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

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Semi-Hilbertian spaces

LetHbe a Hilbert space with inner producth, i.GivenA∈L(H)⁺, leth, i_Athe semi-inner product defined by

hξ, ηi_A :=hAξ, ηi, for everyξ, η ∈ H.

Byk k_Awe denote the semi-norm induced byh, i_A, i.e., kξk_A :=hξ, ξi^1/2_A =kA^1/2ξk ∀ξ∈ H.

(H,h, i_A)is a semi-Hilbertian space.

Our main goal is to study classes of operators on(H,h, i_A).

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Semi-Hilbertian spaces

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Semi-Hilbertian spaces

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Semi-Hilbertian spaces

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Some subalgebras of L(H)

GivenT ∈L(H),S∈L(H)is anA-adjointofTif hTξ, ηi_A =hξ,Sηi_A ∀ξ, η∈ H,

or which is the same, ifSis solution of the equationAX=T^∗A.

Moreover, we shall say thatTisA-selfadjointifAT =T^∗A.Define L_A(H) ={T ∈L(H) : Tadmits A−adjoint}

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Some subalgebras of L(H)

Moreover, we shall say thatTisA-selfadjointifAT =T^∗A.

Define L_A(H) ={T ∈L(H) : Tadmits A−adjoint}

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Some subalgebras of L(H)

Moreover, we shall say thatTisA-selfadjointifAT =T^∗A.Define L_A(H) ={T ∈L(H) : Tadmits A−adjoint}

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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 numberλsuch that CC^∗≤λBB^∗. Moreover, if one of these conditions holds then for every closed subsapceMsuch thatM+^. N(B) =Hthere exists a unique D_M ∈L(H)such that BD_M =C and R(DM)⊆ M.This solution will be called areduced solutionof the equation BX=C.

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Douglas’ theorem

Theorem (Douglas)

2 R(C)⊆R(B).

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Douglas’ theorem

Theorem (Douglas)

2 R(C)⊆R(B).

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Douglas’ theorem

Theorem (Douglas)

2 R(C)⊆R(B).

3 There is a positive numberλsuch that CC^∗ ≤λBB^∗.

Moreover, if one of these conditions holds then for every closed subsapceMsuch thatM+^. N(B) =Hthere exists a unique D_M ∈L(H)such that BD_M =C and R(DM)⊆ M.This solution will be called areduced solutionof the equation BX=C.

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Douglas’ theorem

Theorem (Douglas)

2 R(C)⊆R(B).

3 There is a positive numberλsuch that CC^∗ ≤λBB^∗. Moreover, if one of these conditions holds then for every closed subsapceMsuch thatM+^. N(B) =Hthere exists a unique D_M ∈L(H)such that BD_M =C and R(DM)⊆ M.This solution will be called areduced solutionof the equation BX=C.

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Some subalgebras of L(H)

Therefore,

L_A(H) ={T ∈L(H) : R(T^∗A)⊆R(A)}

In the sequel, givenT ∈LA(H),T^]will denote a reduced solution of AX=T^∗A.

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Some subalgebras of L(H)

Therefore,

L_A(H) ={T ∈L(H) : R(T^∗A)⊆R(A)}

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Some subalgebras of L(H)

Therefore,

L_A(H) ={T ∈L(H) : R(T^∗A)⊆R(A)}

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Some properties of T

^]

LetT ∈LA(H). Then the following statements hold:

1 IfW ∈LA(H)thenTW ∈LA(H)and(TW)^]=W^]T^].

2 T^]∈L_A(H),(T^])^]=Q_M//N(A)T.

3 T^]T andTT^]areA-selfadjoint.

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Some properties of T

^]

2 T^]∈L_A(H),(T^])^]=Q_M//N(A)T.

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Some properties of T

^]

2 T^]∈L_A(H),(T^])^]=Q_M//N(A)T.

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Some subalgebras of L(H)

Let

L_A1/2(H) : = {T ∈L(H) : ∃c>0 kTξk_A≤ckξk_A ∀ξ∈ H}

= {T ∈L(H) : R(T^∗A^1/2)⊆R(A^1/2)} The next inclusion holds:L_A(H)⊆L_A1/2(H).

