Operators on anti-dual pairs: Generalized Schur complement

The goal of this paper is to develop the theory of Schur complementation in the context of operators acting on anti-dual pairs. As a byproduct, we obtain a natural generalization of the parallel sum and parallel difference, as well as the Lebesgue-type decomposition. To demonstrate how this operator approach works in application, we derive the corresponding results for operators acting on rigged Hilbert spaces, and for representable functionals of ${}^{*}$-algebras.


Introduction
Since the first appearance of the name of the "Schur complement" in [10], the theory of partitioned matrices (or block operators) is an active field of research in linear algebra and functional analysis. The direction we are interested in is the problem of completing special operator systems. To formulate the central question in the most classical setting, consider the incomplete system S = [ A B B * ] of positive semidefinite n × n matrices A, B. The task is to find a matrix D for which [ A B B D ] is a positive semidefinite 2n × 2n matrix. If we denote by A B the smallest possible solution, then the Schur complement of D in the block-matrix [ A B B D ] is D − A B . Therefore, to find the Schur complement and to find the minimal operator that makes a system positive is the same problem. In this paper, we focus our attention to the completion problem.
Because of its wide-range applicability in pure and applied mathematics, a number of authors made a lot of efforts to extend the concept of Schur complement for various settings. We mention first the fundamental work of Pekarev andŠmul'jan [14] on the connection between the shorted operator and positive completions of block operators in the context of Hilbert spaces. The corresponding result in Krein spaces has been developed by Contino, Maestripieri, and Marcantognini in [4] (see also [3,12]). The relation between extension, completion, and lifting problems of operators on both Hilbert and Krein spaces has been discussed in [2,3]. A quite general approach was developed by Friedrich, Günther and Klotz in [6]. They introduced a generalized Schur complement for non-negative 2 × 2 block matrices whose entries are linear operators on linear spaces. In their considerations the setting is purely algebraic and therefore topology plays a minor role.
In the present paper we are going to treat the above completion problem in an even more general setting that covers Hilbert, Krein, and linear spaces. Namely, we consider linear operators acting between a locally convex space and its topological anti-dual. The key idea of our approach is the observation that the block matrix completion problem can be formulated as an operator extension problem. This gives rise to invoke our corresponding Krein-von Neumann extension theory developed in [20]. Our aim is two folded: besides of solving the block completion problem in a quite general setting, we want to demonstrate how the developed method can be applied for structures like rigged Hilbert spaces and involutive algebras.
The paper is organized as follows: after collecting the basic notions and notations regarding operators acting on anti-dual pairs, in Subsection 2.1 we recall the construction of the generalized Krein-von Neumann extension. Subsection 2.2, the cornerstone of this paper, is devoted to provide necessary and sufficient conditions to guarantee that an incomplete operator system is positive. In case of positivity, Theorem 2.1 gives an explicit formula for the minimal solution of the completion problem.
Following the method of Pekarev andŠmul'jan, we introduce the notions of parallel sum and difference as an immediate application. This will be done in Subsection 2.3. Furthermore, by means of these notions we extend a very recent result appeared in [18] on Lebesgue-type decompositions. Namely, in Theorem 2.7 we exhibit an alternative description of the absolutely continuous part. Section 3 is fully devoted to applications: to demonstrate how this operator approach works in concrete structures, we derive the corresponding results for operators acting on rigged Hilbert spaces and for representable functionals on involutive algebras.

Positive completions of operator systems
2.1. Operators on anti-dual pairs. Throughout this paper we follow the notations of [20]. For more details we refer the reader to [20,Section 2 and 3]. An anti-dual pair denoted by F, E is a pair of two complex linear spaces intertwined by a separating sesquilinear map ·, · : F × E → C.
If D is a linear subspace of E, a linear operator A : D → F is said to be positive, if Ax, x ≥ 0 for all x ∈ D. In this paper we always assume E and F to be endowed with the corresponding weak topologies σ(E, F ), and σ(F, E), respectively. We will call the anti-dual pair F, E weak-* sequentially complete if the topological vector space (F, σ(F, E)) is sequentially complete. Recall that the (topological) anti-dual spaceĒ * of a barrelled space E is quasi-complete, hence the anti-dual pair Ē * , E is a weak-* sequentially complete. It is furthermore obvious that the algebraic anti-dualĒ ′ is weakly (sequentially) complete, hence the class of weak-* sequentially complete anti-dual pairs includes Hilbert, Banach, Fréchet spaces, and also vector spaces without topology.
