we write b = a♯
Drazin type generalized inverse
Recent result for generalized inverses – 4 / 20
■ Let a ∈ A■ b is Drazin inverse of a ∈ A, in notation b = aD, if:
bab = b, ba = ab, an+1b = an,
for some non-negative integer n (called index of a).■ b is Koliha-Drazin inverse of a, in notation b = ad, if:
bab = b, ba = ab, a(1− ab) is quasinilpotent.
■ ind (a) = 0 ⇒ a is invertible;■ ind (a) = 1 ⇒ a is group invertible, i.e.
aba = a, bab = b, ba = ab
and we write b = a♯
Drazin type generalized inverse
Recent result for generalized inverses – 4 / 20
■ Let a ∈ A■ b is Drazin inverse of a ∈ A, in notation b = aD, if:
bab = b, ba = ab, an+1b = an,
for some non-negative integer n (called index of a).■ b is Koliha-Drazin inverse of a, in notation b = ad, if:
bab = b, ba = ab, a(1− ab) is quasinilpotent.
■ ind (a) = 0 ⇒ a is invertible;■ ind (a) = 1 ⇒ a is group invertible, i.e.
aba = a, bab = b, ba = ab
and we write b = a♯
Idempotents and generalized
inverses
Recent result for generalized inverses – 5 / 20
■ Theorem. Let A be a Banach algebra with involution.Then a ∈ A is MP-invertible if and only if there exists aself-adjoint idempotent p such that
ap = a, a∗a+ 1− p ∈ A−1.
■ Theorem. Let A be a Banach algebra. Then a ∈ A isKoliha-Drazin invertible if and only if there exists anidempotent p such that
ap = pa, a+ p ∈ A−1, ap is quasinilpotent.
Idempotents and generalized
inverses
Recent result for generalized inverses – 5 / 20
■ Theorem. Let A be a Banach algebra with involution.Then a ∈ A is MP-invertible if and only if there exists aself-adjoint idempotent p such that
ap = a, a∗a+ 1− p ∈ A−1.
■ Theorem. Let A be a Banach algebra. Then a ∈ A isKoliha-Drazin invertible if and only if there exists anidempotent p such that
ap = pa, a+ p ∈ A−1, ap is quasinilpotent.
Idempotents and generalized
inverses
Recent result for generalized inverses – 6 / 20
■ (D.Djordjević and Wei (2005)) Let a ∈ A and letp, q ∈ A be a idempotents, an element b ∈ A satisfying
bab = b, ba = p, 1− ab = q
will be called a (p, q)-inverse of a (and we write b = a(2)p,q).
■ (X.Mary (2011)) Let a ∈ A and let d ∈ A, an elementb ∈ A satisfying
bab = b, bA = dA, Ab = Ad
will be called the inverse of a along d (and we writeb = a‖d).
Idempotents and generalized
inverses
Recent result for generalized inverses – 6 / 20
■ (D.Djordjević and Wei (2005)) Let a ∈ A and letp, q ∈ A be a idempotents, an element b ∈ A satisfying
bab = b, ba = p, 1− ab = q
will be called a (p, q)-inverse of a (and we write b = a(2)p,q).
■ (X.Mary (2011)) Let a ∈ A and let d ∈ A, an elementb ∈ A satisfying
bab = b, bA = dA, Ab = Ad
will be called the inverse of a along d (and we writeb = a‖d).
Idempotents and generalized
inverses
Recent result for generalized inverses – 7 / 20
■ (Kantun Montiel (2013)) Let a ∈ A and let p, q ∈ A bea idempotents, an element b ∈ A satisfying
bab = b, baA = pA, (1− ab)A = qA
will be called the image-kernel (p, q)-inverse of a.
■ a is group invertible if and only if it is invertible along a;a is Drazin invertible if and only if it is invertible alongak;a is MP invertible if and only if it is invertible along a∗.
Idempotents and generalized
inverses
Recent result for generalized inverses – 7 / 20
■ (Kantun Montiel (2013)) Let a ∈ A and let p, q ∈ A bea idempotents, an element b ∈ A satisfying
bab = b, baA = pA, (1− ab)A = qA
will be called the image-kernel (p, q)-inverse of a.
■ a is group invertible if and only if it is invertible along a;a is Drazin invertible if and only if it is invertible alongak;a is MP invertible if and only if it is invertible along a∗.
Algebra of bounded linear
operators
Recent result for generalized inverses – 8 / 20
■ Let X be a Banach space, then B(X) denotes the spaceof all bounded linear operators from X to X.
■ For T ∈ B(X), let N(T ), R(T ) and σ(T ) denoterespectively the null space, the range and the spectrumof T .