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Some subalgebras of L(H)

Let

= {T ∈L(H) : R(T^∗A^1/2)⊆R(A^1/2)}

The next inclusion holds:L_A(H)⊆L_A1/2(H).

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Some subalgebras of L(H)

Let

= {T ∈L(H) : R(T^∗A^1/2)⊆R(A^1/2)}

The next inclusion holds:LA(H)⊆L_A1/2(H).

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Classes of operators in L

_A^1/2

(H)

A-contractions andA-isometries

Definition

T∈L(H)is called anA-contractionif

kTξk_A≤ kξk_Afor everyξ∈ H,

or equivalently ifT^∗AT ≤A.If the equality holds thenTis called an A-isometry.

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A-isometries

Remarks

1 LetT ∈LA(H). Then,T is anA-isometry if and only if T^]T =Q_M//N(A).

2 TheA-isometries are not, in general, left invertible.

For example, for everyA∈L(H)⁺not injective,P_R(A)is a not left invertible A-isometry.

Recall that givenT ∈L(H)we say thatT⁰ ∈L(H)is ageneralized inverse ofTifTT⁰T =TandT⁰TT⁰ =T⁰.

Remark

TheA-isometries may not admit generalized inverses.

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A-isometries

Remarks

2 TheA-isometries are not, in general, left invertible.

For example, for everyA∈L(H)⁺not injective,P_R(A)is a not left invertible A-isometry.

Remark

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A-isometries

Remarks

2 TheA-isometries are not, in general, left invertible. For example, for everyA∈L(H)⁺not injective,P_R(A)is a not left invertible A-isometry.

Remark

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A-isometries

Remarks

Remark

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A-isometries

Remarks

Remark

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A-partial isometries

Recall that givenT ∈L(H)the following conditions are equivalent:

1 T is a partial isometry;

2 T^∗is a partial isometry;

3 T^∗Tis a projection;

4 TT^∗T =T;

5 T is a contraction with a contractive generalized inverse.

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A-partial isometries

4 TT^∗T =T;

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A-partial isometries

4 TT^∗T =T;

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A-partial isometries

4 TT^∗T =T;

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A-partial isometries

4 TT^∗T =T;

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A-partial isometries

GivenT ∈L_A(H),

(T^]T)²=T^]T.

From this,kTξk_A=kξk_A∀ξ∈R(T^]T)or, which is the same, kTξk_A=kξk_A∀ξ ∈R(T^]T) +N(A).

It holds that

R(T^]T) +N(A) =N(AT)^⊥^A, whereS^⊥^A ={ξ∈ H: hξ, ηi_A=0∀η∈ S}.

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A-partial isometries

GivenT ∈L_A(H),

(T^]T)²=T^]T.

From this,kTξk_A=kξk_A∀ξ∈R(T^]T)

or, which is the same, kTξk_A=kξk_A∀ξ ∈R(T^]T) +N(A).

It holds that

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A-partial isometries

GivenT ∈L_A(H),

(T^]T)²=T^]T.

From this,kTξk_A=kξk_A∀ξ∈R(T^]T)or, which is the same, kTξk_A=kξk_A∀ξ∈R(T^]T) +N(A).

It holds that

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A-partial isometries

GivenT ∈L_A(H),

(T^]T)²=T^]T.

From this,kTξk_A=kξk_A∀ξ∈R(T^]T)or, which is the same, kTξk_A=kξk_A∀ξ∈R(T^]T) +N(A).

It holds that

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A-partial isometries

Definition

T∈L(H)is called anA-partial isometryif kTξk_A =kξk_A∀ξ ∈N(AT)^⊥^A.

Remarks

1 IfA=I then anA-partial isometry is a partial isometry.

2 EveryA-isometry is anA-partial isometry.

3 IfT ∈LA(H)and(A,R(T^]T))is compatible then:T is an A-partial isometry⇔T^]Tis a projection.

4 IfT ∈L_A(H)is anA-partial isometry thenT^]is anA-partial isometry.

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A-partial isometries

Definition

T∈L(H)is called anA-partial isometryif kTξk_A =kξk_A∀ξ ∈N(AT)^⊥^A. Remarks

1 IfA=Ithen anA-partial isometry is a partial isometry.

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

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A-partial isometries

Definition

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A-partial isometries

Definition

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A-partial isometries

Definition

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A-partial isometries

Definition

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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 is false, in general.

