Extensions of dissipative and symmetric operators

Given a densely defined skew-symmetric operator A0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_0$$\end{document} on a real or complex Hilbert space V, we parameterize all m-dissipative extensions in terms of contractions Φ:H-→H+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi :{H_{{-}}}\rightarrow {H_{{+}}}$$\end{document}, where H-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${H_{{-}}}$$\end{document} and H+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${H_{{+}}}$$\end{document} are Hilbert spaces associated with a boundary quadruple. Such an extension generates a unitary C0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_0$$\end{document}-group if and only if Φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi $$\end{document} is a unitary operator. As a corollary we obtain the parametrization of all selfadjoint extensions of a symmetric operator by unitary operators from H-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${H_{{-}}}$$\end{document} to H+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${H_{{+}}}$$\end{document}. Our results extend the theory of boundary triples initiated by von Neumann and developed by V. I. and M. L. Gorbachuk, J. Behrndt and M. Langer, S. A. Wegner and many others, in the sense that a boundary quadruple always exists (even if the defect indices are different in the symmetric case).


Introduction
A classical subject in Functional Analysis and in Mathematical Physics is the description of all selfadjoint extensions of a symmetric operator. It was J. von Neumann who gave the first result in this direction (see Sect. 5). Generalizations of von Neumann's Theorem led to the theory of boundary triples which are described in detail in the monographs [8] by V.I. and M.L. Gorbachuk and by K. Schmüdgen [15]. Such boundary triples exist whenever the given symmetric operator has at least one selfadjoint extension. Multiplying by the complex number i, the problem can be reformulated as follows. Let A 0 be a densely defined skew-symmetric operator on a Hilbert space V . Describe all extensions B of A 0 which generate a unitary C 0 -group.
More recently, Wegner [17] used boundary triples to parameterize also all extensions B of A 0 which generate a contractive C 0 -semigroup (which does not necessarily consist of unitary operators). However, it turns out that for this task the notion of boundary triples is too narrow and does not cover all cases.
In the present article we introduce boundary quadruples, a notion with weaker assumptions, which covers all cases. Moreover, we give quite short proofs which might make the new setting attractive.
Let us describe in more details some of the main results. Let V be a Hilbert space over K = R or C, and let A 0 be a skew-symmetric operator with dense domain. Then It is remarkable that the purely algebraic assumptions (1.1) and (1.2) imply that the mappings G − and G + are continuous and that the images H − , H + are actually Hilbert spaces. Our main result is the following. Moreover, B generates a unitary C 0 -group if and only if is unitary. In this case we refind the known extension results for symmetric operators in a more general setting.
The literature on boundary triples is very rich. We refer to the references and historical notes in the article [17] by Wegner, the two monographs mentioned above, and also to the monograph of Behrndt, Hassi and de Snoo [6], where extension results are part of an elaborate theory (cf. [6,Corollary 2.1.4.5]), and we refer also to the articles of Behrndt-Langer [4] and Behrndt-Schlosser [5]. In a previous article [2], the present authors studied extensions of derivations in a quite different spirit, the main motivation being non-autonomous evolution equations. Even though some ideas used here have their origins in [2], the present article is completely self-contained. It is organized as follows. In Sect. 2 we investigate dissipative operators including a simple proof of Phillips' Theorem. Boundary quadruples are introduced and investigated in Sect. 3. Extensions which generate a unitary group are characterized in Sect. 4. These results can be transferred to extension results for symmetric operators (Sect. 5). Examples are given in Sect. 6. Section 7 is devoted to the wave equation. Finally we compare boundary triples as they occur in the literature with boundary quadruples in Sect. 8.

Dissipative and skew-symmetric operators
Let V be a Hilbert space over K = R or C. We first recall the definitions of several notions associated with dissipativity. By an operator A on V we always understand a linear mapping, defined on a subspace D(A) of V , its domain, which takes values in V .

Definition 2.1 (a) An operator
We now collect well-known properties to see the link between all the notions defined above.

Proposition 2.2 Let A be an operator on V .
(1) The operator A on V is dissipative if and only if Proof (1) follows from the equivalence Then, letting t → 0, we prove the well-known characterization of dissipativity.
By 1) applied with t = 1, we get x = y, and then A = B.
is a Cauchy sequence and by 1) (x n ) n is also a Cauchy sequence, which converges to say x ∈ V . It follows that Ax n → x−y t . Since the graph of A is closed, Ax = x−y t and thus y = (Id − t A)x.
Phillips [12, Corollary of Theorem 1.1.1] obtained the equivalence between mdissipativity and maximal dissipativity on a Hilbert space using the Cayley transform. We give a simple direct proof (see also [17,Theorem 3.1] for still another argument).

