A generalized Schur complement for non-negative operators on linear space
aa r X i v : . [ m a t h . F A ] A ug A GENERALIZED SCHUR COMPLEMENT FORNON-NEGATIVE OPERATORS ON LINEAR SPACES
J. FRIEDRICH, M. G ¨UNTHER, ∗ and L. KLOTZ Abstract.
Extending the corresponding notion for matrices or bounded lin-ear operators on a Hilbert space we define a generalized Schur complement fora non-negative linear operator mapping a linear space into its dual and derivesome of its properties. Introduction
In case of 2 × × × Date : July 30, 2018 ∗ Corresponding author.2010
Mathematics Subject Classification.
Key words and phrases.
Schur complement, shorted operator, extremal operator.
For bounded linear operators on Hilbert spaces many results concerning thegeneralized Schur complement were obtained by Yu. L. Shmulyan. A large partof them was proved independently or rediscovered later on by several mathe-maticians from western countries. The present paper is strongly influenced byShmulyan’s work and was written to illustrate his contribution to the theory ofgeneralized Schur complement. In this way most of our assertions of Sections 4-6are generalizations of results contained in [17] to non-negative operators on linearspaces. 2.
Basic definitions and notations
Any linear space of the present paper is a space over C , the field of complexnumbers, and its zero element is denoted by 0. For a linear space X , let X ′ denoteits dual space of all antilinear functionals on X and h x ′ , x i X := h x ′ , x i the valueof x ′ ∈ X ′ at x ∈ X . If X ∼ is a subspace of X ′ , an arbitrary x ∈ X defines anelement jx of ( X ∼ ) ′ according to h jx, x ∼ i X ′ := h x ∼ , x i X , x ∼ ∈ X ∼ , where ¯ α stands for the complex conjugate of α ∈ C . Convention (CN):
If for all x ∈ X \ { } there exists x ∼ ∈ X ∼ such that h x ∼ , x i 6 = 0, we shall identify X and its isomorphic image under the map j andwrite h jx, x ∼ i X ′ =: h x, x ∼ i X ′ , x ∈ X, x ∼ ∈ X ∼ . The linear space of all linear operators from X into a linear space Y is denoted by L ( X, Y ), and I is the identity operator in case X = Y . If A ∈ L ( X, Y ) and X is a subspace of X , the symbols ker A , ran A and A ↾ X stand for the null space,range, and restriction of A to X , resp. Set AX := ran A ↾ X . The dual operator A ′ ∈ L ( Y ′ , X ′ ) is defined by the relation h y ′ , Ax i Y = h A ′ y ′ , x i X , x ∈ X , y ′ ∈ Y ′ . Examples.
1. If Z is a linear space and A ∈ L ( X, Y ), B ∈ L ( Y, Z ), then( BA ) ′ = A ′ B ′ .2. If A ∈ L ( X, Y ), then A ′′ ∈ L ( X ′′ , Y ′′ ) and A = A ′′ ↾ X according to (CN).3. If A ∈ L ( X, X ′ ), then A ′ ∈ L ( X ′′ , X ′ ). Taking into account (CN), we get h x , Ax i X ′ = h Ax , x i X and h x , Ax i X ′ = h A ′ x , x i X , hence, h Ax , x i X = h A ′ x , x i X , x , x ∈ X. (2.1)An operator A ∈ L ( X, X ′ ) is called Hermitian, if h Ax , x i = h Ax , x i andnon-negative if h Ax , x i ≥ x , x ∈ X . The sets of all Hermitian and allnon-negative operators are denoted by L h ( X, X ′ ) and L ≥ ( X, X ′ ), resp.. Thepolarization identity implies that A is Hermitian if and only if h Ax, x i is realfor all x ∈ X . Thus L ≥ ( X, X ′ ) ⊆ L h ( X, X ′ ) and the space L h ( X, X ′ ) can beprovided with Loewner’s semi-ordering, i.e. for A, D ∈ L h ( X, X ′ ) we shall write A ≤ D if and only if h Ax, x i ≤ h
Dx, x i , x ∈ X . Recall Cauchy’s inequality |h Ax , x i| ≤ h Ax , x ih Ax , x i , x , x ∈ X, (2.2)if A ∈ L ≥ ( X, X ′ ). GENERALIZED SCHUR COMPLEMENT 3 Square roots
Let H be a complex Hilbert space with norm k · k := k · k H and inner product( · | · ) := ( · | · ) H , which is assumed to be antilinear with respect to the secondcomponent. Let R ∈ L ( X, H ). Identifying H and the space of continuousantilinear functionals on H in the common way, one has H ⊆ H ′ and( h | Rx ) = h R ′ h, x i , x ∈ X, h ∈ H. (3.1)Set R ∗ := R ′ ↾ H . From (3.1) it can be concluded that ker R ∗ is equal to theorthogonal complement of (ran R ) c , where M c denotes the closure of a subset M of a topological space. It follows that R ∗ is one-to-one if and only if ran R is densein H and that ran R ∗ = R ∗ (ran R ) c (3.2)Therefore, we can define a generalized inverse R ∗ [ − of R ∗ by R ∗ [ − x ′ := ( R ∗ ↾ (ran R ) c ) − x ′ , x ′ ∈ ran R ∗ . Lemma 3.1.
Let R ∈ L ( X, H ) . An element x ′ ∈ X ′ belongs to ran R ∗ if andonly if the following conditions are satisfied: (i) If x ∈ ker R , then h x ′ , x i = 0 . (ii) sup x ∈ X |h x ′ , x i| k Rx k H < ∞ (with convention := 0 at the left-hand side).Proof. If x ′ ∈ R ∗ h for some h ∈ H , then |h x ′ , x i| = |h R ∗ h, x i| = | ( h | Rx ) | ≤ k h kk Rx k , which yields (i) and (ii). Conversely, assume that (i) and (ii) are satisfied for some x ′ ∈ X ′ . Set ϕ ( Rx ) := h x ′ , x i , x ∈ X . Because of (i) ϕ is correctly defined and(ii) implies that ϕ is continuous, so that ϕ is a continuous antilinear functionalon ran R . Thus, there exists h ∈ H such that h x ′ , x i = ( h | Rx ) = h R ∗ h, x i for all x ∈ X , which yields x ′ = R ∗ h ∈ ran R ∗ . (cid:3) Definition 3.2.
