A-partial isometries and generalized inverses

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A-partial isometries and generalized inverses M. Laura Arias Instituto Argentino de Matemática- CONICET Argentina Operator theory and related topics Lille 2010 M. Laura Arias A-partial isometries and generalized inverses

Transcript of A-partial isometries and generalized inverses

A-partial isometries and generalized inverses

M. Laura Arias

Instituto Argentino de Matemática- CONICETArgentina

Operator theory and related topicsLille 2010

M. Laura Arias A-partial isometries and generalized inverses

Contents

Definition and basic facts about semi-Hilbertian spaces.

Classes of operators on semi-Hilbertian spaces (A-partialisometries).

Relationship with similar classes of operators on the Hilbertspace induced by a positive operator.

M. Laura Arias A-partial isometries and generalized inverses

Contents

Definition and basic facts about semi-Hilbertian spaces.

Classes of operators on semi-Hilbertian spaces (A-partialisometries).

Relationship with similar classes of operators on the Hilbertspace induced by a positive operator.

M. Laura Arias A-partial isometries and generalized inverses

Contents

Definition and basic facts about semi-Hilbertian spaces.

Classes of operators on semi-Hilbertian spaces (A-partialisometries).

Relationship with similar classes of operators on the Hilbertspace induced by a positive operator.

M. Laura Arias A-partial isometries and generalized inverses

Semi-Hilbertian spaces

LetH be a Hilbert space with inner product 〈 , 〉 . Given A ∈ L(H)+,let 〈 , 〉A the semi-inner product defined by

〈ξ, η〉A := 〈Aξ, η〉 , for every ξ, η ∈ H.

By ‖ ‖A we denote the semi-norm induced by 〈 , 〉A, i.e.,

‖ξ‖A := 〈ξ, ξ〉1/2A = ‖A1/2ξ‖ ∀ξ ∈ H.

(H, 〈 , 〉A) is a semi-Hilbertian space.

Our main goal is to study classes of operators on (H, 〈 , 〉A).

M. Laura Arias A-partial isometries and generalized inverses

Semi-Hilbertian spaces

LetH be a Hilbert space with inner product 〈 , 〉 . Given A ∈ L(H)+,let 〈 , 〉A the semi-inner product defined by

〈ξ, η〉A := 〈Aξ, η〉 , for every ξ, η ∈ H.

By ‖ ‖A we denote the semi-norm induced by 〈 , 〉A, i.e.,

‖ξ‖A := 〈ξ, ξ〉1/2A = ‖A1/2ξ‖ ∀ξ ∈ H.

(H, 〈 , 〉A) is a semi-Hilbertian space.

Our main goal is to study classes of operators on (H, 〈 , 〉A).

M. Laura Arias A-partial isometries and generalized inverses

Semi-Hilbertian spaces

LetH be a Hilbert space with inner product 〈 , 〉 . Given A ∈ L(H)+,let 〈 , 〉A the semi-inner product defined by

〈ξ, η〉A := 〈Aξ, η〉 , for every ξ, η ∈ H.

By ‖ ‖A we denote the semi-norm induced by 〈 , 〉A, i.e.,

‖ξ‖A := 〈ξ, ξ〉1/2A = ‖A1/2ξ‖ ∀ξ ∈ H.

(H, 〈 , 〉A) is a semi-Hilbertian space.

Our main goal is to study classes of operators on (H, 〈 , 〉A).

M. Laura Arias A-partial isometries and generalized inverses

Semi-Hilbertian spaces

LetH be a Hilbert space with inner product 〈 , 〉 . Given A ∈ L(H)+,let 〈 , 〉A the semi-inner product defined by

〈ξ, η〉A := 〈Aξ, η〉 , for every ξ, η ∈ H.

By ‖ ‖A we denote the semi-norm induced by 〈 , 〉A, i.e.,

‖ξ‖A := 〈ξ, ξ〉1/2A = ‖A1/2ξ‖ ∀ξ ∈ H.

(H, 〈 , 〉A) is a semi-Hilbertian space.

Our main goal is to study classes of operators on (H, 〈 , 〉A).

M. Laura Arias A-partial isometries and generalized inverses

Some subalgebras of L(H)

Given T ∈ L(H), S ∈ L(H) is an A-adjoint of T if

〈Tξ, η〉A = 〈ξ, Sη〉A ∀ξ, η ∈ H,

or which is the same, if S is solution of the equation AX = T∗A.

Moreover, we shall say that T is A-selfadjoint if AT = T∗A. Define

LA(H) = {T ∈ L(H) : T admits A−adjoint}

M. Laura Arias A-partial isometries and generalized inverses

Some subalgebras of L(H)

Given T ∈ L(H), S ∈ L(H) is an A-adjoint of T if

〈Tξ, η〉A = 〈ξ, Sη〉A ∀ξ, η ∈ H,

or which is the same, if S is solution of the equation AX = T∗A.Moreover, we shall say that T is A-selfadjoint if AT = T∗A.

