Recent result for generalized inverses of a linear operatoroperator_2014/slides/4_2... · 2014. 8....

Transcript of Recent result for generalized inverses of a linear operatoroperator_2014/slides/4_2... · 2014. 8....

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



(x∗)∗ = x, (x+y)∗ = x∗+y∗, (xy)∗ = y∗x∗, 1∗ = 1, 0∗ = 0.




■ Moore 1920− 1935■ Penrose 1955
MP-inverse



(x∗)∗ = x, (x+y)∗ = x∗+y∗, (xy)∗ = y∗x∗, 1∗ = 1, 0∗ = 0.




■ Moore 1920− 1935■ Penrose 1955
MP-inverse



(x∗)∗ = x, (x+y)∗ = x∗+y∗, (xy)∗ = y∗x∗, 1∗ = 1, 0∗ = 0.




■ Moore 1920− 1935■ Penrose 1955
Generalized inverse


■ 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




aba = a.


bab = b

■ b ∈ a(1) ⇒ bab ∈ a(1,2).
Generalized inverse




aba = a.


bab = b

■ b ∈ a(1) ⇒ bab ∈ a(1,2).
Drazin type generalized inverse


■ 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


■ 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.

inverses


■ 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.

inverses


■ (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).

inverses


■ (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).

inverses


■ (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∗.

inverses


■ (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


■ 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 ) = ∞.

operators





asc (T ) = inf{n : N(T n) = N(T n+1)}

dsc (T ) = inf{n : R(T n) = R(T n+1)};


operators





asc (T ) = inf{n : N(T n) = N(T n+1)}

dsc (T ) = inf{n : R(T n) = R(T n+1)};

Generalized inverses for bounded

linear operators


■ 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 .

linear operators


■ 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 .

linear operators


■ 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.

linear operators


■ 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


■ 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


■ 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.


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 . . . , ).)

■ 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 )♯.

■ 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.


■ 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.


■ 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


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


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.

■ 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†.

■ 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


■ 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)


■ 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).


■ 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).

.

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

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Recent result for generalized inverses of a linear operatoroperator_2014/slides/4_2... · 2014. 8....

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Transcript of Recent result for generalized inverses of a linear operatoroperator_2014/slides/4_2... · 2014. 8....