Let A ∈ L ( X, X ′ ). A pair ( R, H ) of a Hilbert space H and anoperator R ∈ L ( X, H ) is called a square root of A if A = R ∗ R , and a minimalsquare root if, addionally, ran R is dense in H .Note that there exists a square root of A if and only if there exists a minimalone. The following result is basic to our considerations and generalizes the factconcerning the existence of a square root of a non-negative selfadjoint operatorin a Hilbert space. Its well known short proof is recapitulated for convenience ofthe reader. Theorem 3.3.
An operator A ∈ L ( X, X ′ ) possesses a square root if and only ifit is non-negative.Proof. Let A ∈ L ≥ ( X, X ′ ). Cauchy’s inequality (2.2) implies that N := { x ∈ X : h Ax, x i = 0 } is a subspace of X . Define an inner product on the quotient space X/N by( x + N | x + N ) := h Ax , x i , x , x ∈ X, J. FRIEDRICH, M. G ¨UNTHER, and L. KLOTZ and denote the completion of the corresponding inner product space by H . Set Rx := x + N , x ∈ X . It follows R ∈ L ( X, H ), (ran R ) c = H , and h Ax , x i = ( Rx | Rx ) = h R ∗ Rx , x i , x , x ∈ X. Therefore, (
R, H ) is a minimal square root of A . The ”only if”-part of the asser-tion is obvious. (cid:3) The notion of a square root of a non-negative operator acting between spacesmore general than Hilbert spaces was discussed and applied by many authors.Most of them deal with a topological space X and in this case continuity prob-lems arise additionally. Some properties of square roots for operators of specialtype were obtained by Va˘ınberg and Engel’son, cf. [23]. For a Banach space X ,the construction of the proof of the preceding theorem was published as an ap-pendix to [21] and attributed to Chobanyan, see also [15] and [22]. Another butrelated construction was proposed by Sebesty´en [16], cf. [19]. G´orniak [7] andG´orniak and Weron [8] dealt with the existence of a continuous square root if X isa topological linear space. G´orniak, Makagon and Weron [9] investigated squareroots of non-negative operator-valued measures. Pusz and Woronowicz [14] ex-tended the construction of the proof of Theorem 3.3 to pairs of non-negativesequilinear forms, cf. [20] for further generalizations. Lemma 3.4. If A ∈ L ( X, X ′ ) and ( R, H ) is a square root of A , then ker R = ker A = { x ∈ X : h Ax, x i = 0 } . Proof.
The result follows from a chain of conclusions: h Ax, x i = 0 ⇒ h R ∗ Rx, x i = 0 ⇒ ( Rx | Rx ) = 0 ⇒ Rx = 0 , and conversely Rx = 0 ⇒ R ∗ Rx = 0 ⇒ Ax = 0 ⇒ h Ax, x i = 0 . (cid:3) The preceding results can be used to derive a version of a part of Douglas’theorem [5], cf. [18].
Proposition 3.5.
Let
A, D ∈ L ≥ ( X, X ′ ) and ( R A , H A ) and ( R D , H D ) be squareroots of A and D , resp.. The following assertions are equivalent: (i) A ≤ α D for some α ∈ [0 , ∞ ) , (ii) there exists a bounded operator W ∈ L ( H A , H D ) with operator norm k W k ≤ α and such that R ∗ A = R ∗ D W .If (i) or (ii) are satisfied, there exists a unique W so that W ⊆ (ran R D ) c . More-over, ker W = ker R ∗ A for this operator W .Proof. Since R ∗ A = R ∗ D W yields R A = R ∗ A ′ ↾ X = W ′ R ∗ D ′ ↾ X = W ∗ R D by (CN), from(ii) it follows h Ax, x i = k R A x k = k W ∗ R D x k ≤ α k R D x k = α h Dx, x i , x ∈ X, hence, (i). Let W j ∈ L ( H A , H D be such that R ∗ A = R ∗ D W j and ran W j ⊆ (ran R D ) c , j = 1 ,
2. Then ran( W − W ) ⊆ ker R ∗ D and ran( W − W ) ⊆ (ran R D ) c , which shows that W = W . Now assume that (i) is true. One has GENERALIZED SCHUR COMPLEMENT 5 ker R D ⊆ ker R A by Lemma 3.4, hence, ran R ∗ A ⊆ ran R ∗ D by Lemma 3.1. The op-erator W := R ∗ [ − D R ∗ A ∈ L ( H A , H D ) satisfies R ∗ D W = R ∗ A , ker W = ker R ∗ A and ran W ⊆ (ran R D ) c . The inclusion ran R ∗ [ − D ⊆ (ran R D ) c implies that W ∗ h = R ∗′ A ( R ∗ [ − D ) ∗ h = 0 if h is orthogonal to ran R D . Therefore, from k W ∗ R D x k = k R A x k = h Ax, x i ≤ α h Dx, x i = α k R D x k , x ∈ X, one can conclude that k W k = k W ∗ k ≤ α . (cid:3) As a by-product of Proposition 3.5 we obtain the following corollary.
Corollary 3.6.
Let A , D , ( R A , H A ) , and ( R D , H D ) be as in Proposition 3.5. (i) If A ≤ α D for some α ∈ [0 , ∞ ) , then ran R ∗ A ⊆ ran R ∗ D . (ii) If β D ≤ A ≤ α D for some α, β ∈ (0 , ∞ ) , then ran R ∗ A = ran R ∗ D . (iii) If ( S A , G A ) is a square root of A , then ran R ∗ A = ran S ∗ A . Corollary 3.7.