Let F 1 , E 1 and F 2 , E 2 be anti-dual pairs and T : E 1 → F 2 a weakly continuous (that is, σ(E 1 , F 1 )-σ(F 2 , E 2 )-continuous) linear operator. Then the weakly continuous linear operator T * : is called the adjoint of T . The set of everywhere defined weakly continuous linear operators T : E → F is denoted by L (E; F ). Just like in the case of Hilbert spaces, every operator T ∈ L (E; F ) is uniquely determined by its quadratic form x → T x, x .
As it plays an important role in the background, we recall the construction of the Krein-von Neumann extension of a positive operator. For more details see [20,Theorem 3.1]. Assume that F, E is a w * -sequentially complete anti-dual pair and A : D → F is a linear operator such that for any y in E there is M y ≥ 0 satisfying This assumption guarantees that one can build a Hilbert space H A by taking the Hilbert space completion of the inner product space ran A, (· | ·) A , where Again, by (2.1), the canonical embedding operator is weakly continuous, and thus admits a unique continuous extension J to H A by weak- * sequentially completeness. Since J ∈ L (H A ; F ), its adjoint J * belongs to L (E; H A ), and J * x = Ax for all x ∈ dom A. As for any x ∈ dom A we have JJ * x = J(Ax) = Ax, the operator JJ * ∈ L (E; F ) is a positive extension of A.
We will refer to A N := JJ * as the Krein-von Neumann extension of A. For A N we obtained the following formulae (see (3.6) and (3.7) in [20]) JJ * y, y = sup | Ax, y | 2 : x ∈ dom A, Ax, x ≤ 1 (2.4) = sup Ax, y + Ax, y − Ax, x : x ∈ dom A (2.5) 2.2. Generalized Schur complement. Using the technique introduced in the previous subsection, we can extend the corresponding results of Pekarev and Smul'jan [14]. Namely, for given operators A, B, C ∈ L (E; F ) we consider the incomplete operator matrix (or shortly system) [ A B C * ]. Such a system is called positive if there exists a D ∈ L (E; F ) such that [ A B C D ] is positive. Clearly, for an incomplete operator matrix to be positive it is necessary that A ≥ 0 and that C = B * , furthermore every D making [ A B C * ] positive must be positive itself. We may therefore restrict ourselves to investigate incomplete operator matrices of the form A B B * * , where A ≥ 0. The next theorem provides necessary and sufficient conditions for positivity.
Theorem 2.1. Let F 1 , E 1 and F 2 , E 2 be weak- * sequentially complete antidual pairs and let A ∈ L (E 1 ; F 1 ) and B ∈ L (E 1 ; F 2 ) be weakly continuous linear operators such that A ≥ 0. Then the following assertions are equivalent: (i) There is a positive operator C ∈ L (E 2 ; F 2 ) such that the operator matrix (iii) For the canonical embedding operator J : H A → F 1 constructed in (2.3) the following range inclusion holds ran B * ⊆ ran J.
If any of the above conditions is fulfilled, then the linear operator is well defined and weakly continuous. Furthermore, its unique continuous extension S ∈ L (H A ; F 2 ) possesses the property that Proof. We use the column vector notation [ x1 x2 ] instead of (x 1 , x 2 ). (i)⇒(ii): Take y 2 from E 2 . By the Cauchy-Schwarz inequality we have (ii)⇒ (iii): Consider the auxiliary Hilbert space H A and for a fixed vector y 2 ∈ E 2 and define the linear functional From (ii) we infer that ϕ is bounded and therefore the Riesz representation theorem implies that there exists h ∈ H A such that Denote by J the canonical embedding operator (2.
We are going to show that the operator T 0 : . Since every such extension can be written as a matrix of the form T = A B * B C for some C ∈ L (E 2 ; F 2 ), C ≥ 0, this will entail the desired implication.