Define

LA(H) = {T ∈ L(H) : T admits A−adjoint}

M. Laura Arias A-partial isometries and generalized inverses

Some subalgebras of L(H)

Given T ∈ L(H), S ∈ L(H) is an A-adjoint of T if

〈Tξ, η〉A = 〈ξ, Sη〉A ∀ξ, η ∈ H,

or which is the same, if S is solution of the equation AX = T∗A.Moreover, we shall say that T is A-selfadjoint if AT = T∗A. Define

LA(H) = {T ∈ L(H) : T admits A−adjoint}

M. Laura Arias A-partial isometries and generalized inverses

Douglas’ theorem

Theorem (Douglas)

Let B,C ∈ L(H). The following conditions are equivalent:

1 There exists D ∈ L(H) such that BD = C.2 R(C) ⊆ R(B).3 There is a positive number λ such that CC∗ ≤ λBB∗.

Moreover, if one of these conditions holds then for every closedsubsapceM such thatM

.+ N(B) = H there exists a unique

DM ∈ L(H) such that BDM = C and R(DM) ⊆M. This solutionwill be called a reduced solution of the equation BX = C.

M. Laura Arias A-partial isometries and generalized inverses

Douglas’ theorem

Theorem (Douglas)

Let B,C ∈ L(H). The following conditions are equivalent:1 There exists D ∈ L(H) such that BD = C.

2 R(C) ⊆ R(B).3 There is a positive number λ such that CC∗ ≤ λBB∗.

Moreover, if one of these conditions holds then for every closedsubsapceM such thatM

.+ N(B) = H there exists a unique

DM ∈ L(H) such that BDM = C and R(DM) ⊆M. This solutionwill be called a reduced solution of the equation BX = C.

M. Laura Arias A-partial isometries and generalized inverses

Douglas’ theorem

Theorem (Douglas)

Let B,C ∈ L(H). The following conditions are equivalent:1 There exists D ∈ L(H) such that BD = C.2 R(C) ⊆ R(B).

3 There is a positive number λ such that CC∗ ≤ λBB∗.

Moreover, if one of these conditions holds then for every closedsubsapceM such thatM

.+ N(B) = H there exists a unique

DM ∈ L(H) such that BDM = C and R(DM) ⊆M. This solutionwill be called a reduced solution of the equation BX = C.

M. Laura Arias A-partial isometries and generalized inverses

Douglas’ theorem

Theorem (Douglas)

Let B,C ∈ L(H). The following conditions are equivalent:1 There exists D ∈ L(H) such that BD = C.2 R(C) ⊆ R(B).3 There is a positive number λ such that CC∗ ≤ λBB∗.

Moreover, if one of these conditions holds then for every closedsubsapceM such thatM

.+ N(B) = H there exists a unique

DM ∈ L(H) such that BDM = C and R(DM) ⊆M. This solutionwill be called a reduced solution of the equation BX = C.

M. Laura Arias A-partial isometries and generalized inverses

Douglas’ theorem

Theorem (Douglas)

Let B,C ∈ L(H). The following conditions are equivalent:1 There exists D ∈ L(H) such that BD = C.2 R(C) ⊆ R(B).3 There is a positive number λ such that CC∗ ≤ λBB∗.

Moreover, if one of these conditions holds then for every closedsubsapceM such thatM

.+ N(B) = H there exists a unique

DM ∈ L(H) such that BDM = C and R(DM) ⊆M. This solutionwill be called a reduced solution of the equation BX = C.

M. Laura Arias A-partial isometries and generalized inverses

Some subalgebras of L(H)

Given T ∈ L(H), S ∈ L(H) is an A-adjoint of T if

〈Tξ, η〉A = 〈ξ, Sη〉A ∀ξ, η ∈ H,

or which is the same, if S is solution of the equation AX = T∗A.

Therefore,

LA(H) = {T ∈ L(H) : R(T∗A) ⊆ R(A)}

In the sequel, given T ∈ LA(H), T] will denote a reduced solution ofAX = T∗A.

M. Laura Arias A-partial isometries and generalized inverses

Some subalgebras of L(H)

Given T ∈ L(H), S ∈ L(H) is an A-adjoint of T if

〈Tξ, η〉A = 〈ξ, Sη〉A ∀ξ, η ∈ H,

or which is the same, if S is solution of the equation AX = T∗A.Therefore,

LA(H) = {T ∈ L(H) : R(T∗A) ⊆ R(A)}

In the sequel, given T ∈ LA(H), T] will denote a reduced solution ofAX = T∗A.

M. Laura Arias A-partial isometries and generalized inverses

Some subalgebras of L(H)

Given T ∈ L(H), S ∈ L(H) is an A-adjoint of T if

〈Tξ, η〉A = 〈ξ, Sη〉A ∀ξ, η ∈ H,

or which is the same, if S is solution of the equation AX = T∗A.Therefore,

LA(H) = {T ∈ L(H) : R(T∗A) ⊆ R(A)}

In the sequel, given T ∈ LA(H), T] will denote a reduced solution ofAX = T∗A.

M. Laura Arias A-partial isometries and generalized inverses

Some properties of T]

Let T ∈ LA(H). Then the following statements hold:1 If W ∈ LA(H) then TW ∈ LA(H) and (TW)] = W]T].

2 T] ∈ LA(H), (T])] = QM//N(A)T.3 T]T and TT] are A-selfadjoint.

M. Laura Arias A-partial isometries and generalized inverses

Some properties of T]

Let T ∈ LA(H). Then the following statements hold:1 If W ∈ LA(H) then TW ∈ LA(H) and (TW)] = W]T].2 T] ∈ LA(H), (T])] = QM//N(A)T.

3 T]T and TT] are A-selfadjoint.

M. Laura Arias A-partial isometries and generalized inverses

Some properties of T]

Let T ∈ LA(H). Then the following statements hold:1 If W ∈ LA(H) then TW ∈ LA(H) and (TW)] = W]T].2 T] ∈ LA(H), (T])] = QM//N(A)T.3 T]T and TT] are A-selfadjoint.

