Operator representation of sectorial linear relations and applications
- Gerald Wanjala^{1}Email author
https://doi.org/10.1186/s13660-015-0581-z
© Wanjala; licensee Springer. 2015
Received: 7 November 2014
Accepted: 29 January 2015
Published: 20 February 2015
Abstract
Let ℋ be a Hilbert space with inner product \(\langle\cdot, \cdot\rangle\) and let \(\mathcal{T}\) be a non-densely defined linear relation in ℋ with domain \(D(\mathcal{T})\). We prove that if \(\mathcal{T}\) is sectorial then it can be expressed in terms of some sectorial operator A with domain \(D(A) = D(\mathcal{T})\) and that \(\mathcal{T}\) is maximal sectorial if and only if A is maximal sectorial in \(\overline{D(\mathcal{T})}\). The operator A has the property that for every \(u\in D(A)\) and every \(v\in D(\mathcal{T})\) and any \(u^{\prime}\in\mathcal{T}(u)\), \(\langle Au,v\rangle= \langle u^{\prime}, v\rangle\). We use this representation to show that every sectorial linear relation \(\mathcal{T}\) is form closable, meaning that the form associated with \(\mathcal{T}\) has a closed extension. We also prove a result similar to Kato’s first representation theorem for sectorial linear relations.
Unlike the results available in the literature, we do not assume that the graph of the linear relation \(\mathcal{T}\) is a closed subspace of \(\mathcal{H} \times\mathcal{H}\).
Keywords
linear form sectorial linear relation numerical rangeMSC
47A06 47A07 47A12 47B441 Introduction
Instead of considering non-closed subspaces \(\mathcal{S}\) of \(\mathcal{H}\times\mathcal{H}\) one can in general consider, as we do, linear relations in ℋ whose graphs are not necessarily closed subspaces of \(\mathcal{H}\times\mathcal{H}\).
The main objective of this paper is to show that an operator representation of the form (1.1) holds for maximal sectorial linear relations \(\mathcal{T}\) whose graphs are not necessarily closed in \(\mathcal{H} \times\mathcal{H}\). The key idea in obtaining this result is the fact that the numerical range of such a linear relation is a proper subset of the complex plane. We use this result to prove a theorem similar to Kato’s first representation theorem.
Since every densely defined linear relation \(\mathcal{T}\) in a Hilbert space ℋ is an operator, all the relations \(\mathcal{T}\) considered in here are assumed to be non-densely defined. We will use the term operator to mean a linear operator unless stated otherwise.
The paper is organized as follows. In Section 2 we recall some basic definitions and known results on sesquilinear forms. Most of the results given here can be found in [4]. Section 3 is devoted to some background information on linear relations, while Section 4 contains the main results. In particular we show that every maximal sectorial linear relation \(\mathcal{T}\) in ℋ has an operator representation of the form (1.1) and that every sectorial linear relation in ℋ is form closable. Finally we show that a result similar to Kato’s first representation theorem holds in the case of sectorial linear relations.
2 Sesquilinear forms and related results
We are concerned with sesquilinear forms \(t(u,v)\) defined for both u and v belonging to a linear manifold D of a Hilbert ℋ. Hence \(t(u,v)\) is complex-valued and linear in \(u\in D\) for each fixed \(v\in D\) and semilinear in \(v\in D\) for each fixed \(u\in D\). The linear manifold D will be called the domain of t, and we will denote it by \(D(t)\). The form \(t(u,u)\) is called the quadratic form associated with \(t(u,v)\). We denote this form by \(t(u)\). We shall refer to the sesquilinear form \(t(u,v)\) as the form t.
A symmetric sesquilinear form t on a Hilbert space ℋ is called nonnegative (in symbols \(t\geq0\)) if the associated quadratic form \(t(u)\) is nonnegative (\(t(u)\geq0\)) for all u, and positive if \(t(u)>0\) for all \(u\neq0\).
Theorem 2.1
- (i)
\(\bar{t}\) is sectorial,
- (ii)
the numerical range \(\Theta(t)\) of t is a dense subset of the numerical range \(\Theta(\bar{t})\) of \(\bar{t}\),
- (iii)
a vertex α and a semi-angle θ for \(\bar {t}\) can be chosen equal to the corresponding values for t.
Theorem 2.2
Let A be a sectorial operator in a Hilbert space ℋ. Then A is form-closable, that is, the form t defined by (2.3) above is closable.