The anti-dual pair F 1 × F 2 , E 1 × E 2 is obviously weak-* sequentially complete, so our only duty is the verify that T 0 satisfies condition (2.1), as it guarantees the existence of the Krein-von Neumann extension according to [20,Theorem 3 To prove the rest of our statement observe first that inequality (ii) can be written in the form This inequality says that for every fixed y 2 ∈ E 2 , the linear functional This implies that S 0 of (2.6) is well defined and weakly continuous. Let T N denote the smallest (Krein-von Neumann) extension On the other hand, Thus we conclude that the quadratic forms of SS * and C N are identical, so SS * = C N . Finally, applying (2.5) we obtain that The proof is complete. Now we are able to introduce the notion of complement in this setting.
is any positive operator that makes the system In the sequel we work with the complement only in the special case when A, B ∈ L (E; F ) are both positive (and hence self-adjoint) operators on F, E . In that case the quadratic form of A B can be calculated as As a straightforward consequence of Before proceeding with direct applications let us make first two short comments. On the one hand, if we consider a sesquilinear form t : E×E → C on a given complex vector space E, then we may associate a linear operator A t acting between E and its algebraic anti-dualĒ ′ by setting It is clear now that A t is automatically weakly continuous that is positive (resp., selfadjoint) if and only if t is nonnegative semidefinite (resp., Hermitian). On the other hand, if an anti-dual pair F, E and a positive operator A ∈ L (E; F ) is given, we can associate a hermitian form on E by Therefore, using the above theorem, we can define the complement of sesquilinear forms. This notion (together with the parallel sum and parallel difference) was introduced and studied by Hassi, Sebestyén, and de Snoo in [8] and [9]. Note that A : B is the Schur complement of A in the block matrix A+B A A A by definition. Observe also that A : B = B : A which is not clear from the definition but can easily deduced from formula (2.15). Consider now the system A−B A A * . In order to be positive it is necessary (but nut sufficient) that A − B ≥ 0. By Theorem 2.1, positivity of A−B A A * is equivalent with the following condition: for any y ∈ E there is M y ≥ 0 such that If this holds true, then the complement (A − B) A exists and its quadratic form is calculated as follows: Observe that the assumption A ≥ B is not sufficient to guarantee the existence of (A − B) A . Indeed, the supremum above can be infinity if for example A = B. To see this, substitute x = λy for any y ∈ E satisfying Ay, y = 0. Now we are going to apply the above results to retrieve the Lebesgue decomposition developed in [18] in an alternative way, namely, by means of the parallel addition and subtraction. To do so we recall first what a Lebesgue decomposition is. For more details, see [18]. The key results we are going to take advantage of are [8, Theorem 3.5] and [8, Theorem 3.9]. The combination of the two statements (with the notation of (2.13)) says that the limit of the weighted parallel sums can be formulated by means of the parallel sum and the parallel difference Using is a Lebesgue decomposition of A with respect to B. Furthermore, We conclude this section with a corollary which states that A r can be expressed by means of the complement and the parallel sum.
Corollary 2.8. Let F, E be a weak- * sequentially complete anti-dual pair. Assume that A and B are positive operators belonging to L (E; F ). Then Proof. First observe that if C and A are positive operators such that C B does exits, then it can be written as Indeed, using that Bx, y + By, x = Bx, x + By, y − B(x − y), (x − y) , an elementary calculation shows that Clearly, the involution A † := A * | D makes L † (D) a * -algebra. Consider a †-closed subalgebra A of L † (D) that contains the identity operator and equip D with a locally convex topology τ A induced by the semi-norms Clearly, τ A is finer then the norm topology induced by H on D. The canonical embedding of D into H is continuous with dense range and therefore the adjoint of that map embeds H intoD ′ continuously, when endowed with the strong dual topology β ′ := β(D ′ , D): This triplet (D, H,D ′ ) is usually called a rigged Hilbert space or Gelfand triplet. For more details about rigged Hilbert spaces see [1] and [15,Chapter 3]. Suppose now that (D, τ A ) is a barrelled space (for example, a Fréchet space). Every weakly continuous (and in particular, every positive) operator A : D →D ′ is then continuous with respect to the topologies τ A and β ′ . On the converse, every linear operator A that is continuous for τ A and β ′ is weakly continuous. In any case, we shall call A simply continuous. Furthermore, sinceD ′ is weakly quasi-complete, we infer that the anti-dual pair D′ , D is weak-* sequentially complete. This enables us to apply our preceding results in this rigged Hilbert space framework. (i) There is a positive operator C : If any of (i) or (  imply Ax n , x n → 0, (b) singular if for every x ∈ D there exists a sequence (x n ) n∈N in D such that x n − x → 0 and Ax n , x n → 0, (n → +∞).