M. Laura Arias A-partial isometries and generalized inverses

Some subalgebras of L(H)

Let

LA1/2(H) : = {T ∈ L(H) : ∃ c > 0 ‖Tξ‖A ≤ c‖ξ‖A ∀ ξ ∈ H}

= {T ∈ L(H) : R(T∗A1/2) ⊆ R(A1/2)}

The next inclusion holds: LA(H) ⊆ LA1/2(H).

M. Laura Arias A-partial isometries and generalized inverses

Some subalgebras of L(H)

Let

LA1/2(H) : = {T ∈ L(H) : ∃ c > 0 ‖Tξ‖A ≤ c‖ξ‖A ∀ ξ ∈ H}= {T ∈ L(H) : R(T∗A1/2) ⊆ R(A1/2)}

The next inclusion holds: LA(H) ⊆ LA1/2(H).

M. Laura Arias A-partial isometries and generalized inverses

Some subalgebras of L(H)

Let

LA1/2(H) : = {T ∈ L(H) : ∃ c > 0 ‖Tξ‖A ≤ c‖ξ‖A ∀ ξ ∈ H}= {T ∈ L(H) : R(T∗A1/2) ⊆ R(A1/2)}

The next inclusion holds: LA(H) ⊆ LA1/2(H).

M. Laura Arias A-partial isometries and generalized inverses

Classes of operators in LA1/2(H)A-contractions and A-isometries

DefinitionT ∈ L(H) is called an A-contraction if

‖Tξ‖A ≤ ‖ξ‖A for every ξ ∈ H,

or equivalently if T∗AT ≤ A. If the equality holds then T is called anA-isometry.

M. Laura Arias A-partial isometries and generalized inverses

A-isometries

Remarks

1 Let T ∈ LA(H). Then, T is an A-isometry if and only ifT]T = QM//N(A).

2 The A-isometries are not, in general, left invertible.

For example,for every A ∈ L(H)+ not injective, PR(A) is a not left invertibleA-isometry.

Recall that given T ∈ L(H) we say that T ′ ∈ L(H) is a generalizedinverse of T if TT ′T = T and T ′TT ′ = T ′.

RemarkThe A-isometries may not admit generalized inverses.

M. Laura Arias A-partial isometries and generalized inverses

A-isometries

Remarks1 Let T ∈ LA(H). Then, T is an A-isometry if and only if

T]T = QM//N(A).

2 The A-isometries are not, in general, left invertible.

For example,for every A ∈ L(H)+ not injective, PR(A) is a not left invertibleA-isometry.

Recall that given T ∈ L(H) we say that T ′ ∈ L(H) is a generalizedinverse of T if TT ′T = T and T ′TT ′ = T ′.

RemarkThe A-isometries may not admit generalized inverses.

M. Laura Arias A-partial isometries and generalized inverses

A-isometries

Remarks1 Let T ∈ LA(H). Then, T is an A-isometry if and only if

T]T = QM//N(A).

2 The A-isometries are not, in general, left invertible. For example,for every A ∈ L(H)+ not injective, PR(A) is a not left invertibleA-isometry.

Recall that given T ∈ L(H) we say that T ′ ∈ L(H) is a generalizedinverse of T if TT ′T = T and T ′TT ′ = T ′.

RemarkThe A-isometries may not admit generalized inverses.

M. Laura Arias A-partial isometries and generalized inverses

A-isometries

Remarks1 Let T ∈ LA(H). Then, T is an A-isometry if and only if

T]T = QM//N(A).

2 The A-isometries are not, in general, left invertible. For example,for every A ∈ L(H)+ not injective, PR(A) is a not left invertibleA-isometry.

Recall that given T ∈ L(H) we say that T ′ ∈ L(H) is a generalizedinverse of T if TT ′T = T and T ′TT ′ = T ′.

RemarkThe A-isometries may not admit generalized inverses.

M. Laura Arias A-partial isometries and generalized inverses

A-isometries

Remarks1 Let T ∈ LA(H). Then, T is an A-isometry if and only if

T]T = QM//N(A).

2 The A-isometries are not, in general, left invertible. For example,for every A ∈ L(H)+ not injective, PR(A) is a not left invertibleA-isometry.

Recall that given T ∈ L(H) we say that T ′ ∈ L(H) is a generalizedinverse of T if TT ′T = T and T ′TT ′ = T ′.

RemarkThe A-isometries may not admit generalized inverses.

M. Laura Arias A-partial isometries and generalized inverses

A-partial isometries

Recall that given T ∈ L(H) the following conditions are equivalent:1 T is a partial isometry;

2 T∗ is a partial isometry;3 T∗T is a projection;4 TT∗T = T;

5 T is a contraction with a contractive generalized inverse.

M. Laura Arias A-partial isometries and generalized inverses

A-partial isometries

Recall that given T ∈ L(H) the following conditions are equivalent:1 T is a partial isometry;2 T∗ is a partial isometry;

3 T∗T is a projection;4 TT∗T = T;

5 T is a contraction with a contractive generalized inverse.

M. Laura Arias A-partial isometries and generalized inverses

A-partial isometries

Recall that given T ∈ L(H) the following conditions are equivalent:1 T is a partial isometry;2 T∗ is a partial isometry;3 T∗T is a projection;

4 TT∗T = T;

5 T is a contraction with a contractive generalized inverse.

M. Laura Arias A-partial isometries and generalized inverses

A-partial isometries

Recall that given T ∈ L(H) the following conditions are equivalent:1 T is a partial isometry;2 T∗ is a partial isometry;3 T∗T is a projection;4 TT∗T = T;

5 T is a contraction with a contractive generalized inverse.

M. Laura Arias A-partial isometries and generalized inverses

A-partial isometries

Recall that given T ∈ L(H) the following conditions are equivalent:1 T is a partial isometry;2 T∗ is a partial isometry;3 T∗T is a projection;4 TT∗T = T;

5 T is a contraction with a contractive generalized inverse.

M. Laura Arias A-partial isometries and generalized inverses

A-partial isometries

Given T ∈ LA(H),(T]T)2 = T]T.

