Canonical Graph Contractions of Linear Relations on Hilbert Spaces
Given a closed linear relation T between two Hilbert spaces H and K , the corresponding first and second coordinate projections P T and Q T are both linear contractions from T to H , and to K , respectively. In this paper we investigate the features of these graph contractions. We show among other t...
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| Veröffentlicht in: | Complex analysis and operator theory 2021-02, Vol.15 (1), Article 21 |
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| container_title | Complex analysis and operator theory |
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| creator | Tarcsay, Zsigmond Sebestyén, Zoltán |
| description | Given a closed linear relation
T
between two Hilbert spaces
H
and
K
, the corresponding first and second coordinate projections
P
T
and
Q
T
are both linear contractions from
T
to
H
, and to
K
, respectively. In this paper we investigate the features of these graph contractions. We show among other things that
P
T
P
T
∗
=
(
I
+
T
∗
T
)
-
1
, and that
Q
T
Q
T
∗
=
I
-
(
I
+
T
T
∗
)
-
1
. The ranges
ran
P
T
∗
and
ran
Q
T
∗
are proved to be closely related to the so called ‘regular part’ of
T
. The connection of the graph projections to Stone’s decomposition of a closed linear relation is also discussed. |
| doi_str_mv | 10.1007/s11785-020-01066-3 |
| format | Article |
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T
between two Hilbert spaces
H
and
K
, the corresponding first and second coordinate projections
P
T
and
Q
T
are both linear contractions from
T
to
H
, and to
K
, respectively. In this paper we investigate the features of these graph contractions. We show among other things that
P
T
P
T
∗
=
(
I
+
T
∗
T
)
-
1
, and that
Q
T
Q
T
∗
=
I
-
(
I
+
T
T
∗
)
-
1
. The ranges
ran
P
T
∗
and
ran
Q
T
∗
are proved to be closely related to the so called ‘regular part’ of
T
. The connection of the graph projections to Stone’s decomposition of a closed linear relation is also discussed.</description><identifier>ISSN: 1661-8254</identifier><identifier>EISSN: 1661-8262</identifier><identifier>DOI: 10.1007/s11785-020-01066-3</identifier><language>eng</language><publisher>Cham: Springer International Publishing</publisher><subject>Analysis ; Hilbert space ; Mathematics ; Mathematics and Statistics ; Operator Theory</subject><ispartof>Complex analysis and operator theory, 2021-02, Vol.15 (1), Article 21</ispartof><rights>The Author(s) 2021</rights><rights>The Author(s) 2021. This work is published under http://creativecommons.org/licenses/by/4.0/ (the “License”). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License.</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><cites>FETCH-LOGICAL-c314t-90d2dbe8df554c4728c59e9d06ba8e4b3ebfd583cc45a554babc091ce5f951cd3</cites><orcidid>0000-0001-8102-5055</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s11785-020-01066-3$$EPDF$$P50$$Gspringer$$Hfree_for_read</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s11785-020-01066-3$$EHTML$$P50$$Gspringer$$Hfree_for_read</linktohtml><link.rule.ids>314,776,780,27901,27902,41464,42533,51294</link.rule.ids></links><search><creatorcontrib>Tarcsay, Zsigmond</creatorcontrib><creatorcontrib>Sebestyén, Zoltán</creatorcontrib><title>Canonical Graph Contractions of Linear Relations on Hilbert Spaces</title><title>Complex analysis and operator theory</title><addtitle>Complex Anal. Oper. Theory</addtitle><description>Given a closed linear relation
T
between two Hilbert spaces
H
and
K
, the corresponding first and second coordinate projections
P
T
and
Q
T
are both linear contractions from
T
to
H
, and to
K
, respectively. In this paper we investigate the features of these graph contractions. We show among other things that