Let us denote the natural inclusion operator D ֒→D ′ by I D,D ′ : is a Lebesgue-type decomposition. That is, A r is regular, A − A r is singular.

3.2.
Representable functionals. We have already seen some immediate applications of the complement in the operator context. The aim of this section is to show how this general setting can be used in the theory of representable functionals. First we fix the terminology. Let A be a not necessarily unital complex * -algebra. A linear functional f on A is called representable if there is GNS triple, i.e. a Hilbert space H f , a *-homomorphism π f : A → B(H f ) and a vector ξ f ∈ H f such that As it is known, and for every a ∈ A there exists λ a ≥ 0 such that The set of representable functionals will be denoted by A ♯ . The GNS triple mentioned above can be obtained by the following construction: since every representable functional f is positive, the map A : A →Ā * defined by is a positive operator. The range space ran A becomes a pre-Hilbert space if we endow it by the inner product This is indeed an inner product space as the Cauchy-Schwarz inequality guarantees that (Aa | Aa) f = 0 implies Aa = 0 for all a ∈ A . We denote the Hilbert space completion of this space by H f . Next we introduce a densely defined continuous operator π f (x) for all x ∈ A by The continuity of π f (x) is due to (3.4), and we continue to write π f (x) for its unique norm preserving extension. It is easy to verify that π f is a * -representation of A in B(H f ). The cyclic vector of π f is obtained by considering the linear functional Aa → f (a) from H f into C whose continuity is guaranteed by (3.3). The Riesz representation theorem yields a unique vector ξ f ∈ H f satisfying It is straightforward to verify that whence we infer that The following is the main result of this section. It provides a sufficient condition to the existence of the complement of functionals.
Theorem 3.4. Let f, g be linear functionals on a * -algebra A . Suppose that f is representable and that there is a constant C ≥ 0 such that Then there exists a representable positive functional h such that f + g + g * + h is representable and for all a, b in A . Furthermore, there is a smallest h possessing this property.
Proof. Consider the following linear functional on H f It is bounded according to (3.8), hence there exists a unique η g ∈ H f such that Let us define h by setting Clearly, h ∈ A ♯ . We claim that h is the smallest representable functional with property (3.9). To see this let us consider the operators A, B ∈ L (A ;Ā ′ ) defined by Aa, b := f (b * a) and Ba, b := g(b * a) (a, b ∈ A ). Then the incomplete matrix is positive as it fulfills condition (ii) of Theorem 2.1: Aa, a , a, b ∈ A . By Theorem 2.1, there exists a positive operator A B : A →Ā ′ such that A B * B AB is positive, furthermore the quadratic form of A B is calculated as follows: ] ≥ 0, which shows that h satisfies (3.9). To show the minimality of h suppose h ′ ∈ A ♯ fulfills (3.9) as well. Then letting one gets A B * B C ′ ≥ 0 and thus A B ≤ C ′ by Theorem 2.1, which is equivalent to h ≤ h ′ . Finally, a direct calculations shows that which proves that f + g + g * + h ∈ A ♯ .
Definition 3.5. We call the smallest representable functional satisfying (3.9) the complement of f with respect to g, and we denote it by f g . The complement f g can be calculated as where η g ∈ H f is the representing vector of the bounded linear functional .
With the notation of the proof of Theorem 3.4, we have also that Hence we gain two formulae for f g (a * a): from (2.8) we get and similarly, (2.9) yields Recall also that, with notation of Theorem 2.1, we have A B = SS * , where S : H A → F is the unique weakly continuous operator defined by This operator plays a key role in the following simple corollary, which provides an elegant formula for the complement in the case when A possesses a unit element.