From this, ‖Tξ‖A = ‖ξ‖A ∀ξ ∈ R(T]T) or, which is the same,

‖Tξ‖A = ‖ξ‖A ∀ξ ∈ R(T]T) + N(A).

It holds thatR(T]T) + N(A) = N(AT)⊥A ,

where S⊥A = {ξ ∈ H : 〈ξ, η〉A = 0 ∀η ∈ S}.

M. Laura Arias A-partial isometries and generalized inverses

A-partial isometries

Given T ∈ LA(H),(T]T)2 = T]T.

From this, ‖Tξ‖A = ‖ξ‖A ∀ξ ∈ R(T]T)

or, which is the same,

‖Tξ‖A = ‖ξ‖A ∀ξ ∈ R(T]T) + N(A).

It holds thatR(T]T) + N(A) = N(AT)⊥A ,

where S⊥A = {ξ ∈ H : 〈ξ, η〉A = 0 ∀η ∈ S}.

M. Laura Arias A-partial isometries and generalized inverses

A-partial isometries

Given T ∈ LA(H),(T]T)2 = T]T.

From this, ‖Tξ‖A = ‖ξ‖A ∀ξ ∈ R(T]T) or, which is the same,

‖Tξ‖A = ‖ξ‖A ∀ξ ∈ R(T]T) + N(A).

It holds thatR(T]T) + N(A) = N(AT)⊥A ,

where S⊥A = {ξ ∈ H : 〈ξ, η〉A = 0 ∀η ∈ S}.

M. Laura Arias A-partial isometries and generalized inverses

A-partial isometries

Given T ∈ LA(H),(T]T)2 = T]T.

From this, ‖Tξ‖A = ‖ξ‖A ∀ξ ∈ R(T]T) or, which is the same,

‖Tξ‖A = ‖ξ‖A ∀ξ ∈ R(T]T) + N(A).

It holds thatR(T]T) + N(A) = N(AT)⊥A ,

where S⊥A = {ξ ∈ H : 〈ξ, η〉A = 0 ∀η ∈ S}.

M. Laura Arias A-partial isometries and generalized inverses

A-partial isometries

DefinitionT ∈ L(H) is called an A-partial isometry if

‖Tξ‖A = ‖ξ‖A ∀ξ ∈ N(AT)⊥A .

Remarks

1 If A = I then an A-partial isometry is a partial isometry.2 Every A-isometry is an A-partial isometry.3 If T ∈ LA(H) and (A,R(T]T)) is compatible then: T is an

A-partial isometry⇔ T]T is a projection.4 If T ∈ LA(H) is an A-partial isometry then T] is an A-partial

isometry.

M. Laura Arias A-partial isometries and generalized inverses

A-partial isometries

DefinitionT ∈ L(H) is called an A-partial isometry if

‖Tξ‖A = ‖ξ‖A ∀ξ ∈ N(AT)⊥A .

Remarks

1 If A = I then an A-partial isometry is a partial isometry.2 Every A-isometry is an A-partial isometry.3 If T ∈ LA(H) and (A,R(T]T)) is compatible then: T is an

A-partial isometry⇔ T]T is a projection.4 If T ∈ LA(H) is an A-partial isometry then T] is an A-partial

isometry.

M. Laura Arias A-partial isometries and generalized inverses

A-partial isometries

DefinitionT ∈ L(H) is called an A-partial isometry if

‖Tξ‖A = ‖ξ‖A ∀ξ ∈ N(AT)⊥A .

Remarks

1 If A = I then an A-partial isometry is a partial isometry.

2 Every A-isometry is an A-partial isometry.3 If T ∈ LA(H) and (A,R(T]T)) is compatible then: T is an

A-partial isometry⇔ T]T is a projection.4 If T ∈ LA(H) is an A-partial isometry then T] is an A-partial

isometry.

M. Laura Arias A-partial isometries and generalized inverses

A-partial isometries

DefinitionT ∈ L(H) is called an A-partial isometry if

‖Tξ‖A = ‖ξ‖A ∀ξ ∈ N(AT)⊥A .

Remarks

1 If A = I then an A-partial isometry is a partial isometry.2 Every A-isometry is an A-partial isometry.

3 If T ∈ LA(H) and (A,R(T]T)) is compatible then: T is anA-partial isometry⇔ T]T is a projection.

4 If T ∈ LA(H) is an A-partial isometry then T] is an A-partialisometry.

M. Laura Arias A-partial isometries and generalized inverses

A-partial isometries

DefinitionT ∈ L(H) is called an A-partial isometry if

‖Tξ‖A = ‖ξ‖A ∀ξ ∈ N(AT)⊥A .