Let H j be Hilbert spaces and R j ∈ L ( X, H j ) , j = 1 , . If ( R, H ) is a square root of the non-negative operator A := R ∗ R + R ∗ R , then ran R ∗ = ran R ∗ + ran R ∗ .Proof. Let G be the orthogonal sum of H and H and S ∈ L ( X, G ) be definedby S = (cid:16) R R (cid:17) . Since S ∗ = ( R ∗ , R ∗ ) and S ∗ S = A , we get that ran S ∗ = ran R ∗ +ran R ∗ and that ( S, G ) is a square root of A . Now apply Corollary 3.6 (iii). (cid:3) Lemma 3.8.
Let ( R, H ) be a square root and ( S, G ) a minimal square root of A ∈ L ≥ ( X, X ′ ) . There exists an isometry U ∈ L ( G, H ) such that U S = R .Proof. By Lemma 3.4 there exists an operator ˜ U satisfying ˜ U Sx = Rx , x ∈ X .From k ˜ U Sx k = k Rx k = h Ax, x i = k Sx k it follows that ˜ U is isometric and canbe extended to an isometry U ∈ L ( G, H ). (cid:3) Generalized Schur complements and shorted operators ofoperators of positive type
Let X und Y be linear spaces. Definition 4.1.
A pair (
A, B ) of an operator A ∈ L ≥ ( X, X ′ ) and B ∈ L ( Y, X ′ )is called a positive pair if ran B ⊆ ran R ∗ for some square root (and, hence, forall square roots) ( R, H ) of A .The following criterion is an immediate consequence of Lemma 3.1. Lemma 4.2.
Let A ∈ L ≥ ( X, X ′ ) and B ∈ L ( Y, X ′ ) . The pair ( A, B ) is apositive pair if and only if for all y ∈ Y the following conditions are satisfied: (i) If x ∈ ker A , then h By, x i = 0 . (ii) sup x ∈ X |h By, x i| h Ax, x i < ∞ (with convention := 0 ). According to (CN), the space X can be considered as a subspace of the domainof B ′ . To abbreviate the notation we set B ∼ := B ′ ↾ X . J. FRIEDRICH, M. G ¨UNTHER, and L. KLOTZ
Note that h By, x i = h B ∼ x, y i , x ∈ X , y ∈ Y . Thus condition (i) of Lemma 4.2 isequivalent to the inclusion ker A ⊆ ker B ∼ .If ( A, B ) is a positive pair and (
R, H ) is a square root of A , the operators T := R ∗ [ − B (4.1)and ω ( A, B ) := T ∗ T = ( R ∗ [ − B ) ∗ R ∗ [ − B (4.2)can be defined. Note that B = R ∗ T . The following lemma is obvious. Lemma 4.3. If ( A, B ) is a positive pair and ( R, H ) is a square root of A , thenfor all y ∈ Y inf {k T y − Rx k : x ∈ X } = 0 . Equivalently, ran T ⊆ (ran R ) c . (cid:3) Recall that the dual space of X × Y can be written as a Cartesian product( X × Y ) ′ = X ′ × Y ′ , where h· , ·i X × Y = h· , ·i X + h· , ·i Y . Also, it should not causeconfusion if we identify the subspace X × { } of X × Y with X . An operator A of L (cid:0) X × Y, ( X × Y ) ′ (cid:1) can be represented as a 2 × A = (cid:16) A BC D (cid:17) , where A ∈ L ( X, X ′ ), B, C ∈ L ( Y, X ′ ), D ∈ L ( Y, Y ′ ). It is not hard to see that A isHermitian if and only if A and D are Hermitian and C = B ∼ . To abbreviate thenotation we set L h (cid:0) X × Y, ( X × Y ) ′ (cid:1) =: L h and L ≥ (cid:0) X × Y, ( X × Y ) ′ (cid:1) =: L ≥ . Definition 4.4.
An operator (cid:16)
A BB ∼ D (cid:17) ∈ L h is called an operator of positivetype if ( A, B ) is a positive pair. The set of operators of positive type is denotedby L + (cid:0) X × Y, ( X × Y ) ′ (cid:1) =: L + . Definition 4.5.
Let A = (cid:16) A BB ∼ D (cid:17) ∈ L + . The operator σ ( A ) := D − ω ( A, B )is called a generalized Schur complement of A and the operator S ( A ) := (cid:18) σ ( A ) (cid:19) is called a shorted operator.The following result is a generalization of [2, Corollary 1 to Theorem 3]. Proposition 4.6. If A = (cid:16) A BB ∼ D (cid:17) ∈ L + , then ran A ∩ Y ′ ⊆ ran S ( A ) .Proof. Let y ′ ∈ Y ′ be such that A (cid:16) xy (cid:17) = (cid:16) y ′ (cid:17) for some (cid:16) xy (cid:17) ∈ X × Y . Let ( R, H )be a minimal square root of A . Since A = (cid:18) R ∗ R R ∗ TT ∗ R T ∗ T (cid:19) + S ( A ) , one has R ∗ Rx + R ∗ T y = 0, hence, Rx + T y = 0 and T ∗ Rx + T ∗ T y = 0, whichyields (cid:18) y ′ (cid:19) = S ( A ) (cid:18) xy (cid:19) ∈ ran S ( A ) . (cid:3) GENERALIZED SCHUR COMPLEMENT 7
The next result is a simple but useful consequence of Lemma 4.3.
Proposition 4.7.
