Operators on anti-dual pairs: self-adjoint extensions and the Strong Parrott Theorem

The aim of this paper is to develop an approach to obtain self-adjoint extensions of symmetric operators acting on anti-dual pairs. The main advantage of such a result is that it can be applied for structures not carrying a Hilbert space structure or a normable topology. In fact, we will show how hermitian extensions of linear functionals of involutive algebras can be governed by means of their induced operators. As an operator theoretic application, we provide a direct generalization of Parrott's theorem on contractive completion of $2$ by $2$ block operator-valued matrices. To exhibit the applicability in noncommutative integration, we characterize hermitian extendibility of symmetric functionals defined on a left ideal of a $C^{*}$-algebra.


Introduction
The question whether a self-adjoint extension exists arises naturally in various situations when a partially defined (bounded or unbounded) symmetric operator is given. For classical results we refer the reader to [2,6,8,16] and the references therein, for more recent results see for example [3,11]. In our previous paper [19], we have developed a Krein-von Neumann type extension theory for positive operators acting on anti-dual pairs. That technique is general enough to not only overcome the lack of a Hilbert space structure, but also the lack of a normable topology. Our running example in [19] -illustrating the applicability of the general setting -came from noncommutative integration theory. Namely, we have demonstrated how functional extensions can be governed by their induced operators. The aim of the present paper is to continue these investigations and to discuss the problem of self-adjoint extendibility.
Below we briefly describe the content. Section 1 contains a short overview of concepts and earlier results that help the reader to follow the proofs. In particular, we briefly sketch the construction of the generalized Krein-von Neumann extension that serves as the basis for our treatment. In Section 2 we examine the extension problem in our anti-dual pair setting. The main result Theorem 2.1 generalizes Krein's theorem on the existence of a norm preserving self-adjoint extension of a bounded symmetric operator [10,Theorem 5.33]. Due to the lack of norm, we are going to consider extensions bounded by a fixed positive operator A. It will turn out that extensions preserving the A-bound form an operator interval. As a nice application of Theorem 2.1, in Section 3 we will generalize Yamada's recent result [21], which is an extension of the Strong Parrott Theorem [7,12]. We will close the paper by demonstrating that Theorem 2.1 on self-adjoint extensions is an effective generalization. Namely, we shall see in Section 4 how this result can be applied to obtain hermitian extensions of functionals of an involutive algebra.

Preliminaries
In this section we summarize shortly all the notions and tools to make the presentation self-contained. For more details we refer the reader to [19,Section 2 and 3]. An anti-dual pair denoted by F, E is a system of two complex vector spaces E and F intertwined by a separating sesquilinear map ·, · : F × E → C, i.e., ·, · is linear in its first, and conjugate linear in its second argument. Let D be a linear subspace of E. We call a linear operator A : D → F symmetric, if Ax, y = Ay, x holds for all x, y ∈ D. In analog with the Hilbert space case, an operator A : Just as in the dual pair case, we endow E and F with the corresponding weak topologies σ(E, F ), resp. σ(F, E), induced by the families { f, · | f ∈ F }, resp. { ·, x | x ∈ E}. Both σ(E, F ) and σ(F, E) are locally convex Hausdorff topologies with duality properties Here F ′ andĒ ′ refer to the topological dual and anti-dual space of F and E, respectively, and the vectors f ∈ F and x ∈ E are identified with f, · , and ·, x , 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. One of the useful properties of weak topologies is the following: for a topological vector space (V, τ ), a linear operator T : V → F is continuous with respect to τ and σ(F, E) if and only if the linear functionals are continuous for every x ∈ E. This fact and (1.1) enables us to define the adjoint (that is, the topological transpose) of a weakly continuous operator. Let F 1 , E 1 and F 2 , E 2 be anti-dual pairs and T : E 1 → F 2 a weakly continuous linear operator. Then the weakly continuous linear operator T * : is called the adjoint of T . The set of everywhere defined weakly continuous (i.e., σ(E, F )-σ(F, E) continuous) linear operators T : E → F will be denoted by L (E, F ). In the case when H and K are Hilbert spaces, L (H, K) coincides with the set B(H, K) of bounded linear operators from H to K. An operator T ∈ L (E, F ) is called self-adjoint if T * = T . Obviously, everywhere defined symmetric operators (and hence everywhere defined positive operators) are automatically weakly continuous and self-adjoint. Now we proceed by recalling the construction of the Krein-von Neumann extension of a positive operator. We will use the notations of this section without further notice. For more details see [19,Theorem 3.1 (iv) ⇒ (i)]. Let F, E be a w * -sequentially complete anti-dual pair and let A : E ⊇ dom A → F be a positive operator with domain dom A. Assume further that for any y in E there is M y ≥ 0 such that 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 (1.2), the canonical embedding operator is weakly continuous, and thus admits a unique continuous extension J to H A by w * -sequentially completeness of F .
it follows that J * x = Ax for all x ∈ dom A. As for any x ∈ dom A we have 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. We remark that the extension result above is closely related to the theory of reproducing kernels (see for example [13,14,21]). In fact, one can say that the operator A : E ⊇ D → F is a restriction of a reproducing kernel if and only if (1.2) holds. Finally we mention that our assumption on F to be w * -sequentially complete is weaker than quasi-completeness imposed by Schwartz in [13].