Theorem 2.3 (Phillips) Let V be a Hilbert space and A an operator on V . Then A is m-dissipative if and only if A is maximal dissipative and D(A) is dense.
Then A ⊂ A and By Zorn's lemma, each densely defined dissipative operator has a maximal dissipative extension.
By the Lumer-Phillips Theorem [1, Theorem 3.4.5 and Proposition 3.3.8], an operator A on V generates a C 0 -semigroup of contractions if and only if A is m-dissipative. Now we introduce the basic objects of this article, a skew-symmetric operator which can be characterized as follows. We continue to consider the real and complex case simultaneously.

Proposition 2.4
Let A be an operator on V . The following assertions are equivalent: This proves (iii) in the real case. If K = C, then replacing u by λu we see that If B is a densely defined operator on V , the adjoint B * of B is defined as follows. For Using this definition we see that a densely defined operator A 0 on V is skew-symmetric if and only if A 0 ⊂ (−A 0 ) * . Our aim is to describe all dissipative extensions of such an operator A 0 . It turns out that they all are restrictions of (−A 0 ) * . A proof using Cayley transform is given in [ Hence, for t > 0, Dividing by t and letting t → 0, we get Since the above inequality holds for ±x, it follows that If K = R, since x ∈ D(A 0 ) is arbitrary, this implies that y ∈ D(A * 0 ) and By = (−A 0 ) * y. If K = C, replacing t by λt with λ ∈ C, the above argument shows that for all λ ∈ C and x ∈ D(A 0 ). Choosing λ = 1 and then λ = i, we get A 0 x, y V + x, By V = 0, and the conclusion follows as in the case K = R.

Boundary quadruples and m-dissipative restrictions
In this section we introduce and study the basic notion of this article. Let V be a Hilbert space over K = R or C and let A 0 be a densely defined skew-symmetric operator on V . Let A = (−A 0 ) * . Then, as a consequence of the definition of the adjoint for all u, w ∈ D(A), and We will see below that these assumptions imply that H − and H + are actually complete and that G − , G + are continuous (with respect to the graph norm on D(A)). Boundary quadruples do always exist. Below are two possible constructions which always work. However, for concrete examples, other choices might be convenient. We let Similarly
We consider D(A) with the graph norm Then D(A) is a Hilbert space with · D(A) as corresponding norm. We first prove that G − and G + are automatically continuous for this norm.
where H − , H + are the completions of H − and H + respectively. It suffices to show that G has a closed graph. For that let Note that b is symmetric and hence automatically continuous as a consequence of the Closed Graph Theorem. Thus . Thus Passing to limits this remains true for all z -∈ H − , z + ∈ H + .
Next we show that H − and H + are complete. We could deduce it from [2, Proposition 4.10] but we prefer to give a direct proof.  In particular,

Proof a) We show that G − D(A) is closed in the completion
In particular,

Example 3.7 (Second example) The second construction depends on the following decomposition
where A 0 is the closure of A 0 (which is again skew-symmetric). A proof can be found in [ We give a proof to be as self-contained as possible, and we treat again the real and complex case simultaneously. First we note the decomposition which is a Hilbert direct sum since Ran(Id .
with the notation defined above. In fact, using that, Let (H − , H + , G − , G + ) be a boundary quadruple, which is fixed for the remainder of this section. We will decribe all m-dissipative extensions of A 0 in terms of this quadruple.
Recall that A 0 is closable and that A 0 is skew-symmetric again. The domain of A 0 may be described by the boundary quadruple as follows.
Hence u ∈ ker G + ∩ ker G − . Since G + and G − are continuous for the graph norm on D(A), it follows that Theorem 3.10 Let B be an operator on V such that A 0 ⊂ B. The following assertions are equivalent:

To say that
: each operator A is dissipative. Before proving Theorem 3.10 we show that A determines . More precisely the following holds.

Proof of Theorem 3.10
Let B be a dissipative extension of A 0 . Then by Theorem 2.5,  Proof This follows from Theorem 3.10 and Proposition 3.11.
Let us consider some special cases.

Proposition 3.13
The following assertions are equivalent: This follows from Proposition 3.9. Proposition 3.13 describes when both operators G + , G − are zero; it turns out that this is independent of the boundary quadruple. Next we consider the case when G + = 0.
As a consequence A 0 is also the only extension of A 0 generating a C 0 -semigroup. We conclude this section by establishing isomorphisms between boundary quadruples.   Since G is surjective, it follows that CGu = * C Gu for all u ∈ D(A). Consequently, C = * C . (ii) ⇒ (i) This is shown by reversing the arguments leading to "(i) implies (ii)".