Let A = (cid:16) A BB ∼ D (cid:17) ∈ L + . For (cid:16) xy (cid:17) ∈ X × Y , (cid:28) S ( A ) (cid:18) xy (cid:19) , (cid:18) xy (cid:19)(cid:29) = inf z ∈ X (cid:28) A (cid:18) x − zy (cid:19) (cid:18) x − zy (cid:19)(cid:29) . (4.3) Particularly, σ ( A ) and S ( A ) do not depend on the choice of the square root of A .Proof. Since (4.3) is independent of x ∈ X , it is enough to prove it for x = 0.From Lemma 4.3 it followsinf z ∈ X (cid:28) A (cid:18) − zy (cid:19) , (cid:18) − zy (cid:19)(cid:29) = inf z ∈ X (cid:28)(cid:18) R ∗ R R ∗ TT ∗ R T ∗ T (cid:19) (cid:18) − zy (cid:19) , (cid:18) − zy (cid:19)(cid:29) + (cid:28) S ( A ) (cid:18) y (cid:19) , (cid:18) y (cid:19)(cid:29) = inf z ∈ X k T y − Rz k + (cid:28) S ( A ) (cid:18) y (cid:19) , (cid:18) y (cid:19)(cid:29) = (cid:28) S ( A ) (cid:18) y (cid:19) , (cid:18) y (cid:19)(cid:29) . (cid:3) Corollary 4.8. (i) If A ∈ L + , then S ( A ) ≤ A and ker A ⊆ ker S ( A ) .(ii) If A , A ∈ L + and A ≤ A , then S ( A ) ≤ S ( A ) .Proof. The first assertion of (i) as well as (ii) are immediately clear from Propo-sition 4.7. To prove the second assertion of (i), let (cid:16) xy (cid:17) ∈ ker A . If z ∈ X , onehas (cid:28) A (cid:18) x − zy (cid:19) , (cid:18) x − zy (cid:19)(cid:29) = h Az, z i ≥ , which implies that the infimum at the right hand side of (4.3) is equal to 0. Since S ( A ) ≤ A and (cid:28)(cid:0) A − S ( A ) (cid:1) (cid:18) xy (cid:19) , (cid:18) xy (cid:19)(cid:29) = 0 , it follows (cid:16) xy (cid:17) ∈ ker (cid:0) A − S ( A ) (cid:1) by Lemma 3.4, hence, (cid:16) xy (cid:17) ∈ ker S ( A ). (cid:3) Corollary 4.9. If A ∈ L ≥ , then S ( A ) ∈ L ≥ . (cid:3) Further applications of square roots
First we express the generalized Schur complement of an operator of L ≥ withthe aid of its square root and derive a range description, cf. [2, Corollary 4 toTheorem 1]. Let A ∈ L ≥ and ( R, H ) be a square root of A . Let L be theorthogonal complement of ( RX ) c and P be the orthoprojection onto L . Notethat L can be characterized by L = { h ∈ H : R ∗ h ∈ Y ′ } , which yields R ∗ L =ran R ∗ ∩ Y ′ . Proposition 5.1. If A ∈ L ≥ , then S ( A ) = R ∗ P R . J. FRIEDRICH, M. G ¨UNTHER, and L. KLOTZ
Proof.
Let (cid:16) xy (cid:17) ∈ X × Y . An application of (4.3) gives (cid:28) S ( A ) (cid:18) xy (cid:19) , (cid:18) xy (cid:19)(cid:29) = inf z ∈ X (cid:13)(cid:13)(cid:13)(cid:13) R (cid:18) xy (cid:19) − R (cid:18) z (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) , which shows that D S ( A ) (cid:16) xy (cid:17) , (cid:16) xy (cid:17)E is the squared distance of R (cid:16) xy (cid:17) to RX .Therefore, (cid:28) S ( A ) (cid:18) xy (cid:19) , (cid:18) xy (cid:19)(cid:29) = (cid:13)(cid:13)(cid:13)(cid:13) P R (cid:18) xy (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) = (cid:28) R ∗ P R (cid:18) xy (cid:19) , (cid:18) xy (cid:19)(cid:29) and the assertion follows from the polarization identity. (cid:3) Proposition 5.2. If A ∈ L ≥ and ( R, H ) and ( S, G ) are square roots of A and S ( A ) , resp., then ran S ∗ = ran R ∗ ∩ Y ′ .Proof. Setting R X := R ↾ X and R Y := R ↾ Y , we get A = (cid:18) R ∗ X R ∗ Y (cid:19) ( R X R Y ) = (cid:18) R ∗ X R X R ∗ X R Y R ∗ Y R X R ∗ Y R Y (cid:19) , hence, σ ( A ) = R ∗ Y R Y − ( R ∗ [ − X R ∗ X R Y ) ∗ R ∗ [ − X R ∗ X R Y = R ∗ Y P R Y since R ∗ [ − X R ∗ X = I − P . Thus S ( A ) = R ∗ P R and (
P R, H ) is a square root of S ( A ). If (cid:16) y ′ (cid:17) ∈ X ′ × Y ′ is such that R ∗ h = (cid:16) y ′ (cid:17) for some h ∈ H , then R ∗ X h = 0,hence, P h = h and ( P R ) ∗ h = R ∗ h = (cid:16) y ′ (cid:17) , which implies that ran R ∗ ∩ Y ′ ⊆ ran( P R ) ∗ = ran S ∗ by Corollary 3.6 (iii). Since, obviously, ran S ∗ ⊆ Y ′ andran S ∗ ⊆ ran R ∗ by Corollaries 4.8 (i) and 3.6 (i), the assertion is proved. (cid:3) Our next result is a generalization of the Crabtree-Haynsworth quotient formula[4]. To give it a nice form let us denote σ ( A ) =: A /A . Proposition 5.3.
Let X , Y , and Z be linear spaces, D := A B B X B ∼ D B Y B ∼ X B ∼ Y D ∈ L ≥ ( X × Y × Z, X ′ × Y ′ × Z ′ ) , and A := (cid:16) A BB ∼ D (cid:17) . The operator A /A is the left upper corner of D /A and D /A . A /A = D / A . Proof.