Remarks

1 If A = I then an A-partial isometry is a partial isometry.2 Every A-isometry is an A-partial isometry.3 If T ∈ LA(H) and (A,R(T]T)) is compatible then: T is an

A-partial isometry⇔ T]T is a projection.

4 If T ∈ LA(H) is an A-partial isometry then T] is an A-partialisometry.

M. Laura Arias A-partial isometries and generalized inverses

A-partial isometries

DefinitionT ∈ L(H) is called an A-partial isometry if

‖Tξ‖A = ‖ξ‖A ∀ξ ∈ N(AT)⊥A .

Remarks

1 If A = I then an A-partial isometry is a partial isometry.2 Every A-isometry is an A-partial isometry.3 If T ∈ LA(H) and (A,R(T]T)) is compatible then: T is an

A-partial isometry⇔ T]T is a projection.4 If T ∈ LA(H) is an A-partial isometry then T] is an A-partial

isometry.

M. Laura Arias A-partial isometries and generalized inverses

A-partial isometries and generalized inverses

Proposition

Let T ∈ LA(H). The following assertions hold:

1 If TT]T = T then T]TT] = T].

2 If TT]T = T then T is an A-partial isometry. The converse isfalse, in general.

Theorem (Corach et al, 2005)

Given T ∈ L(H) with closed range, there exists T ′ ∈ L(H) such that

TT ′T = T, T ′TT ′ = T ′, ATT ′ = (TT ′)∗A, AT ′T = (T ′T)∗A

if and only if the pairs (A,R(T)) and (A,N(T)) are complatible.

M. Laura Arias A-partial isometries and generalized inverses

A-partial isometries and generalized inverses

Proposition

Let T ∈ LA(H). The following assertions hold:1 If TT]T = T then T]TT] = T].

2 If TT]T = T then T is an A-partial isometry. The converse isfalse, in general.

Theorem (Corach et al, 2005)

Given T ∈ L(H) with closed range, there exists T ′ ∈ L(H) such that

TT ′T = T, T ′TT ′ = T ′, ATT ′ = (TT ′)∗A, AT ′T = (T ′T)∗A

if and only if the pairs (A,R(T)) and (A,N(T)) are complatible.

M. Laura Arias A-partial isometries and generalized inverses

A-partial isometries and generalized inverses

Proposition

Let T ∈ LA(H). The following assertions hold:1 If TT]T = T then T]TT] = T].

2 If TT]T = T then T is an A-partial isometry. The converse isfalse, in general.

Theorem (Corach et al, 2005)

Given T ∈ L(H) with closed range, there exists T ′ ∈ L(H) such that

TT ′T = T, T ′TT ′ = T ′, ATT ′ = (TT ′)∗A, AT ′T = (T ′T)∗A

if and only if the pairs (A,R(T)) and (A,N(T)) are complatible.

M. Laura Arias A-partial isometries and generalized inverses

A-partial isometries and generalized inverses

Proposition

Let T ∈ LA(H). The following assertions hold:1 If TT]T = T then T]TT] = T].

2 If TT]T = T then T is an A-partial isometry. The converse isfalse, in general.

Theorem (Corach et al, 2005)

Given T ∈ L(H) with closed range, there exists T ′ ∈ L(H) such that

TT ′T = T, T ′TT ′ = T ′, ATT ′ = (TT ′)∗A, AT ′T = (T ′T)∗A

if and only if the pairs (A,R(T)) and (A,N(T)) are complatible.

M. Laura Arias A-partial isometries and generalized inverses

A-partial isometries and generalized inverses

Proposition

Let T ∈ LA(H) such that T admits A-generalized inverse. Thefollowing conditions are equivalent:

1 T is an A-partial isometry such thatH = R(T]) + N(T) for someT] ;

2 T is an A-contraction and there is an A-contraction S ∈ LA(H)with cos0(R(S),N(A)) < 1 such that TST = T and STS = S.

M. Laura Arias A-partial isometries and generalized inverses

A-partial isometries and generalized inverses

Proposition

Let T ∈ LA(H) such that T admits A-generalized inverse. Thefollowing conditions are equivalent:

1 T is an A-partial isometry such thatH = R(T]) + N(T) for someT] ;

2 T is an A-contraction and there is an A-contraction S ∈ LA(H)with cos0(R(S),N(A)) < 1 such that TST = T and STS = S.

M. Laura Arias A-partial isometries and generalized inverses

A-partial isometries and generalized inverses

Proposition

Let T ∈ LA(H) such that T admits A-generalized inverse. Thefollowing conditions are equivalent:

1 T is an A-partial isometry such thatH = R(T]) + N(T) for someT] ;

2 T is an A-contraction and there is an A-contraction S ∈ LA(H)with cos0(R(S),N(A)) < 1 such that TST = T and STS = S.

M. Laura Arias A-partial isometries and generalized inverses

Hilbert space induced by a positive operator

Definition

Let A ∈ L(H)+. A pair (K,Π) is called a Hilbert space induced byA if:

1 K is a Hilbert space.2 Π : H → K is a linear operator.3 R(Π) is dense in K.4 〈Πξ,Πη〉 = 〈Aξ, η〉 ∀ξ, η ∈ H.

Theorem (Cojuhari et al, 2005)

Given A ∈ L(H)+ there exists a Hilbert space induced by A and it isunique, modulo unitary equivalence.