Unitary groups
Let V be a Hilbert space over K = R or C. In this section we want to determine all extensions of a densely defined skew-symmetric operator which generate a unitary group. We recall the following two well-known facts. An operator B on V generates a C 0 -group (U (t)) t∈R if and only if ±B generate C 0 -semigroups (U ± (t)) t≥0 . In that case, U (t) = U + (t) and U (−t) = U − (t) for t ≥ 0. Moreover, if B generates a C 0semigroup (S(t)) t≥0 , then B * generates the C 0 -semigroup (S(t) * ) t≥0 . By a unitary C 0 -group we mean a C 0 -group of unitary operators. The following is well-known.
This proves the claim. One similarly shows that x + = x + for all x + ∈ H + . Thus is unitary.

Selfadjoint extensions of symmetric operators
In this section we consider a complex Hilbert space V . An operator S on V is called symmetric if

If S is densely defined, then S is symmetric if and only if
It is immediate that an operator S is symmetric if and only if i S is skew-symmetric. Selfadjointness can be characterized by Stone's Theorem.
. Now the existence follows from Example 3.6 and Proposition 3.8.

Lemma 6.1 One has
This can be proved as in the scalar case, see [7, Theorem 8.2 and Corollary 8.9] and [3, Sec. 8.5] for a vector-valued version.

Lemma 6.2 Let u, v
In that case the semigroup (S(t)) t≥0 generated by B is given by This B is the only generator of a C 0 -semigroup which is an extension of A 0 . H 1 ((a, ∞), H ). In that case the semigroup (S(t)) t≥0 generated by B is given by There are infinitely many other non-contractive C 0 -semigroups having as generator an extension of A 0 . H 1 (R, H ) and A 0 generates a unitary C 0 -group given by ( This is a closed subspace of L 2 ((a, b), H ). Define the operator A 0 on V by Thus u ∈ C ([a, b], H ) and u(a) = u(b) = 0, u(c) ∈ H 1 ∩ H 2 for each u ∈ D(A 0 ). , c), and H 1 0 ((a, c), H 1 ) is dense in L 2 ((a, c), H 1 ) and H 1 0 ((c, b), H 2 ) is dense in L 2 ((c, b), H 2 ). The operator A 0 is skew-symmetric. This is a consequence of Lemma 6.2 since lim x→c,x<c v(x) = lim x→c,x>c v(x) for all v ∈ D(A 0 ). Moreover, A = −A * 0 is given by: H 1 ((a, b), H ) : u(t) ∈ H 1 , t ≤ c and u(t) ∈ H 2 , t ≥ c} = u(b) and G − u = u(a). Then we see that (G − D(A), G + D(A), G − , G + ) is a boundary quadruple for A 0 and G + D(A) = H 2 , G − D(A) = H 1 . We omit the details of the proof.

Theorem 6.5 Let T be an operator such that
The following assertions are equivalent: ((a, b), H ) :  (a, b), H ) : We find the three classical cases.
In each of these three cases the operator T given by If 1 , 2 ∈ L (H ) are contractions then the operator i T with domain (6.3) given by For the proof of Theorems 6.5 and 6.6 we need the following.
is arbitrary, this implies that u ∈ H 1 ((a, b), C) and −u (s) = c + b s u (t)dt for almost alls.
Thus the sum u is in ker G − + ker G + . Now, Theorem 6.5 is a direct consequence of Theorem 3.10, and Theorem 6.6 follows from Theorem 5.5.

The wave equation
In this section we treat the wave equation in terms of quadruples. We are most grateful to Nathanael Skrepek who informed us on the papers [9,10] by Kurula and Zwart, where boundary triples are used for similar, but different results. We refer to [16] for further results on the Maxwell equations. Let ⊂ R d be a bounded, open set with Lipschitz boundary. We consider the skew-symmetric operator A 0 on V := L 2 ( ) × L 2 ( ) d given by Then A := (−A 0 ) * is given by where H div ( ) := {u ∈ L 2 ( ) d : divu ∈ L 2 ( )}. By := ∂ we denote the boundary of ∂ and by L 2 ( ) the Lebesgue space with respect to the surface measure. There exists a unique operator tr ∈ L (H 1 ( ), L 2 ( )) such that tru = u | if u ∈ H 1 ( ) ∩ C ( ). We write u := tru for all u ∈ H 1 ( ), and call u the trace of u. Then ker tr = H 1 0 ( ), the closure of C ∞ c ( ) in H 1 ( ). The space H 1/2 ( ) := tr H 1 ( ) is a Hilbert space for the following norm. Let g ∈ H 1/2 ( ). Then there exists a unique u ∈ (ker tr) ⊥ = H 1 0 ( ) ⊥ such that u = g.

Relation to the literature
In the monographs [8] by V. I. and M. L. Gorbachuk and [15] by Schmüdgen, boundary triples are used to parameterize selfadjoint extensions of a densely defined symmetric operator. Wegner [17] uses them to investigate m-dissipative extensions of a densely defined skew-symmetric operator, as we do in the present article. However we use boundary quadruples. We now explain the relation of our results to those of Wegner and those presented in the two monographs [8] and [15]. We start with a general algebraic property of triples.
if and only if