Let (
R, H ) be a minimal square root of A , R X := R ↾ X , R Y := R ↾ Y , E := ( R ∗ ) − (cid:18) B X B Y (cid:19) , GENERALIZED SCHUR COMPLEMENT 9 hence, R ∗ X E = B X , R ∗ Y E = B Y . From R ∗ [ − X R ∗ X = I − P one obtains D /A = (cid:18) R ∗ Y R Y R ∗ Y EE ∗ R Y D (cid:19) − (cid:0) R ∗ [ − X ( R ∗ X R Y , R ∗ X E ) (cid:1) ∗ R ∗ [ − X ( R ∗ X R Y , R ∗ X E )= (cid:18) R ∗ Y P R Y R ∗ Y P EE ∗ P R Y D − E ∗ ( I − P ) E (cid:19) and A /A = R ∗ Y R Y − (cid:0) R ∗ [ − X R ∗ X R Y (cid:1) ∗ R ∗ [ − X R ∗ X R Y = R ∗ Y P R Y , which shows that A /A is the left upper corner of D /A . Since ( P R Y , H ) is asquare root of A /A , one can compute D /A . A /A = D − E ∗ ( I − P ) E − (cid:0) ( P R Y ) ∗ [ − R ∗ Y P E (cid:1) ∗ ( P R Y ) ∗ [ − R ∗ Y P E = D − E ∗ ( I − P ) E − E ∗ QE, where Q denotes the orthoprojection onto (ran P R Y ) c . Comparing this with D /A = D − (cid:0) ( R ∗ ) − R ∗ E (cid:1) ∗ ( R ∗ ) − R ∗ E = D − E ∗ E, we can conclude that the assertion will be proved if we can show that the restric-tion of I − P + Q to ran R is the identity. If h ∈ ran R , then h = R X x + R Y y = R X x + ( I − P ) R Y y + P R Y y for some (cid:16) xy (cid:17) ∈ X × Y . Since R X x + ( I − P ) R Y y ∈ (ran R X ) c , there exists asequence { x n } n ∈ N of elements of X such that lim n →∞ R X x n = R X x + ( I − P ) R Y y .For h n := R X x n + P R Y y , we have( I − P + Q ) h n = ( I − P + Q )( R X x n + P R Y y ) = R X x n + P R Y y = h n and therefore( I − P + Q ) h = lim n →∞ ( I − P + Q ) h n = lim n →∞ ( R X x n + P R Y y ) = R X x + R Y y = h. (cid:3) We conclude this section with a criterion for non-negativity of operators of L h . Proposition 5.4.
Let A = (cid:16) A BB ∼ D (cid:17) ∈ L h . The operator A is non-negative ifand only if the following two conditions are satisfied: (i) The operators A and D are non-negative. (ii) For any square roots ( R A , H A ) and ( R D , H D ) of A and D , resp., thereexists a contraction K ∈ L ( H D , H A ) such that B = R ∗ A KR and ran K ⊆ (ran R A ) c .Proof. If A is non-negative, assertion (i) is trivial. To prove (ii) let ( R, H ) be asquare root of A and R X := R ↾ X , R Y = R ↾ Y , hence A = (cid:18) R ∗ X R X R ∗ X R Y R ∗ Y R X R ∗ Y R Y (cid:19) . Let ( S A , G A ) and ( S D , G D ) be minimal square roots of A and D , resp.. Accordingto Lemma 3.8 there exist isometries U A ∈ L ( G A , H A ), V A ∈ L ( G A , H ), U D ∈ L ( G D , H D ), V D ∈ L ( G D , H ) satisfying U A S A = R A , V A S A = R X , U D S D = R D , V D S D = R Y . It follows B = R ∗ X R Y = R ∗ A U A V ∗ A V D U ∗ D R D = R ∗ A KR D , where K := U A V ∗ A V D U ∗ D ∈ L ( H D , H A ) is a contraction with ran K ⊆ (ran R A ) c .Conversely, if (i) and (ii) are satisfied, then A = (cid:18) R ∗ A R A R ∗ A KR D ( R ∗ A KR D ) ∼ R ∗ D R D (cid:19) . Since ( R ∗ A KR D ) ∼ = ( R ′ A KR D ) ′ ↾ X = R ′ D K ′ R ′′ A ↾ X = R ∗ D K ∗ R A , one obtains A = (cid:18) R ∗ A R ∗ D (cid:19) (cid:18) I KK ∗ I (cid:19) (cid:18) R A R D (cid:19) , which implies that A is non-negative. (cid:3) Albert’s theorem
An application of Proposition 5.4 leads to a generalization of an importantcriterion for non-negativity [1], which is often called Albert’s theorem in matrixtheory. It should be mentioned that Shmulyan [17, Theorem 1.7] had proved asimilar assertion even for bounded operators in Hilbert spaces ten years earlier.We also mention the papers [3] and [10].
Theorem 6.1.
An operator A = (cid:16) A BB ∼ D (cid:17) ∈ L h is non-negative if and only ifit is of positive type and σ ( A ) is non-negative.Proof. If A ∈ L ≥ , Proposition 5.4 implies that A is of positive type and R ∗ [ − A B = KR D for some contraction K ∈ L ( H D , H A ). It follows σ ( A ) = R ∗ D R D − ( KR D ) ∗ KR D = R ∗ D ( I − K ∗ K ) R D ≥ A ∈ L + and σ ( A ) ∈ L ≥ ( Y, Y ′ ). If ( R A , H A ) and ( R D , H D ) aresquare roots of A and D , resp., one has k R ∗ [ − A By k = h ω ( A, B ) y, y i ≤ h Dy, y i = k R D y k , y ∈ Y, which yields KR D = R ∗ [ − B , hence, R ∗ A KR D = B for some contraction K ∈ L ( H D , H A ). An application of Proposition 5.4 completes the proof. (cid:3) The preceding theorem can be used to study the set L ≥ as well as the set L + and to establish interrelations between these two sets. A first result is theinclusion L ≥ ⊆ L + . For a positive pair ( A, B ) set A ex := (cid:18) A BB ∼ ω ( A, B ) (cid:19) ∈ L + . Corollary 6.2.
