### Abstract

Let ℋ be a Hilbert space with inner product 〈⋅,⋅〉 and let T be a non-densely defined linear relation in ℋ with domain D(T). We prove that if T is sectorial then it can be expressed in terms of some sectorial operator A with domain D(A)=D(T) and that T is maximal sectorial if and only if A is maximal sectorial in D(T)¯. The operator A has the property that for every u∈D(A) and every v∈D(T) and any u′∈T(u), 〈Au,v〉=〈u′,v〉. We use this representation to show that every sectorial linear relation T is form closable, meaning that the form associated with 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 T is a closed subspace of H×H.

Original language | English |
---|---|

Pages (from-to) | 1-16 |

Number of pages | 16 |

Journal | Journal of Inequalities and Applications |

Volume | 2015 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2015 |

### Fingerprint

### Keywords

- linear form
- numerical range
- sectorial linear relation

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics
- Discrete Mathematics and Combinatorics

### Cite this

**Operator representation of sectorial linear relations and applications.** / Wanjala, Gerald.

Research output: Contribution to journal › Article

*Journal of Inequalities and Applications*, vol. 2015, no. 1, pp. 1-16. https://doi.org/10.1186/s13660-015-0581-z

}

TY - JOUR

T1 - Operator representation of sectorial linear relations and applications

AU - Wanjala, Gerald

PY - 2015

Y1 - 2015

N2 - Let ℋ be a Hilbert space with inner product 〈⋅,⋅〉 and let T be a non-densely defined linear relation in ℋ with domain D(T). We prove that if T is sectorial then it can be expressed in terms of some sectorial operator A with domain D(A)=D(T) and that T is maximal sectorial if and only if A is maximal sectorial in D(T)¯. The operator A has the property that for every u∈D(A) and every v∈D(T) and any u′∈T(u), 〈Au,v〉=〈u′,v〉. We use this representation to show that every sectorial linear relation T is form closable, meaning that the form associated with 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 T is a closed subspace of H×H.

AB - Let ℋ be a Hilbert space with inner product 〈⋅,⋅〉 and let T be a non-densely defined linear relation in ℋ with domain D(T). We prove that if T is sectorial then it can be expressed in terms of some sectorial operator A with domain D(A)=D(T) and that T is maximal sectorial if and only if A is maximal sectorial in D(T)¯. The operator A has the property that for every u∈D(A) and every v∈D(T) and any u′∈T(u), 〈Au,v〉=〈u′,v〉. We use this representation to show that every sectorial linear relation T is form closable, meaning that the form associated with 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 T is a closed subspace of H×H.

KW - linear form

KW - numerical range

KW - sectorial linear relation

UR - http://www.scopus.com/inward/record.url?scp=84923227268&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84923227268&partnerID=8YFLogxK

U2 - 10.1186/s13660-015-0581-z

DO - 10.1186/s13660-015-0581-z

M3 - Article

VL - 2015

SP - 1

EP - 16

JO - Journal of Inequalities and Applications

JF - Journal of Inequalities and Applications

SN - 1025-5834

IS - 1

ER -