Corollary 3.6. Let f, g be linear functionals on a * -algebra A . Suppose that f ∈ A ♯ and g satisfies (3.8).
Then the complement f g of f with respect to g is of the form f g = Sη g , i.e.
If A is unital, then f g = A B 1, i.e., Proof. First observe that S * a = π f (a)η g holds for all a ∈ A . Indeed, according to Theorem 3.4, we have that g(a) = (π f (a)ξ f | η g ) f , and thus Using this, we easily conclude that which implies (3.14). Furthermore, S * a = π f (a)η g implies which obviously implies (3.15) whenever 1 ∈ A .
As we have seen in the previous section, the existence of the complement guarantees the existence of some important operations like the parallel sum and parallel difference. Theorem 3.4 now allows us to extend these concepts for representable functionals.
We begin with the parallel sum. Consider two representable functionals f, g ∈ A ♯ , then f + g ∈ A ♯ and |f (a)| 2 ≤ M (f (a * a) + g(a * a)) a ∈ A holds for suitable M . Thus the complement (f + g) f ∈ A ♯ exists and as we conclude from Theorem 3.4 that (f + g) f exists and f ≥ (f + g) f . Therefore the following definition is correct.
Definition 3.7. Let f and g be representable functionals on the * -algebra A .
Then the parallel sum f : g defined by (f : g) := f − (f + g) f is a representable functional as well. From (3.13) it is easy to check that f : g satisfies For a completely different approach to the parallel sum see [17]. Now we proceed with the notion of parallel difference.
Definition 3.8. Let f and g be representable functionals on the * -algebra A , and assume that f ≥ g and the complement (f − g) f ∈ A ♯ exists. Then the parallel difference g ÷ f ∈ A ♯ is defined by g ÷ f := (f − g) f − f .
As it was pointed out in the operator setting, the assumption f ≥ g is not enough for the existence of g ÷ f . However, Theorem 3.4 offers a sufficient condition as follows: assume that |f (a)| 2 ≤ C · (f (a * a) − g(a * a)), a ∈ A (3. 18) holds for some C ≥ 0. Then (3.18) implies that f −g ∈ A ♯ and that the complement (f − g) f ∈ A ♯ exists.
To conclude the paper, we establish a Lebesgue decomposition theorem for representable functionals by means of the parallel sum and difference. Definition 3.9. Let f and g be representable functionals on the * -algebra A . We say that f is g-absolutely continuous (f ≪ g) if for every sequence (a n ) n∈N of A such that g(a * n a n ) → 0 and f ((a n −a m ) * (a n −a m )) → 0 it follows that f (a * n a n ) → 0. Furthermore, f and g are singular (f ⊥ g) with respect to each other if h = 0 is the only representable functional such that h ≤ f and h ≤ g. A decomposition f = f 1 + f 2 is called a Lebesgue-type decomposition of f with respect to g if f 1 ≪ g and f 2 ⊥ g.
For the sake of simplicity, let us introduce the following temporary notation: if a representable functional h is given, we denote its induced operator by A h . Accordingly, it is obvious that A f : A g = A f :g and A f ÷ A g = A f ÷g due to (3.16) and (3.17). This simple observation leads us to the following Lebesgue type decomposition theorem (for earlier versions of the Lebesgue decomposition of representable functionals see [7,11,16,18,19,21], and the references therein).
Theorem 3.10. Let A be a * -algebra, and assume that f, g ∈ A ♯ . Then the functional f r = (f : g) ÷ g is representable and g-absolutely continuous. Furthermore, the decomposition is a Lebesgue-type decomposition of f with respect to g. Here the absolutely continuous part f r can be written as Zs. Tarcsay, Department of Applied Analysis and Computational Mathematics, Eötvös Loránd University, Pázmány Péter sétány 1/c., Budapest H-1117, Hungary T. Titkos, Alfréd Rényi Institute of Mathematics, Reáltanoda utca 13-15., Budapest H-1053, Hungary, and, BBS University of Applied Sciences, Alkotmány u. 9., Budapest H-1054, Hungary