Partial isometries and pseudoinverses in semi-Hilbertian spaces Mar - PowerPoint PPT Presentation

Partial isometries and pseudoinverses in semi-Hilbertian spaces Mar´ ıa Celeste Gonzalez Joint work with Guillermina Fongi Instituto Argentino de Matem´ atica (IAM-CONICET) Universidad Nacional de General Sarmiento Argentina YMC*A 2016 - University of M¨ unster July 26, 2016 1 / 11

Introduction Consider a Hilbert H space with inner product � , � and A a positive semidefinited operator in B ( H ). Consider the semi-inner product � , � A on H defined by � ξ, η � A = � A ξ, η � . Definition ( H , � , � A ) is called a semi-Hilbertian space. ◮ If A is positive and injective then ( H , � , � A ) is a pre-Hilbert space. ◮ If A is positive and invertible then ( H , � , � A ) is a Hilbert space. 2 / 11

Partial isometries Definition T ∈ B ( H ) is a partial isometry if � T ξ � = � ξ � for all ξ ∈ N ( T ) ⊥ . Proposition The following equivalent conditions are well-known for T ∈ B ( H ): 1. T is a partial isometry; 2. T ∗ is a partial isometry; 3. T ∗ T is a projection; 4. TT ∗ is a projection; 5. T ∗ TT ∗ = T ∗ ; 6. TT ∗ T = T (i.e., T ∗ is a generalized inverse of T ); 7. T ∗ = T † , where T † is the Moore-Penrose inverse of T ; 8. Given η ∈ H , T ∗ η is the unique least square solution with minimal norm of the equation T ξ = η for all ξ ∈ H ; 3 / 11

Moore Penrose inverse and Douglas theorem Definition The Moore-Penrose inverse of T ∈ B ( H ) is the densely defined operator T † : R ( T ) ⊕ R ( T ) ⊥ → N ( T ) ⊥ , such that T † | R ( T ) = ( T | N ( T ) ⊥ ) − 1 and N ( T † ) = R ( T ) ⊥ . Theorem (R. Douglas 1966) Let A , B ∈ B ( H ). The following conditions are equivalent: 1. the equation AX = B has a solution in B ( H ); 2. R ( B ) ⊆ R ( A ); 3. there exists λ > 0 such that BB ∗ ≤ λ AA ∗ . If one of these conditions holds then there exists a unique solution D ∈ B ( H ) such that R ( D ) ⊆ N ( A ) ⊥ ; Moreover, D = A † B . 4 / 11

A -adjoint operators Definition Given T ∈ B ( H ), W is an A -adjoint of T if � T ξ, η � A = � ξ, W η � A for all ξ, η ∈ H Remarks ◮ T admits an A -adjoint ↔ AX = T ∗ A has solution ↔ R ( T ∗ A ) ⊆ R ( A ). ◮ T can have none, one or infinitely many A -adjoint operators. ◮ If T admits an A -adjoint then T ♯ = A † T ∗ A is an A -adjoint. ◮ R ( T ♯ ) ⊆ R ( A ). ◮ T ♯ is bounded even though A † might be unbounded. Notation B A ( H ) = { T ∈ B ( H ) : T admits an A -adjoint operator } 5 / 11

A -partial isometries Definition T ∈ B A ( H ) is an A -p.i. if � T ξ � A = � ξ � A , for all ξ ∈ R ( T ♯ T ) . Proposition (Fongi, G. 2016) If T ∈ B A ( H ) then the following assertions are equivalent: 1. T is an A -partial isometry; 2. T ♯ is an A -partial isometry; 3. T ♯ T is an A -selfadjoint projection; 4. TT ♯ is an A -selfadjoint projection; 5. T ♯ TT ♯ = T ♯ ; a. ATT ♯ T = AT . Remarks ◮ The above proposition is not valid for every A -adjoint of T . ◮ R ( T ) is not closed, in general. ◮ R ( T ♯ ) is closed. 6 / 11