Two operators A ∈ L ( X, X ′ ) and B ∈ L ( Y, X ′ ) form a positivepair if and only if the set A := n A ∈ L ≥ : A = (cid:16) A BB ∼ D (cid:17) for some D ∈ L ≥ ( Y, Y ′ ) o GENERALIZED SCHUR COMPLEMENT 11 is non-empty. If ( A, B ) is a positive pair, the operator A ex is the minimal elementof A . (cid:3) Corollary 6.3. If A ∈ L ≥ , the set A := (cid:8) A ∈ L ≥ : A ≤ A and X ⊆ ker A (cid:9) is non-empty and S ( A ) is its maximal element.Proof. Corollaries 3.8 (i) and 3.9 imply that S ( A ) ∈ A . If A = (cid:16) A BB ∼ D (cid:17) and A ∈ A , then A has representation A = (cid:16) D (cid:17) and D − ω ( A, B ) − D ≥ A ≤ S ( A ) by Theorem 6.1. (cid:3) Corollary 6.4.
Let A = (cid:16) A BB ∼ D (cid:17) ∈ L h . The operator A belongs to L + if andonly if there exists an operator A ∈ L h satisfying X ⊆ ker A and A ≤ A .Proof. If A ∈ L + , the operator A := S ( A ) has all properties claimed. Con-versely, if there exists an operator A satisfying all conditions, it has the form A = (cid:16) D (cid:17) , where D ∈ L ( Y, Y ′ ) and A − A = (cid:16) A BB ∼ D − D (cid:17) ∈ L ≥ . Itfollows from Theorem 6.1 that ( A, B ) is a positive pair, hence, A ∈ L + . (cid:3) Another application of Theorem 6.1 gives an expression of the supremum oc-curing in Lemma 4.2.
Corollary 6.5. If ( A, B ) is a positive pair, then sup x ∈ X |h By, x i| h Ax, x i = h ω ( A, B ) y, y i , y ∈ Y. (6.1) Proof.
Let y ∈ Y . Since A ex ∈ L ≥ by Corollary 6.2, one has |h By, x i| ≤ h Ax, x ih ω ( A, B ) y, y i , which yields |h By, x i| h Ax, x i ≤ h ω ( A, B ) y, y i , x ∈ X, if one takes into account the convention := 0. Thus, (6.1) has been proved if T y = R ∗ [ − By = 0, where ( R, H ) is a minimal square root of A . Now assume that T y = 0. There exists a sequence { x n } n ∈ N of elements of X such that Rx n = 0, n ∈ N , and lim n →∞ Rx n = T y . It followslim n →∞ |h By, x n i| h Ax n , x n i = lim n →∞ |h R ∗ T y, x n i| h R ∗ Rx n , x n i = lim n →∞ | ( T y | Rx n ) | k Rx n k = k T y k = h ω ( A, B ) y, y i . (cid:3) Corollary 6.6.
Let ( A j , B j ) with A j ∈ ( X, X ′ ) , B j ∈ ( Y, X ′ ) , j = 1 , , be positivepairs. Then ( A + A , B + B ) is a positive pair and ω ( A + A , B + B ) ≤ ω ( A , B ) + ω ( A , B ) . (6.2) Proof.
Since the operators ( A j ) ex , j = 1 , (cid:18) A + A B + B B ∼ + B ∼ ω ( A , B ) + ω ( A , B ) (cid:19) ∈ L ≥ , hence, (6.2) by Corollary 6.2. (cid:3) Corollary 6.7. If A j ∈ L + , j = 1 , , then A + A ∈ L + and S ( A ) + S ( A ) ≤ S ( A + A ) . (cid:3) A subset A of L h is called bounded below if there exists A ∈ L h such that A ≤ A for all A ∈ A . An operator A ∈ L h is called an infimum of A if thefollowing conditions are satisfied:a) A ≤ A for all A ∈ A ,b) A ≤ A for all A ∈ L h such that A ≤ A , A ∈ A .If an infimum of A exists, it is unique. Recall that any set A , which is boundedfrom below and directed downwards (i.e. for all A , A ∈ A there exists A ∈ A such that A ≤ A and A ≤ A ), possesses an infimum. Particularly, if { A n } n ∈ N is a decreasing sequence of operators of L h , which is bounded from below, thereexists an infimum A and h A z , z i = lim n →∞ h A n z , z i for all z , z ∈ X × Y . Corollary 6.8.
Let A be a subset of L h , which has an infimum A . The operator A belongs to L + if and only if the set S ( A ) := { S ( A ) : A ∈ A } is boundedfrom below. In this case S ( A ) is the infimum of S ( A ) .Proof. If A ∈ L + , the set S ( A ) is bounded from below since S ( A ) ≤ S ( A ), A ∈ A , by Corollary 4.8 (ii). Conversely, assume that there exists A ∈ L h such that A ≤ S ( A ) for all A ∈ A . It follows − A ≥ − S ( A ), which yields − A ∈ L + by Corollary 6.4 and S ( − A ) ≥ S ( − S ( A )) = − S ( A ), hence, − S ( − A ) ≤ S ( A ) ≤ A , A ∈ A , by Corollary 4.8. One obtains − S ( − A ) ≤ A and therefore A ∈ L + by Corollary 6.4. Moreover, S ( A ) ≤ S ( A ), A ∈ A , and A = − ( − A ) ≤ − S ( − A ) = S (cid:0) − S ( − A ) (cid:1) ≤ S ( A )by Corollary 4.8, which implies that S ( A ) is the infimum of S ( A ). (cid:3) Extremal operators
An operator A ∈ L + was called an extremal operator by M.G. Kre˘ın [11] if S ( A ) = 0. Since A = S ( A ) + A ex , an operator is extremal if and only if it hasthe form A = A ex = (cid:18) A BB ∼ ω ( A, B ) (cid:19) for some positive pair ( A, B ). Particularly, any extremal operator is non-negative.Applying Proposition 4.7 we can give several criteria for an operator to be ex-tremal.
Lemma 7.1.
Let A ∈ L ≥ . The following assertions are equivalent: (i) The operator is extremal.