Let t be a closed sectorial form in a Hilbert space ℋ, \(D^{\prime}\) be a linear subspace of \(D(t)\), and let \(t^{\prime}\) be the restriction of t to \(D^{\prime}\). The subspace \(D^{\prime}\) is called a core of t if the closure of \(t^{\prime}\) is t, that is, \(\bar{t}^{\prime}=t\).
See [4, p.317] for the following remark.
Remark 2.3
If t is bounded, then \(D^{\prime}\) is a core of t if and only if \(D^{\prime}\) is dense in \(D(t)\).
Let ℋ be a Hilbert space and let \(A:D(A)\to\overline{D(A)}\) be a sectorial operator in ℋ. We say that A is maximal sectorial in \(\overline{D(A)}\) if it has no proper sectorial extension \(A_{1}\) in \(\overline{D(A)}\). Maximal sectorial operators are useful in operator representations of sectorial forms as seen from the following theorem which is referred to in the literature as Kato’s first representation theorem.
Theorem 2.4
- (i)\(D(A) \subset D(t)\) andfor every \(u\in D(A)\) and \(v\in D(t)\);$$ t(u,v) = \langle Au, v\rangle $$
- (ii)
\(D(A)\) is a core of t;
- (iii)if \(u\in D(t)\), \(w\in\mathcal{H}\) andholds for every v belonging to a core of t, then \(u\in D(A)\) and \(Au=w\).$$t(u,v)=\langle w,v\rangle $$
3 Relations on sets
3.1 Preliminaries
For a detailed study of relations, we refer to [1–3] and [5].
3.2 Linear relations
- (1)
\(T(x) + \mathcal{T}(z) = \mathcal{T}(x+z)\),
- (2)
\(\mathrm{\alpha}\mathcal{T}(x) = \mathcal{T}(\mathrm{\alpha}x)\).
Let \(\mathcal{T}\) be a symmetric linear relation in ℋ. We say that T is semi-bounded below by a real number α if \(\langle k, h\rangle\geq \mathrm{\alpha}\langle h, h \rangle\) for all \(( h, k) \in G(\mathcal{T})\). It is said to be semi-bounded above by a real number β if \(\langle k, h\rangle\leq \mathrm{\beta}\langle h, h \rangle\) for all \((h, k) \in G(\mathcal{T})\). We say that \(\mathcal{T}\) is semi-bounded if it is either bounded below or above.
We conclude this section with the following three theorems which are taken from [5].
Theorem 3.1
- (i)
\(\mathcal{T}\) is a linear relation.
- (ii)
\(G(\mathcal{T})\) is a linear subspace of \(X\times Y\).
- (iii)
\(\mathcal{T}^{-1}\) is a linear relation.
- (iv)
\(G(\mathcal{T}^{-1})\) is a linear subspace of \(Y\times X\).
Corollary 3.2
- (i)Then \(\mathcal{T}\) is a linear relation if and only ifholds for all \(x_{1}, x_{2} \in D(\mathcal{T})\) and some nonzero scalars α and β.$$\mathcal{T}(\alpha x_{1} + \beta x_{2}) = \alpha \mathcal{T}(x_{1}) + \beta \mathcal{T}(x_{2}) $$
- (ii)
If \(\mathcal{T}\) is a linear relation, then \(\mathcal{T}(0)\) and \(\mathcal{T}^{-1}(0)\) are linear subspaces.
For a linear relation \(\mathcal{T}\), the subspace \(\mathcal{T}^{-1}(0)\) is called the null space (or kernel) of \(\mathcal{T}\) and is denoted by \(N(\mathcal{T})\).
Theorem 3.3
- (i)
\(\mathcal{T}\) is single-valued if and only if \(T(0)= \{0\}\);
- (ii)
if \(\mathcal{S}\) and \(\mathcal{T}\) are two linear relations in a Hilbert space ℋ such that \(G(\mathcal{S})\subset G(\mathcal{T})\), then \(\mathcal{T}\) is an extension of \(\mathcal{S}\) if and only if \(\mathcal{S}(0)=\mathcal{T}(0)\).
