- Research
- Open Access
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 range
MSC
- 47A06
- 47A07
- 47A12
- 47B44
1 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
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
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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