Theorem (Corach et al, 2005)

Given T ∈L(H)with closed range, there exists T⁰∈L(H)such that TT⁰T =T, T⁰TT⁰=T⁰, ATT⁰ = (TT⁰)^∗A, AT⁰T = (T⁰T)^∗A if and only if the pairs(A,R(T))and(A,N(T))are complatible.

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A-partial isometries and generalized inverses

Proposition

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

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

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A-partial isometries and generalized inverses

Proposition

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

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A-partial isometries and generalized inverses

Proposition

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

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A-partial isometries and generalized inverses

Proposition

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

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

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

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A-partial isometries and generalized inverses

Proposition

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A-partial isometries and generalized inverses

Proposition

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Hilbert space induced by a positive operator

Definition

LetA∈L(H)⁺. A pair(K,Π)is called aHilbert space induced by Aif:

1 Kis a Hilbert space.

2 Π :H → Kis a linear operator.

3 R(Π)is dense inK.

4 hΠξ,Πηi=hAξ, ηi ∀ξ, η∈ H.

Theorem (Cojuhari et al, 2005)

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

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Hilbert space induced by a positive operator

Definition

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Hilbert space induced by a positive operator

Definition

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Hilbert space induced by a positive operator

Definition

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Hilbert space induced by a positive operator

Definition

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Hilbert space induced by a positive operator

Definition

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Hilbert space induced by a positive operator

LetR(A^1/2)be equipped with the inner product defined by (A^1/2ξ,A^1/2η) :=D

P_R(A)ξ,P_R(A)ηE

∀ξ, η∈ H.

R(A^1/2) = (R(A^1/2),(, ))is a Hilbert space induced byA

Can we describeL(R(A^1/2))by means ofL(H)?

How doesL_A1/2(H)relate toL(R(A^1/2))?

Do the previous classes of operators inL_A1/2(H)relate to similar classes ofL(R(A^1/2))?

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Hilbert space induced by a positive operator

P_R(A)ξ,P_R(A)ηE

∀ξ, η∈ H.

R(A^1/2) = (R(A^1/2),(, ))is a Hilbert space induced byA Can we describeL(R(A^1/2))by means ofL(H)?

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Hilbert space induced by a positive operator

P_R(A)ξ,P_R(A)ηE

∀ξ, η∈ H.

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Hilbert space induced by a positive operator

P_R(A)ξ,P_R(A)ηE

∀ξ, η∈ H.

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The algebra L(R(A

^1/2

))

Proposition

LetT˜:R(A^1/2)→R(A^1/2)be a linear operator. Then there exists a unique linear operator V:H → Hsuch that R(V)⊆R(A)and

A^1/2V = ˜TA^1/2.

Moreover,T is bounded in˜ R(A^1/2)if and only if V is bounded inH.

Moreover,kTk˜ _R(A1/2)=kVk.

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The algebra L(R(A

^1/2

))

Theorem (Krein)Let L be an inner product space with an additional Banach normk k_B and let T :L→L be a linear operator such that hTξ, ηi=hξ,Tηifor allξ, η∈L. If T isk k_B-bounded then it is also k kL-bounded.

Theorem

LetT˜:R(A^1/2)→R(A^1/2)and Z:R(A^1/2)→R(A^1/2)be linear operators such that

DT˜(A^1/2ξ),A^1/2ηE

=D

A^1/2ξ,Z(A^1/2η)E for everyξ, η∈ H. IfT is bounded in˜ R(A^1/2)thenT is bounded in˜ H.

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The algebra L(R(A

^1/2

))

Theorem (Krein)Let L be an inner product space with an additional Banach normk k_B and let T :L→L be a linear operator such that hTξ, ηi=hξ,Tηifor allξ, η∈L. If T isk k_B-bounded then it is also k kL-bounded.