Theorem 3.4
4 Operator representation of sectorial linear relations and applications
Let \(\mathcal{T}\) be a sectorial linear relation in a Hilbert space ℋ. As in the case of sectorial operators, we say that \(\mathcal{T}\) is maximal sectorial in ℋ if there does not exist a sectorial linear relation \(\mathcal{T}_{1}\) in ℋ such that \(G(\mathcal{T})\subset G(\mathcal{T}_{1})\).
Lemma 4.1
Proof
We know that the numerical range \(\Theta(\mathcal{T})\) of a sectorial linear relation \(\mathcal{T}\) satisfies the condition \(\Theta(\mathcal{T}) \ne\mathbb{C}\). The following theorem is particularly useful when dealing with this type of linear relations.
Theorem 4.2
- (i)
If \(\Theta(\mathcal{T}) \ne\mathbb{C}\) then \(\mathcal{T}(0)\perp\overline{D(\mathcal{T})}\).
- (ii)
If \(\mathcal{T}\) is a maximal sectorial linear relation, then \(\mathcal{T}(0) = \overline{D(\mathcal{T})}^{\perp}\).
Proof
(ii) Since \(\Theta(\mathcal{T}) \ne\mathbb{C}\), it follows from part (i) that \(\mathcal{T}(0) \subset\overline{D(\mathcal{T})}^{\perp}\).
Now assume that there exists \(h\in\overline{D(\mathcal{T})}^{\perp}\) such that \(h\notin\mathcal{T}(0)\). Lemma 4.1 implies that there exists a sectorial linear relation \(\widetilde{\mathcal{T}}\) in ℋ such that \(G(\mathcal{T}) \subset G(\widetilde{\mathcal{T}})\), contradicting the maximality of \(\mathcal{T}\). Hence \(\overline {D(\mathcal{T})}^{\perp}\subset\mathcal{T}(0)\). It therefore follows that \(\mathcal{T}(0)= \overline{D(\mathcal{T})}^{\perp}\). □
Corollary 4.3
Let \(\mathcal{T}\) be a densely defined sectorial linear relation in a Hilbert space ℋ. Then \(\mathcal{T}\) is an operator.
The lemma below is helpful in defining the sectorial form associated with a sectorial linear relation \(\mathcal{T}\).
Lemma 4.4
Proof
Lemma 4.5
Proof
Theorem 4.6
- (i)\(\mathcal{T}\) is sectorial if and only if there exists a sectorial operator A in ℋ with \(D(A)=D(\mathcal{T})\) and \(R(A) \subset\overline{D(A)}\) such thatfor all \(x\in D(\mathcal{T})\).$$ \mathcal{T}(x) \subset\overline{D(\mathcal{T})}^{\perp}+ Ax $$(4.17)
- (ii)\(\mathcal{T}\) is maximal sectorial if and only if the operator A is maximal sectorial in \(\overline{D(\mathcal{T})}\) andfor all \(x\in D(\mathcal{T})=D(A)\).$$ \mathcal{T}(x) = \mathcal{T}(0) + Ax $$(4.18)
Proof
Theorem 4.7
Note that the form t given by (4.31) is well defined since the inner product \(\langle u^{\prime}, v\rangle\) is independent of the choice of the vector \(u^{\prime}\in\mathcal{T}(u)\) by Lemma 4.4.
Proof
In the next theorem we show that every closed sectorial sesquilinear form in a Hilbert space ℋ has an associated sectorial linear relation in ℋ.
Theorem 4.8
- (a)
- (i)There exists a sectorial linear relation \(\mathcal{T}\) in ℋ such that \(D(\mathcal{T}) \subset D(t)\), \(\mathcal{T}(0)=\overline{D(t)}^{\perp}\) andfor every \(u\in D(\mathcal{T})\) and \(v\in D(t)\), where \(u^{\prime}\) is an arbitrary vector in \(\mathcal{T}(u)\).$$ t(u,v)= \bigl\langle u^{\prime}, v\bigr\rangle $$(4.33)
- (ii)If \(u\in D(t)\), \(w\in\overline{D(t)}\) and \(t(u,v)=\langle w,v\rangle\) holds for every v belonging to a core of t, then \(u\in D(\mathcal{T})\) and$$\mathcal{T}(u)=\overline{D(t)}^{\perp}+ w. $$
- (iii)
The linear relation \(\mathcal{T}\) in (i) is unique.
- (i)
- (b)
If t is bounded, the relation \(\mathcal{T}\) in (a) is maximal sectorial.
Proof
Declarations
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Authors’ Affiliations
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