M. Laura Arias A-partial isometries and generalized inverses

Hilbert space induced by a positive operator

Definition

Let A ∈ L(H)+. A pair (K,Π) is called a Hilbert space induced byA if:

1 K is a Hilbert space.

2 Π : H → K is a linear operator.3 R(Π) is dense in K.4 〈Πξ,Πη〉 = 〈Aξ, η〉 ∀ξ, η ∈ H.

Theorem (Cojuhari et al, 2005)

Given A ∈ L(H)+ there exists a Hilbert space induced by A and it isunique, modulo unitary equivalence.

M. Laura Arias A-partial isometries and generalized inverses

Hilbert space induced by a positive operator

Definition

Let A ∈ L(H)+. A pair (K,Π) is called a Hilbert space induced byA if:

1 K is a Hilbert space.2 Π : H → K is a linear operator.

3 R(Π) is dense in K.4 〈Πξ,Πη〉 = 〈Aξ, η〉 ∀ξ, η ∈ H.

Theorem (Cojuhari et al, 2005)

Given A ∈ L(H)+ there exists a Hilbert space induced by A and it isunique, modulo unitary equivalence.

M. Laura Arias A-partial isometries and generalized inverses

Hilbert space induced by a positive operator

Definition

Let A ∈ L(H)+. A pair (K,Π) is called a Hilbert space induced byA if:

1 K is a Hilbert space.2 Π : H → K is a linear operator.3 R(Π) is dense in K.

4 〈Πξ,Πη〉 = 〈Aξ, η〉 ∀ξ, η ∈ H.

Theorem (Cojuhari et al, 2005)

Given A ∈ L(H)+ there exists a Hilbert space induced by A and it isunique, modulo unitary equivalence.

M. Laura Arias A-partial isometries and generalized inverses

Hilbert space induced by a positive operator

Definition

Let A ∈ L(H)+. A pair (K,Π) is called a Hilbert space induced byA if:

1 K is a Hilbert space.2 Π : H → K is a linear operator.3 R(Π) is dense in K.4 〈Πξ,Πη〉 = 〈Aξ, η〉 ∀ξ, η ∈ H.

Theorem (Cojuhari et al, 2005)

Given A ∈ L(H)+ there exists a Hilbert space induced by A and it isunique, modulo unitary equivalence.

M. Laura Arias A-partial isometries and generalized inverses

Hilbert space induced by a positive operator

Definition

Let A ∈ L(H)+. A pair (K,Π) is called a Hilbert space induced byA if:

1 K is a Hilbert space.2 Π : H → K is a linear operator.3 R(Π) is dense in K.4 〈Πξ,Πη〉 = 〈Aξ, η〉 ∀ξ, η ∈ H.

Theorem (Cojuhari et al, 2005)

Given A ∈ L(H)+ there exists a Hilbert space induced by A and it isunique, modulo unitary equivalence.

M. Laura Arias A-partial isometries and generalized inverses

Hilbert space induced by a positive operator

Let R(A1/2) be equipped with the inner product defined by

(A1/2ξ,A1/2η) :=⟨

PR(A)ξ,PR(A)η⟩∀ξ, η ∈ H.

R(A1/2) = (R(A1/2), ( , )) is a Hilbert space induced by A

Can we describe L(R(A1/2)) by means of L(H)?

How does LA1/2(H) relate to L(R(A1/2))?

Do the previous classes of operators in LA1/2(H) relate to similarclasses of L(R(A1/2))?

M. Laura Arias A-partial isometries and generalized inverses

Hilbert space induced by a positive operator

Let R(A1/2) be equipped with the inner product defined by

(A1/2ξ,A1/2η) :=⟨

PR(A)ξ,PR(A)η⟩∀ξ, η ∈ H.

R(A1/2) = (R(A1/2), ( , )) is a Hilbert space induced by A

Can we describe L(R(A1/2)) by means of L(H)?

How does LA1/2(H) relate to L(R(A1/2))?

Do the previous classes of operators in LA1/2(H) relate to similarclasses of L(R(A1/2))?

M. Laura Arias A-partial isometries and generalized inverses

Hilbert space induced by a positive operator

Let R(A1/2) be equipped with the inner product defined by

(A1/2ξ,A1/2η) :=⟨

PR(A)ξ,PR(A)η⟩∀ξ, η ∈ H.

R(A1/2) = (R(A1/2), ( , )) is a Hilbert space induced by A

Can we describe L(R(A1/2)) by means of L(H)?

How does LA1/2(H) relate to L(R(A1/2))?

Do the previous classes of operators in LA1/2(H) relate to similarclasses of L(R(A1/2))?

M. Laura Arias A-partial isometries and generalized inverses

Hilbert space induced by a positive operator

Let R(A1/2) be equipped with the inner product defined by

(A1/2ξ,A1/2η) :=⟨

PR(A)ξ,PR(A)η⟩∀ξ, η ∈ H.

R(A1/2) = (R(A1/2), ( , )) is a Hilbert space induced by A

Can we describe L(R(A1/2)) by means of L(H)?

How does LA1/2(H) relate to L(R(A1/2))?

Do the previous classes of operators in LA1/2(H) relate to similarclasses of L(R(A1/2))?

M. Laura Arias A-partial isometries and generalized inverses

The algebra L(R(A1/2))

Proposition

Let T : R(A1/2)→ R(A1/2) be a linear operator. Then there exists aunique linear operator V : H → H such that R(V) ⊆ R(A) and

A1/2V = TA1/2.

Moreover, T is bounded in R(A1/2) if and only if V is bounded inH.Moreover, ‖T‖R(A1/2) = ‖V‖.

