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

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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, ηA = Aξ, 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.

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Partial isometries

Definition

T ∈ B(H) is a partial isometry if Tξ = 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 Tξ = η for all ξ ∈ H;

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Moore Penrose inverse and Douglas theorem

Definition

The Moore-Penrose inverse of T ∈ B(H) is the densely defined

• perator

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∗ ≤ λAA∗.

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.

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A-adjoint operators

Definition

Given T ∈ B(H), W is an A-adjoint of T if Tξ, ηA = ξ, W ηA 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

BA(H) = {T ∈ B(H) : T admits an A-adjoint operator}

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A-partial isometries

Definition

T ∈ BA(H) is an A-p.i. if TξA = ξA, for all ξ ∈ R(T ♯T).

Proposition (Fongi, G. 2016)

If T ∈ BA(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.

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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 Tξ = η.

Remarks

◮ If T ∈ BA(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.

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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 Tξ = η. Then,

T(T †η) − η = min{Tξ − η : ∀ξ ∈ H} and T †η = min{θ : θ is l.s.s. of Tξ = η}.

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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 Tξ = η. An operator T ′ ∈ B(H) is an:

◮ A-inverse of T if: η − TT ′ηA ≤ η − TξA ∀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 Tξ = η}.

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A-partial isometries and pseudoinverses

Theorem (Fongi, G. (2016))

T ∈ BA(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. PN(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).

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A-partial isometries and pseudoinverses

Theorem (Fongi, G. (2016))

T ∈ BA(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. PN(AT)⊥T ♯ is an (A, I)-inverse of T.

T ∗η is the l.s.s. with minimal norm of Tξ = 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).

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A-partial isometries and generalized inverses

Which are the A-partial isometries T that satisfy TT ♯T = T?

Theorem (Fongi, G. (2016))

Let T ∈ BA(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.

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