Self-adjoint extensions of symmetric operators
M.G. Krein proved in [10] that every bounded symmetric Hilbert space operator possesses a norm preserving self-adjoint extension. The problem of constructing self-adjoint extensions of a symmetric operator arises in our anti-dual pair setting naturally. Since we cannot speak about norm preservation due to the lack of norm, we need to find a suitable notion to generalize Krein's theorem. Observe that the norm of a self-adjoint operator S ∈ B(H) can be expressed by means of the partial order induced by positivity. Namely, S is the smallest constant α ≥ 0 such that −αI ≤ S ≤ αI. Based on this observation, a symmetric op- holds. The smallest constant α is called the A-bound of S 0 and is denoted by α A (S 0 ). We will call the extension In the next theorem, which is the main result of this section, we will present a sufficient condition to guarantee for a symmetric linear operator that it possesses a self-adjoint extension. Moreover, we describe the set of all A-bound preserving extensions of a given symmetric operator.
Theorem 2.1. Let F, E be a weak- * sequentially complete anti-dual pair and let S 0 : dom S 0 → F be a symmetric operator, i.e., Proof. Introduce the following linear manifold and fix an x ∈ dom S 0 . Let us define the conjugate linear functional f x as Observe that f x is continuous, because holds for some α by A-boundedness. According to the Riesz representation theorem, there exists a unique representing vector ζ x ∈ H A such that is well defined and linear. By (2.3) we have hence S 0 is symmetric. Introduce the operators Clearly, T m and T M are both positive operators on dom S 0 . Furthermore, we have for all h ∈ dom S 0 that By [19,Theorem 4.2], there exist two minimal positive extensions A m , A M ∈ B(H A ) of T m and T M , respectively. Note also that Finally, for x ∈ dom S 0 and y ∈ E, A similar calculation shows that S 0 ⊂ S M holds as well.
To prove (2.2) let S ∈ [S m , S M ] and take x ∈ dom S 0 and y ∈ E. Since S−S m ≥ 0 it follows that hence there is a symmetric operator S ∈ B(H A ) with S ≤ 3α A (S 0 ) such that S = J SJ * . It is clear that S m ≤ S ≤ S M , and thus α A (S 0 ) = S = α A (S). Assume conversely that S 0 ⊂ S is any self-adjoint extension such that α A (S) = α A (S 0 ). Then it is clear that The proof is complete.
In the following corollary we recover the classical result of Krein on self-adjoint norm-preserving extensions.  [19,Corollary 4.3]. This clearly gives (2.5).

A generalized Strong Parrott Theorem
The aim of this section is to generalize Parrott's famous theorem [12] on contractive extensions of 2 by 2 block operator-valued matrices, which is one of the crucial results in extension and dilation theory. As an application, we will deduce Yamada's recent result [21,Theorem 4] on the extension of the Strong Parrott Theorem [7,20].
Using the generalized Parrott theorem above, we obtain a new proof for a recentresult of A. Yamada [21,Theorem 4].
Corollary 3.2. Let F 1 , E 1 1 , F 2 , E 2 2 be anti-dual pairs and let H, K be Hilbert spaces. For S 1 ∈ L (E 1 , H), S 2 ∈ L (E 1 , K), T 1 ∈ L (H, F 2 ) and T 2 ∈ L (K, F 2 ) the following conditions are equivalent: i.e., X makes the following diagram commutative: Proof. Implication (ii)⇒(i) is straightforward, so we only prove that (i) implies (ii). Consider the anti dual pairs (H | H) and (K | K) and the operators From (i) we see that X 0 , X 1 are well defined contractions such that ), for every x 1 ∈ E 1 and x 2 ∈ E 2 . Hence the pair X 0 , X 1 fulfills the conditions of Theorem 3.1 with A 1 = I H , A 2 = I K and α 1 = α 2 = 1. Consequently, there exists X ∈ B(H, K), X ≤ 1 such that X 0 ⊂ X, X 1 ⊂ X * , and therefore XS 1 = S 2 and X * T * 2 = T * 1 which yields (ii). Yamada's work itself generalizes Parrott's theorem [12] and the Strong Parrott Theorem (see [1,4,7]). So we get the following classical result automatically. Corollary 3.3. Let H and K be Hilbert spaces, H 1 ⊆ H and K 1 ⊆ K be closed linear subspaces, and denote by P K1 the orthogonal projection onto K 1 . For given contractions T 1 : H 1 → K and T ′ 1 : H → K 1 the following conditions are equivalent: exists a contraction T ∈ B(H, K) such that