M. Laura Arias A-partial isometries and generalized inverses

The algebra L(R(A1/2))

Theorem (Krein) Let L be an inner product space with an additionalBanach norm ‖ ‖B and let T : L→ L be a linear operator such that〈Tξ, η〉 = 〈ξ,Tη〉 for all ξ, η ∈ L. If T is ‖ ‖B-bounded then it is also‖ ‖L-bounded.

Theorem

Let T : R(A1/2)→ R(A1/2) and Z : R(A1/2)→ R(A1/2) be linearoperators such that

⟨T(A1/2ξ),A1/2η

⟩=

⟨A1/2ξ,Z(A1/2η)

⟩for

every ξ, η ∈ H. If T is bounded in R(A1/2) then T is bounded inH.

M. Laura Arias A-partial isometries and generalized inverses

The algebra L(R(A1/2))

Theorem (Krein) Let L be an inner product space with an additionalBanach norm ‖ ‖B and let T : L→ L be a linear operator such that〈Tξ, η〉 = 〈ξ,Tη〉 for all ξ, η ∈ L. If T is ‖ ‖B-bounded then it is also‖ ‖L-bounded.

Theorem

Let T : R(A1/2)→ R(A1/2) and Z : R(A1/2)→ R(A1/2) be linearoperators such that

⟨T(A1/2ξ),A1/2η

⟩=

⟨A1/2ξ,Z(A1/2η)

⟩for

every ξ, η ∈ H. If T is bounded in R(A1/2) then T is bounded inH.

M. Laura Arias A-partial isometries and generalized inverses

How does LA1/2(H) relate to L(R(A1/2))?

H T //

?��

H

?��

R(A1/2)T // R(A1/2)

Given T ∈ L(H), under which conditions does T ∈ L(R(A1/2)) existsuch that WAT = TWA?

M. Laura Arias A-partial isometries and generalized inverses

How does LA1/2(H) relate to L(R(A1/2))?

H T //

ZA��

HZA

��R(A1/2)

T // R(A1/2)

Definition

Let ZA : H → R(A1/2) defined by ZAξ = A1/2ξ.

Note that for every ξ, η ∈ H it holds

(ZAξ,ZAη) = (A1/2ξ,A1/2η) =⟨

PR(A)ξ,PR(A)η⟩6= 〈ξ, η〉A .

Given T ∈ L(H), under which conditions does T ∈ L(R(A1/2)) existsuch that WAT = TWA?

M. Laura Arias A-partial isometries and generalized inverses

How does LA1/2(H) relate to L(R(A1/2))?

H T //

WA��

HWA

��R(A1/2)

T // R(A1/2)

Definition

Let WA : H → R(A1/2) defined by WAξ = Aξ.

Note that for every ξ, η ∈ H it holds

(WAξ,WAη) = (Aξ,Aη) =⟨

A1/2ξ,A1/2η⟩

= 〈ξ, η〉A .

Given T ∈ L(H), under which conditions does T ∈ L(R(A1/2)) existsuch that WAT = TWA?

M. Laura Arias A-partial isometries and generalized inverses

How does LA1/2(H) relate to L(R(A1/2))?

H T //

WA��

HWA

��R(A1/2)

T // R(A1/2)

Definition

Let WA : H → R(A1/2) defined by WAξ = Aξ.

Note that for every ξ, η ∈ H it holds

(WAξ,WAη) = (Aξ,Aη) =⟨

A1/2ξ,A1/2η⟩

= 〈ξ, η〉A .

Given T ∈ L(H), under which conditions does T ∈ L(R(A1/2)) existsuch that WAT = TWA?

M. Laura Arias A-partial isometries and generalized inverses

How does LA1/2(H) relate to L(R(A1/2))?

Proposition

Consider T ∈ L(H). Then, there exists T ∈ L(R(A1/2)) such thatTWA = WAT if and only if T ∈ LA1/2(H). In such case T is unique.

Proposition

Given T ∈ L(R(A1/2)) there exists T ∈ L(H) such that WAT = TWA

if and only if TR(A) ⊆ R(A). In such case, there exists a uniqueT ∈ LA1/2(H) such that R(T) ⊆ R(A).

M. Laura Arias A-partial isometries and generalized inverses

How does LA1/2(H) relate to L(R(A1/2))?

Proposition

Consider T ∈ L(H). Then, there exists T ∈ L(R(A1/2)) such thatTWA = WAT if and only if T ∈ LA1/2(H). In such case T is unique.

Proposition

Given T ∈ L(R(A1/2)) there exists T ∈ L(H) such that WAT = TWA

if and only if TR(A) ⊆ R(A). In such case, there exists a uniqueT ∈ LA1/2(H) such that R(T) ⊆ R(A).

M. Laura Arias A-partial isometries and generalized inverses

How does LA1/2(H) relate to L(R(A1/2))?

Thus, the following mappings are well defined:

α : LA1/2(H) −→ L(R(A1/2)), T 7−→ T,

where TWAξ = WATξ for all ξ ∈ H.

β : L(R(A1/2)) −→ LA1/2(H), T 7−→ T

where TWAξ = WATξ for all ξ ∈ H and R(T) ⊆ R(A).• ‖α(T)‖R(A1/2) = ‖T‖A and ‖β(T)‖A = ‖T‖R(A1/2).• The compositions αβ and βα can be explicitly computed asαβ : L(R(A1/2)) −→ L(R(A1/2)), αβ(T) = Tandβα : LA1/2(H) −→ LA1/2(H), βα(T) = PR(A)TPR(A).