A -partial isometries and generalized inverses Going back to partial isometries, recall that the following conditions are equivalent: 1. T is a partial isometry; 6. TT ∗ T = T ; 7. T ∗ = T † ; 8. T ∗ η is the l.s.s. with minimal norm of the equation T ξ = η . Remarks ◮ If T ∈ B A ( H ) and TT ♯ T = T then T is an A -partial isometry. The converse is false. (Arias, Mbekhta (2013)) ◮ Given T ∈ B ( H ), if TXT = T has solution then T has closed range. Therefore, we will deal with operators with closed range. 7 / 11

Equivalences for the Moore-Penrose Inverse Theorem (Moore 1920, Penrose 1955, Groestch 1977) Given T ∈ B ( H ) be with closed range, the following conditions are equivalent: I. T † is the Moore-Penrose inverse of T ; II. T † satisfies the four equations: TXT = T ; XTX = X ; TX = ( TX ) ∗ ; XT = ( XT ) ∗ . III. Given η ∈ H , consider the equation T ξ = η . Then, � T ( T † η ) − η � = min {� T ξ − η � : ∀ ξ ∈ H} and � T † η � = min {� θ � : θ is l.s.s. of T ξ = η } . 8 / 11

Moore-Penrose inverse in ( H , � , � A ) Definition (II.) (Corach, Maestripieri (2005)) An operator T ′ ∈ B ( H ) is an A -generalized inverse of T ∈ B ( H ) if: TT ′ T = T ; T ′ TT ′ = T ′ ; A ( TT ′ ) = ( TT ′ ) ∗ A ; A ( T ′ T ) = ( T ′ T ) ∗ A . Definition (III.) (Corach, Fongi, Maestripieri (2013)) Given η ∈ H , consider the equation T ξ = η . An operator T ′ ∈ B ( H ) is an: ◮ A -inverse of T if: � η − TT ′ η � A ≤ � η − T ξ � A ∀ ξ ∈ H . III. ( A , I )-inverse of T if it is an A -inverse of T such that for each η ∈ H , � T ′ η � = min {� θ � : θ is an A -l.s.s. of T ξ = η } . 9 / 11

A -partial isometries and pseudoinverses Theorem (Fongi, G. (2016)) T ∈ B A ( H ) with closed range. Consider the following statements: 1. T is an A -partial isometry; 2. TT ♯ T = T ; 3. T ♯ is an A -generalized inverse of T ; 4. T ♯ is an A -inverse of T ; 5. P N ( AT ) ⊥ T ♯ is an ( A , I )-inverse of T . The following relationship between the above assertions holds: 2 ⇔ 3 ⇒ 4 ⇔ 5 ⇔ 1. But 4 �⇒ 3 (so 1 �⇒ 2). Remark 2 ⇔ 3 was proved by Arias, Mbekhta (2013). 10 / 11

A -partial isometries and pseudoinverses Theorem (Fongi, G. (2016)) T ∈ B A ( H ) with closed range. Consider the following statements: 1. T is an A -partial isometry; ( T is a partial isometry) 2. TT ♯ T = T ; ( TT ∗ T = T ;) 3. T ♯ is an A -generalized inverse of T ; ( T ∗ = T † ;) 4. T ♯ is an A -inverse of T ; 5. P N ( AT ) ⊥ T ♯ is an ( A , I )-inverse of T . T ∗ η is the l.s.s. with minimal norm of T ξ = η The following relationship between the above assertions holds: 2 ⇔ 3 ⇒ 4 ⇔ 5 ⇔ 1. But 4 �⇒ 3 (so 1 �⇒ 2). Remark 2 ⇔ 3 was proved by Arias, Mbekhta (2013). 10 / 11

A -partial isometries and generalized inverses Which are the A -partial isometries T that satisfy TT ♯ T = T ? Theorem (Fongi, G. (2016)) Let T ∈ B A ( H ) be an A -partial isometry. The following assertion are equivalent: ◮ TT ♯ T = T ; ◮ H = R ( T ) ˙ + N ( T ♯ ); ◮ H = R ( T ♯ ) ˙ + N ( T ); ◮ R ( T ) ∩ N ( A ) = { 0 } and R ( T ) is closed. 11 / 11