Theorem

LetT˜:R(A^1/2)→R(A^1/2)and Z:R(A^1/2)→R(A^1/2)be linear operators such that

DT˜(A^1/2ξ),A^1/2ηE

=D

A^1/2ξ,Z(A^1/2η)E for everyξ, η∈ H. IfT is bounded in˜ R(A^1/2)thenT is bounded in˜ H.

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How does L

_A^1/2

(H) relate to L(R(A

^1/2

))?

H ^T ^//

?

H

?

R(A^1/2) ^T^˜ ^//R(A^1/2)

GivenT ∈L(H),under which conditions doesT˜ ∈L(R(A^1/2))exist such thatW_AT = ˜TW_A?

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How does L

_A^1/2

(H) relate to L(R(A

^1/2

))?

H ^T ^//

Z_A

H

Z_A

R(A^1/2) ^T^˜ ^//R(A^1/2)

Definition

LetZ_A:H →R(A^1/2)defined byZ_Aξ =A^1/2ξ.

Note that for everyξ, η∈ Hit holds (ZAξ,Z_Aη) = (A^1/2ξ,A^1/2η) = D

P_R(A)ξ,P_R(A)η E

6=hξ, ηi_A.

GivenT ∈L(H),under which conditions doesT˜ ∈L(R(A^1/2))exist such thatWAT = ˜TWA?

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How does L

_A^1/2

(H) relate to L(R(A

^1/2

))?

H ^T ^//

W_A

H

W_A

R(A^1/2) ^T^˜ ^//R(A^1/2)

Definition

LetW_A :H →R(A^1/2)defined byW_Aξ =Aξ.

Note that for everyξ, η∈ Hit holds (WAξ,W_Aη) = (Aξ,Aη) =

D

A^1/2ξ,A^1/2η E

=hξ, ηi_A.

GivenT ∈L(H),under which conditions doesT˜ ∈L(R(A^1/2))exist such thatWAT = ˜TWA?

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How does L

_A^1/2

(H) relate to L(R(A

^1/2

))?

H ^T ^//

W_A

H

W_A

R(A^1/2) ^T^˜ ^//R(A^1/2)

Definition

LetW_A :H →R(A^1/2)defined byW_Aξ =Aξ.

Note that for everyξ, η∈ Hit holds (WAξ,W_Aη) = (Aξ,Aη) =

D

A^1/2ξ,A^1/2η E

=hξ, ηi_A. GivenT ∈L(H),under which conditions doesT˜∈L(R(A^1/2))exist such thatWAT = ˜TWA?

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How does L

_A^1/2

(H) relate to L(R(A

^1/2

))?

Proposition

Consider T ∈L(H). Then, there existsT˜ ∈L(R(A^1/2))such that TW˜ _A=W_AT if and only if T ∈L_A1/2(H). In such caseT is unique.˜

Proposition

GivenT˜ ∈L(R(A^1/2))there exists T ∈L(H)such that W_AT = ˜TW_A if and only ifTR(A)˜ ⊆R(A).In such case, there exists a unique T∈L_A1/2(H)such that R(T)⊆R(A).

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How does L

_A^1/2

(H) relate to L(R(A

^1/2

))?

Proposition

Consider T ∈L(H). Then, there existsT˜ ∈L(R(A^1/2))such that TW˜ _A=W_AT if and only if T ∈L_A1/2(H). In such caseT is unique.˜

Proposition

GivenT˜ ∈L(R(A^1/2))there exists T ∈L(H)such that W_AT = ˜TW_A if and only ifTR(A)˜ ⊆R(A).In such case, there exists a unique T∈L_A1/2(H)such that R(T)⊆R(A).

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How does L

_A^1/2

(H) relate to L(R(A

^1/2

))?

Thus, the following mappings are well defined:

α:L_A1/2(H)−→L(R(A˜ ^1/2)),T 7−→T,˜ whereTW˜ Aξ=WATξfor allξ∈ H.

β: ˜L(R(A^1/2))−→L_A1/2(H),T˜ 7−→T whereTW˜ Aξ=WATξfor allξ ∈ HandR(T)⊆R(A).

• kα(T)k_R(A1/2)=kTk_Aandkβ(˜T)k_A=kTk˜ _R(A1/2).