■ The ascent, notated by asc (T ), and the descent,notated by dsc (T ), of T are given by
asc (T ) = inf{n : N(T n) = N(T n+1)}
dsc (T ) = inf{n : R(T n) = R(T n+1)};
if no such n exists, then asc (T ) = ∞, respectivelydsc (T ) = ∞.
Algebra of bounded linear
operators
Recent result for generalized inverses – 8 / 20
■ Let X be a Banach space, then B(X) denotes the spaceof all bounded linear operators from X to X.
■ For T ∈ B(X), let N(T ), R(T ) and σ(T ) denoterespectively the null space, the range and the spectrumof T .
■ The ascent, notated by asc (T ), and the descent,notated by dsc (T ), of T are given by
asc (T ) = inf{n : N(T n) = N(T n+1)}
dsc (T ) = inf{n : R(T n) = R(T n+1)};
if no such n exists, then asc (T ) = ∞, respectivelydsc (T ) = ∞.
Algebra of bounded linear
operators
Recent result for generalized inverses – 8 / 20
■ Let X be a Banach space, then B(X) denotes the spaceof all bounded linear operators from X to X.
■ For T ∈ B(X), let N(T ), R(T ) and σ(T ) denoterespectively the null space, the range and the spectrumof T .
■ The ascent, notated by asc (T ), and the descent,notated by dsc (T ), of T are given by
asc (T ) = inf{n : N(T n) = N(T n+1)}
dsc (T ) = inf{n : R(T n) = R(T n+1)};
if no such n exists, then asc (T ) = ∞, respectivelydsc (T ) = ∞.
Generalized inverses for bounded
linear operators
Recent result for generalized inverses – 9 / 20
■ Let A ∈ B(X). An operator B ∈ B(X) is an innerinverse for A if A = ABA holds, in this case we say thatA is inner regular and write B = A(1). If B = BABholds, then A is outer regular, B is an outer inverse forA and we write B = A(2).
■ Neither inner nor outer inverses are unique. However, ifwe fix the range and nullspace of the outer inverse, sayR(B) = M and N (B) = N where M and N are givensubspaces of X, then when the outer inverse exists, it isunique, and we denote it by A
(2)M,N .
Generalized inverses for bounded
linear operators
Recent result for generalized inverses – 9 / 20
■ Let A ∈ B(X). An operator B ∈ B(X) is an innerinverse for A if A = ABA holds, in this case we say thatA is inner regular and write B = A(1). If B = BABholds, then A is outer regular, B is an outer inverse forA and we write B = A(2).
■ Neither inner nor outer inverses are unique. However, ifwe fix the range and nullspace of the outer inverse, sayR(B) = M and N (B) = N where M and N are givensubspaces of X, then when the outer inverse exists, it isunique, and we denote it by A
(2)M,N .
Generalized inverses for bounded
linear operators
Recent result for generalized inverses – 10 / 20
■ If A(2)M,N exists, then both of M and N are closed and
complemented, and there exist two bounded projectionsP and Q such that R(P ) = M and R(Q) = N . In thisway, the outer inverse is linked to two idempotentoperators. There is some advantage in considering onlyone operator, instead of two. In this way, we haveinterest in an operator T ∈ B(X) such thatR(T ) = R(B) and N (T ) = N (B).
■ D.S. Djordjević and V. Rakočević, Lectures ongeneralized inverses, University of Nǐs, Faculty ofSciences and Mathematics, Nǐs, 2008.
Generalized inverses for bounded
linear operators
Recent result for generalized inverses – 10 / 20
■ If A(2)M,N exists, then both of M and N are closed and
complemented, and there exist two bounded projectionsP and Q such that R(P ) = M and R(Q) = N . In thisway, the outer inverse is linked to two idempotentoperators. There is some advantage in considering onlyone operator, instead of two. In this way, we haveinterest in an operator T ∈ B(X) such thatR(T ) = R(B) and N (T ) = N (B).
■ D.S. Djordjević and V. Rakočević, Lectures ongeneralized inverses, University of Nǐs, Faculty ofSciences and Mathematics, Nǐs, 2008.
Generalized inverse along an
operator
Recent result for generalized inverses – 11 / 20
■ Definition. (G.Kantun Montiel, S.Dj.) LetA, T ∈ B(X). We say that A is invertible along T ifthere exists B ∈ B(X) such that B = BAB and
B = LT, T = V B,
B = TU, T = BW,
for some L,U, V,W ∈ B(X). In this case, we writeB = A‖T .