GENERALIZED SCHUR COMPLEMENT 13 (ii)
For all (cid:16) xy (cid:17) ∈ X × Y and arbitrary ε > there exists z ∈ X such that D A (cid:16) x − zy (cid:17) , (cid:16) x − zy (cid:17)E < ε. (iii) For any square root ( R, H ) of A the spaces ( RX ) c and (ran R ) c coincide. (iv) For any square root ( R, H ) of A one has ran R ∗ ∩ Y ′ = { } .Proof. The equivalence of (i) and (ii) is an immediate consequence of (4.3). Toprove (i) ⇔ (iii), choose a minimal square root ( R, H ) of A and let L and P bedefined as in Proposition 5.1. Then S ( A ) = R ∗ P R = 0 if and only if P = 0or, equivalently, L = { } , which in turn is equivalent to ( RX ) c = (ran R ) c . Theequivalence of (iii) and (iv) follows from the equality R ∗ L = ran R ∗ ∩ Y ′ . (cid:3) Let (
A, B ) be a positive pair and (
R, H ) a square root of A . Recall the notation(4.1) of the operator T := R ∗ [ − B . Moreover, let P B be the orthoprojection onto(ran T ) c . Since the operator A ex is non-negative, from Theorem 6.1 one canconclude that ( ω ( A, B ) , B ∼ ) is a positive pair as well changing the roles of X and Y . Thus, the operators T ∗ [ − B ∼ and ω (cid:0) ω ( A, B ) , B ∼ (cid:1) = (cid:2) T ∗ [ − B ∼ (cid:3) ∗ T ∗ [ − B ∼ can be defined. Lemma 7.2.
The equalities T ∗ [ − B ∼ = P B R and ω (cid:0) ω ( A, B ) , B ∼ (cid:1) = R ∗ P B R hold true.Proof. The second equality is an immediate consequence of the first one. To provethe first equality we shall show that (cid:0) T ∗ [ − B ∼ x | h (cid:1) = ( P B Rx | h ) for all x ∈ X and h ∈ H . (7.1)Since ran T ∗ [ − B ∼ ⊆ (ran T ) c it is enough to prove (7.1) for x ∈ X and h ∈ ran T .If h = T y for some y ∈ Y , we get (cid:0) T ∗ [ − B ∼ x | h (cid:1) = (cid:0) T ∗ [ − B ∼ x | T y (cid:1) = (cid:10) T ∗ T ∗ [ − B ∼ x, y (cid:11) = h B ∼ x, y i and ( P B Rx | h ) = ( P B Rx | T y ) = ( Rx | T y )= h R ∗ T y, x i = h By, x i = h B ∼ x, y i , hence, (7.1). (cid:3) From Corollary 6.2 it follows that ω (cid:0) ω ( A, B ) , B ∼ (cid:1) is a minimal element of theset n A ∈ L ( X, X ′ ) : (cid:16) A BB ∼ ω ( A, B ) (cid:17) ∈ L ≥ o Note also that ω (cid:16) ω (cid:0) ω ( A, B ) , B ∼ (cid:1) , B (cid:17) = ω ( A, B ) , cf. [12, Proposition 1.4 (A)]. We call an extremal operator A ex = (cid:16) A BB ∼ ω ( A, B ) (cid:17) doubly extremal if ω (cid:0) ω ( A, B ) , B ∼ (cid:1) = A . In the case of bounded operators on Hilbert spaces the remaining results of thepresent section were proved by Pekarev and Shmulyan [13] and partly rediscoveredby Niemiec [12]. We mention that Niemiec’s proofs are based on Douglas’ theoremand do not make explicit use of 2 × Proposition 7.3.
An operator A ex is doubly extremal if and only if ker T ∗ = ker R ∗ (7.2) for any square root ( R, H ) of A .Proof. According to Lemma 7.2, A is doubly extremal if and only if R ∗ P B R = R ∗ R . If ker T ∗ = ker R ∗ or, equivalently, (ran T ) c = (ran R ) c , it follows P B R = R , hence, R ∗ P B R = R ∗ R . Conversely, assume R ∗ P B R = R ∗ R , which yields k P B Rx k = k Rx k , x ∈ X , hence, (ran R ) c ⊆ ran P B and ker P B ⊆ ker R ∗ . Sinceran P B = (ran T ) c or ker P B = ker T ∗ , we getker T ∗ ⊆ ker R ∗ . (7.3)On the other hand, if h ∈ ker R ∗ , then h is orthogonal to (ran R ) c and0 = (cid:0) h | T y (cid:1) = (cid:10) T ∗ h, y (cid:11) , y ∈ Y, thus, ker R ∗ ⊆ ker T ∗ . Taking into account (7.3) we obtain the desired equality. (cid:3) The preceding assertion shows that the equality (7.2) does not depend on thechoice of the square root (
R, H ) of A and that in the case of a minimal square rootthe operator A ex is doubly extremal if and only if ker T ∗ = { } . Moreover, writing(7.2) in the equivalent form (ran T ) c = (ran R ) c we obtain a generalization of [13,Theorem 1.6]. It means that A ex is doubly extremal if and only if the inverseimage of ran B under the map R ∗ is dense in H . Corollary 7.4. If ran R ∗ = ran B , the operator A ex is doubly extremal. (cid:3) To give another criterion for A ex to be doubly extremal we equip the space Y ′ with σ ( Y ′ , Y )-topology, i.e. the smallest topology such that for arbitrary y ∈ Y ,the functional y ′
7→ h y ′ , y i is continuous on Y ′ . Let ( A, B ) be a positive pair and(
R, H ) a square root of A . Denote by H the subspace of all h ∈ H such thatthere exists a sequence { x n } n ∈ N of elements of X with the following properties:a) lim n →∞ Rx n = h with respect to the norm topology of H ,b) lim n →∞ B ∼ x n = 0 with respect to the σ ( Y ′ , Y )-topology. Lemma 7.5.
The space H is equal to (ran R ) c ∩ ker T ∗ .Proof. An element h ∈ H belongs to H if and only if there exists a sequence { x n } n ∈ N of elements of X such that lim n →∞ Rx n = h and for all y ∈ Y , (cid:10) T ∗ h, y (cid:11) = lim n →∞ (cid:0) Rx n | T y (cid:1) = lim n →∞ ( x n | By ) = lim n →∞ ( B ∼ x n | y ) = 0 . (cid:3) Proposition 7.6.