M. Laura Arias A-partial isometries and generalized inverses

How does LA1/2(H) relate to L(R(A1/2))?

Thus, the following mappings are well defined:

α : LA1/2(H) −→ L(R(A1/2)), T 7−→ T,

where TWAξ = WATξ for all ξ ∈ H.

β : L(R(A1/2)) −→ LA1/2(H), T 7−→ T

where TWAξ = WATξ for all ξ ∈ H and R(T) ⊆ R(A).

• ‖α(T)‖R(A1/2) = ‖T‖A and ‖β(T)‖A = ‖T‖R(A1/2).• The compositions αβ and βα can be explicitly computed asαβ : L(R(A1/2)) −→ L(R(A1/2)), αβ(T) = Tandβα : LA1/2(H) −→ LA1/2(H), βα(T) = PR(A)TPR(A).

M. Laura Arias A-partial isometries and generalized inverses

How does LA1/2(H) relate to L(R(A1/2))?

Thus, the following mappings are well defined:

α : LA1/2(H) −→ L(R(A1/2)), T 7−→ T,

where TWAξ = WATξ for all ξ ∈ H.

β : L(R(A1/2)) −→ LA1/2(H), T 7−→ T

where TWAξ = WATξ for all ξ ∈ H and R(T) ⊆ R(A).• ‖α(T)‖R(A1/2) = ‖T‖A and ‖β(T)‖A = ‖T‖R(A1/2).

• The compositions αβ and βα can be explicitly computed asαβ : L(R(A1/2)) −→ L(R(A1/2)), αβ(T) = Tandβα : LA1/2(H) −→ LA1/2(H), βα(T) = PR(A)TPR(A).

M. Laura Arias A-partial isometries and generalized inverses

How does LA1/2(H) relate to L(R(A1/2))?

Thus, the following mappings are well defined:

α : LA1/2(H) −→ L(R(A1/2)), T 7−→ T,

where TWAξ = WATξ for all ξ ∈ H.

β : L(R(A1/2)) −→ LA1/2(H), T 7−→ T

where TWAξ = WATξ for all ξ ∈ H and R(T) ⊆ R(A).• ‖α(T)‖R(A1/2) = ‖T‖A and ‖β(T)‖A = ‖T‖R(A1/2).• The compositions αβ and βα can be explicitly computed asαβ : L(R(A1/2)) −→ L(R(A1/2)), αβ(T) = T

andβα : LA1/2(H) −→ LA1/2(H), βα(T) = PR(A)TPR(A).

M. Laura Arias A-partial isometries and generalized inverses

How does LA1/2(H) relate to L(R(A1/2))?

Thus, the following mappings are well defined:

α : LA1/2(H) −→ L(R(A1/2)), T 7−→ T,

where TWAξ = WATξ for all ξ ∈ H.

β : L(R(A1/2)) −→ LA1/2(H), T 7−→ T

where TWAξ = WATξ for all ξ ∈ H and R(T) ⊆ R(A).• ‖α(T)‖R(A1/2) = ‖T‖A and ‖β(T)‖A = ‖T‖R(A1/2).• The compositions αβ and βα can be explicitly computed asαβ : L(R(A1/2)) −→ L(R(A1/2)), αβ(T) = Tandβα : LA1/2(H) −→ LA1/2(H), βα(T) = PR(A)TPR(A).

M. Laura Arias A-partial isometries and generalized inverses

Do the previous classes of operators in LA1/2(H) relate tosimilar classes of L(R(A1/2))?

By means of α, we obtain the next relationship between classes inLA1/2(H) and similar classes of L(R(A1/2)).

TheoremThe following equalities hold:

1 α(CA(H)) = C(R(A1/2)).

2 α(IA(H)) = I(R(A1/2)).

3 α(JA(H)) = J (R(A1/2)).

M. Laura Arias A-partial isometries and generalized inverses

Do the previous classes of operators in LA1/2(H) relate tosimilar classes of L(R(A1/2))?

By means of α, we obtain the next relationship between classes inLA1/2(H) and similar classes of L(R(A1/2)).

TheoremThe following equalities hold:

1 α(CA(H)) = C(R(A1/2)).

2 α(IA(H)) = I(R(A1/2)).

3 α(JA(H)) = J (R(A1/2)).

M. Laura Arias A-partial isometries and generalized inverses

Do the previous classes of operators in LA1/2(H) relate tosimilar classes of L(R(A1/2))?

By means of α, we obtain the next relationship between classes inLA1/2(H) and similar classes of L(R(A1/2)).

TheoremThe following equalities hold:

1 α(CA(H)) = C(R(A1/2)).

2 α(IA(H)) = I(R(A1/2)).

3 α(JA(H)) = J (R(A1/2)).

M. Laura Arias A-partial isometries and generalized inverses

Do the previous classes of operators in LA1/2(H) relate tosimilar classes of L(R(A1/2))?

By means of α, we obtain the next relationship between classes inLA1/2(H) and similar classes of L(R(A1/2)).

TheoremThe following equalities hold:

1 α(CA(H)) = C(R(A1/2)).

2 α(IA(H)) = I(R(A1/2)).

3 α(JA(H)) = J (R(A1/2)).

M. Laura Arias A-partial isometries and generalized inverses

Thank you.

M. Laura Arias A-partial isometries and generalized inverses