•The compositionsαβandβαcan be explicitly computed as αβ: ˜L(R(A^1/2))−→˜L(R(A^1/2)), αβ(˜T) = ˜T

and

βα:L_A1/2(H)−→L_A1/2(H), βα(T) =P_R(A)TP_R(A).

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How does L

_A^1/2

(H) relate to L(R(A

^1/2

))?

β: ˜L(R(A^1/2))−→L_A1/2(H),T˜ 7−→T whereTW˜ Aξ=WATξfor allξ∈ HandR(T)⊆R(A).

and

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How does L

_A^1/2

(H) relate to L(R(A

^1/2

))?

and

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How does L

_A^1/2

(H) relate to L(R(A

^1/2

))?

and

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How does L

_A^1/2

(H) relate to L(R(A

^1/2

))?

and

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Do the previous classes of operators in L

_A^1/2

(H) relate to similar classes of L(R(A

^1/2

))?

By means ofα,we obtain the next relationship between classes in L_A1/2(H)and similar classes ofL(R(A^1/2)).

Theorem

The following equalities hold:

1 α(C_A(H)) =C(R(Ae ^1/2)).

2 α(I_A(H)) =I(R(Ae ^1/2)).

3 α(J_A(H)) =Je(R(A^1/2)).

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Do the previous classes of operators in L

_A^1/2

(H) relate to similar classes of L(R(A

^1/2

))?

Theorem

1 α(C_A(H)) =C(R(Ae ^1/2)).

2 α(I_A(H)) =I(R(Ae ^1/2)).

3 α(J_A(H)) =Je(R(A^1/2)).

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Do the previous classes of operators in L

_A^1/2

(H) relate to similar classes of L(R(A

^1/2

))?

Theorem

1 α(C_A(H)) =C(R(Ae ^1/2)).

2 α(I_A(H)) =I(R(Ae ^1/2)).

3 α(J_A(H)) =Je(R(A^1/2)).

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Do the previous classes of operators in L

_A^1/2

(H) relate to similar classes of L(R(A

^1/2

))?

Theorem

1 α(C_A(H)) =C(R(Ae ^1/2)).

2 α(I_A(H)) =I(R(Ae ^1/2)).

3 α(J_A(H)) =Je(R(A^1/2)).

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A-partial isometries and generalized inverses

A-partial isometries and generalized inverses

Contents

Contents

Contents

Semi-Hilbertian spaces

Semi-Hilbertian spaces

Semi-Hilbertian spaces

Semi-Hilbertian spaces

Some subalgebras of L(H)

Some subalgebras of L(H)

Some subalgebras of L(H)

Douglas’ theorem

Douglas’ theorem

Douglas’ theorem

Douglas’ theorem

Douglas’ theorem

Some subalgebras of L(H)

Some subalgebras of L(H)

Some subalgebras of L(H)

Some properties of T

Some properties of T

Some properties of T

Some subalgebras of L(H)

Some subalgebras of L(H)

Some subalgebras of L(H)

Classes of operators in L

(H)

A-isometries

A-isometries

A-isometries

A-isometries

A-isometries

A-partial isometries

A-partial isometries

A-partial isometries

A-partial isometries

A-partial isometries

A-partial isometries

A-partial isometries

A-partial isometries

A-partial isometries

A-partial isometries

A-partial isometries

A-partial isometries

A-partial isometries

A-partial isometries

A-partial isometries

A-partial isometries and generalized inverses

A-partial isometries and generalized inverses

A-partial isometries and generalized inverses

A-partial isometries and generalized inverses

A-partial isometries and generalized inverses

A-partial isometries and generalized inverses

A-partial isometries and generalized inverses

Hilbert space induced by a positive operator

Hilbert space induced by a positive operator

Hilbert space induced by a positive operator

Hilbert space induced by a positive operator

Hilbert space induced by a positive operator

Hilbert space induced by a positive operator

Hilbert space induced by a positive operator

Hilbert space induced by a positive operator

Hilbert space induced by a positive operator

Hilbert space induced by a positive operator

The algebra L(R(A

))

The algebra L(R(A

))

The algebra L(R(A

))

How does L

(H) relate to L(R(A

))?

How does L

(H) relate to L(R(A

))?

How does L

(H) relate to L(R(A

))?