An operator A ex is doubly extremal if and only if H = { } . GENERALIZED SCHUR COMPLEMENT 15
Proof.
Since H = { } if and only if ker R ∗ = ker T ∗ by Lemma 7.5, the assertionfollows from Proposition 7.3. (cid:3) Corollary 7.7.
If an operator A ex is doubly extremal, then ker A = ker B ∼ . If ker A = ker B ∼ and ran R is closed, then A ex is doubly extremal.Proof. If A ex is doubly extremal, then (cid:0) T ∗ [ − B ∼ , H (cid:1) is a square root of A , hence,ker B ∼ ⊆ ker T ∗ [ − B ∼ = ker A by Lemma 3.4. The first assertion of the corollaryfollows since ker A ⊆ ker B ∼ by Lemma 4.2. Now assume that ker A = ker B ∼ and ran R is closed. If h ∈ H , there exist x ∈ X and a sequence { x n } n ∈ N ofelements of X such that h = Rx = lim n →∞ Rx n and lim n →∞ h B ∼ x n , y i = 0 for all y ∈ Y . It follows h B ∼ x, y i = (cid:10) R ∗ T y, x (cid:11) = (cid:0) Rx | T y (cid:1) = lim n →∞ (cid:0) Rx n | T y (cid:1) = lim n →∞ h By, x n i = lim n →∞ h B ∼ x n , y i = 0 , y ∈ Y, which implies that x ∈ ker B ∼ = ker A = ker R and h = 0. An application ofProposition 7.6 completes the proof. (cid:3) References
1. Arthur Albert,
Conditions for positive and nonnegative definiteness in terms of pseudoin-verses , SIAM J. Appl. Math. (1969), 434–440.2. W. N. Anderson, Jr. and G. E. Trapp, Shorted operators. II , SIAM J. Appl. Math. (1975), 60–71.3. T. Andˆo, Truncated moment problems for operators , Acta Sci. Math. (Szeged) (1970),319–334.4. Douglas E. Crabtree and Emilie V. Haynsworth, An identity for the Schur complement ofa matrix , Proc. Amer. Math. Soc. (1969), 364–366.5. R. G. Douglas, On majorization, factorization, and range inclusion of operators on Hilbertspace , Proc. Amer. Math. Soc. (1966), 413–415.6. Bernd Fritzsche, Bernd Kirstein, and Lutz Klotz, Completion of non-negative block opera-tors in Banach spaces , Positivity (1999), no. 4, 389–397.7. J. G´orniak, Locally convex spaces with factorization property , Colloq. Math. (1984),no. 1, 69–79.8. J. G´orniak and A. Weron, Aronszajn-Kolmogorov type theorems for positive definite kernelsin locally convex spaces , Studia Math. (1980/81), no. 3, 235–246.9. Janusz G´orniak, Andrzej Makagon, and Aleksander Weron, An explicit form of dilation the-orems for semispectral measures , Prediction theory and harmonic analysis, North-Holland,Amsterdam-New York, 1983, pp. 85–111.10. Hans-Peter H¨oschel, ¨Uber die Pseudoinverse eines zerlegten positiven linearen Operators ,Math. Nachr. (1976), 167–172.11. M. G. Kre˘ın, The theory of self-adjoint extensions of semi-bounded Hermitian transforma-tions and its applications. I , Rec. Math. [Mat. Sbornik] N.S. (1947), 431–495.12. Piotr Niemiec,
Generalized absolute values and polar decompositions of a bounded operator ,Integral Equations Operator Theory (2011), no. 2, 151–160.13. `E. L. Pekarev and Yu. L. Shmulyan, Parallel addition and parallel subtraction of operators ,Izv. Akad. Nauk SSSR Ser. Mat. (1976), no. 2, 366–387, 470.14. W. Pusz and S. L. Woronowicz, Functional calculus for sesquilinear forms and the purifi-cation map , Rep. Mathematical Phys. (1975), no. 2, 159–170.
15. Laurent Schwartz,
Sous-espaces hilbertiens d’espaces vectoriels topologiques et noyaux as-soci´es (noyaux reproduisants) , J. Analyse Math. (1964), 115–256.16. Zolt´an Sebesty´en, Operator extensions on Hilbert space , Acta Sci. Math. (Szeged) (1993),no. 1-4, 233–248.17. Yu. L. Shmulyan, An operator Hellinger integral , Mat. Sb. (N.S.)
49 (91) (1959), 381–430.18. ,
Two-sided division in the ring of operators , Mat. Zametki (1967), 605–610.19. Zs. Tarcsay, On the parallel sum of positive operators, forms, and functionals , Acta Math.Hungar. (2015), no. 2, 408–426.20. T. Titkos,
On means of nonnegative sesquilinear forms , Acta Math. Hungar. (2014),no. 2, 515–533.21. N. N. Vakhaniya,
Probabilistic distributions in linear spaces , Sakharth. SSR Mecn. Akad.Gamothvl. Centr. ˇSrom. (1971), no. 3, 155, Russian, English translation: North-HollandPublishing Co., New York-Amsterdam 1981.22. N. N. Vakhaniya, V. I. Tarieladze, and S. A. Chobanyan, Probability Distributions in banachSpaces , Nauka, Moscow, 1985, Russian, English translation: D. Reidel Publishing Co,Dordrecht 1987.23. M. M. Va˘ınberg,
Variational method and method of monotone operators in the theory ofnonlinear equations , Nauka, Moscow, 1972, Russian, English Translation: Halsted Press,New York-Toronto 1973.24. Fuzhen Zhang (ed.),
The Schur Complement and its Applications , Numerical Methods andAlgorithms, vol. 4, Springer-Verlag, New York, 2005. Stauffenbergstr. 10, D-04509 Delitzsch, Germany.
E-mail address : [email protected] Mathematisches Institut, Universit¨at Leipzig, PF 10 09 20, D-04009 Leipzig,Germany.
E-mail address ::