Hermitian extensions of linear functionals
Positive functionals play an important role in the representation theory of algebras. Extension of such functionals has been investigated in many different settings. For example, if f is a positive linear functional defined on a closed ideal in a C *algebra, then f always admits an extension with the same norm (see [5,II.6.4.16]). Positive functionals defined on left-ideals of the full operator algebra possessing normal extension were characterized in [18], while positive extendibility of positive functionals defined on left ideals of general * -algebras was studied in [19]. The aim of this section is to demonstrate how our anti-dual pair setting can be used to construct hermitian extensions of linear functionals in the unital * -algebra setting.
Let A be a unital *-algebra with unit 1. If I ⊆ A is a left ideal then we call a linear functional g : A linear functional g ∈ A * is called hermitian if it satisfies It is easy to check that a linear functional g ∈ A * is hermitian if and only if it is symmetric. Note that for a *-algebra without unit element this equivalence does not longer hold. Assume that a positive linear functional f : A → C is given. We say that the symmetric functional g : holds for some α > 0. The f -bound α f (g) is defined as the smallest constant α that fulfills (4.1). If ℓ : I → C is a linear functional then the correspondence defines a linear operator L : I →Ā * . Clearly, L is positive if ℓ is positive and L is symmetric if ℓ is so. Suppose now that f ∈ A * is a positive functional and denote by A the positive operator associated with f , i.e., Ax, y = f (y * x), x, y ∈ A .
Let H A denote the corresponding Hilbert space, that is, H A is the completion of ran A endowed with the inner product (1.3). Observe that in that case we have Consider the canonical embedding J : H A →Ā * in (1.4) and recall its useful properties J * x = Ax, x ∈ A , and JJ * = A. Assume in addition that for some M x ≥ 0. This assures that the operators π f (x) defined by are continuous on H A by norm bound M 1/2 x . Thus, for every x ∈ A we can extend π f (x) to an element of B(H A ). It is then immediate that π f : A positive functional satisfying (4.2) (and hence (4.3)) will be called representable [15].
consists of all hermitian f -bound preserving extensions of g 0 : Proof. Along the lines of the proof of Theorem 2.1, let us introduce a symmetric operator S 0 on dom S 0 := {J * a | a ∈ I } such that A straightforward calculation shows that S 0 : dom S 0 → H A is bounded with norm S 0 = α f (g 0 ). For x, y ∈ A and a ∈ I we have hence we infer that dom S 0 is π f -invariant and π f (x)S 0 ⊂ S 0 π f (x) for all x ∈ A . By Corollary 2.2, it follows that there exist two norm preserving self-adjoint extensions S m , S M ∈ B(H A ) of S 0 such that whenever x is self-adjoint, and hence also for every x ∈ A . We claim that the functionals fulfill all conditions of the statement. First we observe that g m and g M are hermitian: indeed, by (4.4) we have for every x ∈ A that A similar argument shows that g M is hermitian. Next, observe that g m and g M extend g 0 , because for every a ∈ I g m (a) = (S m J * a | J * 1) A = (S 0 J * a | J * 1) A = g(a) holds, and similarly, g M (a) = g(a). Finally, we have whence it follows readily that g m and g M have f -bound α f (g 0 ). Let now g ∈ A * be an arbitrary hermitian extension of g 0 having f -bound α f (g 0 ). Then there is a self-adjoint operator S ∈ B(H A ), S = α f (g 0 ) such that (SJ * x | J * y) A = g(y * x), x, y ∈ A .
It is clear that S 0 ⊂ S and therefore S m ≤ S ≤ S M , due to Corollary 2.2. As a straightforward consequence we conclude that g m ≤ g ≤ g M . Suppose conversely that g ∈ A * is a hermitian functional such that g m ≤ g ≤ g M . First observe that g is f -bounded as it satisfies |g(y * x)| ≤ (g − g m )(x * x) 1/2 (g − g m )(y * y) 1/2 + |g m (y * x)| ≤ 3α f (g 0 )f (x * x) 1/2 f (y * y) 1/2 .
Hence there exists a self-adjoint operator S ∈ B(H A ), such that g(y * x) = (JSJ * x | y) A , x, y ∈ A .
From g m ≤ g ≤ g M it follows that S m ≤ S ≤ S M , and therefore S 0 ⊂ S by Corollary 2.2. Consequently, g 0 (a) = (S 0 J * a | J * 1) A = (SJ * a | J * 1) A = g(a), a ∈ I , thus g 0 ⊂ g. The proof is complete.
We remark that Theorem 4.1 provides only a sufficient condition for the existence of hermitian extensions. On C * -algebras, the statement of Theorem 4.1 may be improved in two ways: first, the condition on f of being representable may be replaced by the formally weaker one of being positive. On the other hand, the existence of a dominating positive functional is both necessary and sufficient. Proof. On a C * -algebra, every positive functional f is representable. If g 0 is symmetric and f -bounded then g 0 has a hermitian extension g that is of the form g(a) = (SJ * a | J * 1), a ∈ A , where S is a bounded self-adjoint operator on H A . It follows therefore that g is continuous. For the converse, assume that g ∈ A * is a continuous hermitian extension of g 0 . Let g = g + − g − be the Hahn-Jordan decomposition of g, with g + , g − ∈ A * positive functionals (see [9]). Letting f := g + + g − , it is easy to check that g 0 is f -bounded with bound α f (g 0 ) = 4.