■ Theorem. Let A, T ∈ B(X) be nonzero operators. Thefollowing statements are equivalent:
1. B is the inverse of A along T .
2. B is an outer inverse of A such that R(B) = R(T )and N (B) = N (T ).
Generalized inverse along an
operator
Recent result for generalized inverses – 11 / 20
■ Definition. (G.Kantun Montiel, S.Dj.) LetA, T ∈ B(X). We say that A is invertible along T ifthere exists B ∈ B(X) such that B = BAB and
B = LT, T = V B,
B = TU, T = BW,
for some L,U, V,W ∈ B(X). In this case, we writeB = A‖T .
■ Theorem. Let A, T ∈ B(X) be nonzero operators. Thefollowing statements are equivalent:
1. B is the inverse of A along T .
2. B is an outer inverse of A such that R(B) = R(T )and N (B) = N (T ).
Example 1.
Recent result for generalized inverses – 12 / 20
Consider X = ℓ2(N) the space of square-summablesequences, and let A, T ∈ B(X) be defined by
Ax := (12x2,13x3,
14x4, . . .),
Tx := (0, x2, x1, 0, 0, . . .).
for x = (x1, x2, x3, x4, . . .). Then A is invertible along Twith B = A‖T defined by
Bx = (0, 2x1, 3x2, 0, 0, . . .),
(by Definition taking
Lx = (0, 2x3, 3x2, 0, . . .), Ux = (3x22x1, 0, . . .),
V x = (0, 13x3,12x2, 0, . . .), Wx = (x1, x2, 0 . . . , ).)
Recent result for generalized inverses – 13 / 20
■ Theorem. Let A, T ∈ B(X) be nonzero operators. Thefollowing statements are equivalent.
1. A is invertible along T .
2. R(T ) and N (T ) are closed and complementedsubspaces of X, A(R(T )) = R(AT ) is closed,R(AT )⊕N (T ) = X and the reductionA|R(T ) : R(T ) → R(AT ) is invertible.
■ Theorem. Let A, T ∈ B(X). The following statementsare equivalent:
1. The operator A is invertible along T ;
2. R(T ) = R(TA) and TA is group invertible;
3. N (T ) = N (AT ) and AT is group invertible.
Moreover, we have A‖T = (TA)♯T = T (AT )♯.
Recent result for generalized inverses – 13 / 20
■ Theorem. Let A, T ∈ B(X) be nonzero operators. Thefollowing statements are equivalent.
1. A is invertible along T .
2. R(T ) and N (T ) are closed and complementedsubspaces of X, A(R(T )) = R(AT ) is closed,R(AT )⊕N (T ) = X and the reductionA|R(T ) : R(T ) → R(AT ) is invertible.
■ Theorem. Let A, T ∈ B(X). The following statementsare equivalent:
1. The operator A is invertible along T ;
2. R(T ) = R(TA) and TA is group invertible;
3. N (T ) = N (AT ) and AT is group invertible.
Moreover, we have A‖T = (TA)♯T = T (AT )♯.
Example 2.
Recent result for generalized inverses – 14 / 20
■ Consider X and A, T ∈ B(X) as in Example 1. ThenAT is group invertible with(AT )♯x = (3x2, 2x1, 0, 0, . . .). Also, A is invertible alongT and it is easy to check that B = A‖T = T (AT )♯.
■ Let X = ℓ2(N) and let A, T ∈ B(X) be defined by
Ax := (x1, 2x2, 0, 0, ...), Tx := (x1,12x2,
13x3, ...)
for x = (x1, x2, x3, x4, . . .). Then, we have
ATx = (x1, x2, 0, 0, ..).
Thus, AT is group invertible with (AT )♯ = AT .However, T is compact with infinite range and henceR(T ) is not closed, which by Theorem 1. implies A isnot invertible along T .
Example 2.
Recent result for generalized inverses – 14 / 20
■ Consider X and A, T ∈ B(X) as in Example 1. ThenAT is group invertible with(AT )♯x = (3x2, 2x1, 0, 0, . . .). Also, A is invertible alongT and it is easy to check that B = A‖T = T (AT )♯.
■ Let X = ℓ2(N) and let A, T ∈ B(X) be defined by
Ax := (x1, 2x2, 0, 0, ...), Tx := (x1,12x2,
13x3, ...)
for x = (x1, x2, x3, x4, . . .). Then, we have
ATx = (x1, x2, 0, 0, ..).
Thus, AT is group invertible with (AT )♯ = AT .However, T is compact with infinite range and henceR(T ) is not closed, which by Theorem 1. implies A isnot invertible along T .
Generalized inverse trough
decomposition of operator
Recent result for generalized inverses – 15 / 20
Any nonzero operator has a nonzero outer generalizedinverse. On the other hand, an operator A ∈ B(X) is innerregular if and only if N (A) and R(A) are closed andcomplemented subspaces of X. In this case, let M and Nbe a subspaces in X such that X = N ⊕N (A) andX = R(A)⊕M . Then A has the following matrix form:
A =
[
A1 00 0
]
:
[
N
N (A)
]
→
[
R(A)M
]
,
where A1 ∈ B(N,R(A)) is invertible.
Generalized inverse trough
decomposition of operator
Recent result for generalized inverses – 16 / 20
Furthermore, if B ∈ B(X) is an inner inverse of A such thatR(BA) = N and N (AB) = M , then B has the followingmatrix form:
B =
[
A−11 00 W
]
:
[
R(A)M
]
→
[
N
N (A)
]
,
where W is an arbitrary bounded linear operator from M toN (A).Note that this implies that an inner inverse is not uniqueeven if we fix its range and nullspace.
Recent result for generalized inverses – 17 / 20
■ Theorem. Let A be invertible along T . Then A has thefollowing matrix form:
A =
[
A1 00 A2
]
:
[
R(T )M
]
→
[
R(AT )N (T )
]
, (1)
with A1 invertible.
■ Theorem. Let A be invertible along T with B = A‖T .
1. If T is invertible, B = A−1.
2. If R(T ) = R(A), N (T ) = N (A), then B = A♯.
3. If R(T ) = R(Ak), N (T ) = N (Ak) for some k ∈ N,then B = AD.
4. If A, T ∈ B(H), R(T ) = R(A∗), N (T ) = N (A∗),then B = A†.
Recent result for generalized inverses – 17 / 20
■ Theorem. Let A be invertible along T . Then A has thefollowing matrix form:
A =
[
A1 00 A2
]
:
[
R(T )M
]
→
[
R(AT )N (T )
]
, (1)
with A1 invertible.
■ Theorem. Let A be invertible along T with B = A‖T .
1. If T is invertible, B = A−1.
2. If R(T ) = R(A), N (T ) = N (A), then B = A♯.
3. If R(T ) = R(Ak), N (T ) = N (Ak) for some k ∈ N,then B = AD.
4. If A, T ∈ B(H), R(T ) = R(A∗), N (T ) = N (A∗),then B = A†.
Commuting outer inverses
Recent result for generalized inverses – 18 / 20
■ Theorem. Let A, T ∈ B(X) be a par of commutingoperator. Then A is invertible along T if and only if
X = R(T )⊕N (T ) and
A =
[
A1 00 A2
]
:
[
R(T )N (T )
]
→
[
R(T )N (T )
]
, (2)
where A1 is invertible.
In this case
A‖T =
[
A−11 00 0
]
:
[
R(T )N (T )
]
→
[
R(T )N (T )
]
. (3)
Commuting outer inverses
Recent result for generalized inverses – 19 / 20
■ A subset Λ ⊂ σ(A) is said to be a spectral set of A if Λand σ(A) \ Λ are both closed in C. For a spectral set Λof A, PΛ(A) the spectral projection associated with A isdefined by
PΛ(A) := −1
2πi
∫
C
Rλ(A)dλ,
where C is a Cauchy contour that separates Λ fromσ(A) \ Λ.
■ Theorem. Let A ∈ B(X) and Λ be a spectral set for A.If 0 /∈ Λ then A is invertible along PΛ(A).
Commuting outer inverses
Recent result for generalized inverses – 19 / 20
■ A subset Λ ⊂ σ(A) is said to be a spectral set of A if Λand σ(A) \ Λ are both closed in C. For a spectral set Λof A, PΛ(A) the spectral projection associated with A isdefined by
PΛ(A) := −1
2πi
∫
C
Rλ(A)dλ,
where C is a Cauchy contour that separates Λ fromσ(A) \ Λ.
■ Theorem. Let A ∈ B(X) and Λ be a spectral set for A.If 0 /∈ Λ then A is invertible along PΛ(A).
Recent result for generalized inverses – 20 / 20
.
Thanks
MP-inverseGeneralized inverseDrazin type generalized inverseIdempotents and generalized inversesIdempotents and generalized inversesIdempotents and generalized inversesAlgebra of bounded linear operatorsGeneralized inverses for bounded linear operatorsGeneralized inverses for bounded linear operatorsGeneralized inverse along an operatorExample 1.Example 2.Generalized inverse trough decomposition of operatorGeneralized inverse trough decomposition of operatorCommuting outer inversesCommuting outer inverses
pdstartclock: pdclock: time: 10:18:46