P
T
P
T
∗
=
(
I
+
T
∗
T
)
-
1
, and that
Q
T
Q
T
∗
=
I
-
(
I
+
T
T
∗
)
-
1
. The ranges
ran
P
T
∗
and
ran
Q
T
∗
are proved to be closely related to the so called ‘regular part’ of
T
. The connection of the graph projections to Stone’s decomposition of a closed linear relation is also discussed.</description><subject>Analysis</subject><subject>Hilbert space</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Operator Theory</subject><issn>1661-8254</issn><issn>1661-8262</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2021</creationdate><recordtype>article</recordtype><sourceid>C6C</sourceid><recordid>eNp9kE1LxDAURYMoOI7-AVcB19F8N1lq0RlhQPBjHZI01Q41qUln4b-32kF3rt7jce59cAA4J_iSYFxdFUIqJRCmGGGCpUTsACyIlAQpKunh7y74MTgpZYuxxJXWC3BT25hi520PV9kOb7BOcczWj12KBaYWbroYbIaPobf7W4Trrnchj_BpsD6UU3DU2r6Es_1cgpe72-d6jTYPq_v6eoM8I3xEGje0cUE1rRDc84oqL3TQDZbOqsAdC65thGLec2EnxFnnsSY-iFYL4hu2BBdz75DTxy6U0WzTLsfppaG80lxWiumJojPlcyolh9YMuXu3-dMQbL5dmdmVmVyZH1eGTSE2h8oEx9eQ_6r_SX0BH9RsbQ</recordid><startdate>20210201</startdate><enddate>20210201</enddate><creator>Tarcsay, Zsigmond</creator><creator>Sebestyén, Zoltán</creator><general>Springer International Publishing</general><general>Springer Nature B.V</general><scope>C6C</scope><scope>AAYXX</scope><scope>CITATION</scope><orcidid>https://orcid.org/0000-0001-8102-5055</orcidid></search><sort><creationdate>20210201</creationdate><title>Canonical Graph Contractions of Linear Relations on Hilbert Spaces</title><author>Tarcsay, Zsigmond ; Sebestyén, Zoltán</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c314t-90d2dbe8df554c4728c59e9d06ba8e4b3ebfd583cc45a554babc091ce5f951cd3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2021</creationdate><topic>Analysis</topic><topic>Hilbert space</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Operator Theory</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Tarcsay, Zsigmond</creatorcontrib><creatorcontrib>Sebestyén, Zoltán</creatorcontrib><collection>Springer Nature OA/Free Journals</collection><collection>CrossRef</collection><jtitle>Complex analysis and operator theory</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Tarcsay, Zsigmond</au><au>Sebestyén, Zoltán</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Canonical Graph Contractions of Linear Relations on Hilbert Spaces</atitle><jtitle>Complex analysis and operator theory</jtitle><stitle>Complex Anal. Oper. Theory</stitle><date>2021-02-01</date><risdate>2021</risdate><volume>15</volume><issue>1</issue><artnum>21</artnum><issn>1661-8254</issn><eissn>1661-8262</eissn><abstract>Given a closed linear relation
T
between two Hilbert spaces
H
and
K
, the corresponding first and second coordinate projections
P
T
and
Q
T
are both linear contractions from
T
to
H
, and to
K
, respectively. In this paper we investigate the features of these graph contractions. We show among other things that
P
T
P
T
∗
=
(
I
+
T
∗
T
)
-
1
, and that
Q
T
Q
T
∗
=
I
-
(
I
+
T
T
∗
)
-
1
. The ranges
ran
P
T
∗
and
ran
Q
T
∗
are proved to be closely related to the so called ‘regular part’ of
T
. The connection of the graph projections to Stone’s decomposition of a closed linear relation is also discussed.</abstract><cop>Cham</cop><pub>Springer International Publishing</pub><doi>10.1007/s11785-020-01066-3</doi><orcidid>https://orcid.org/0000-0001-8102-5055</orcidid><oa>free_for_read</oa></addata></record> |
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| ispartof | Complex analysis and operator theory, 2021-02, Vol.15 (1), Article 21 |
| issn | 1661-8254 1661-8262 |
| language | eng |
| recordid | cdi_proquest_journals_2479467839 |
| source | SpringerLink Journals |
| subjects | Analysis Hilbert space Mathematics Mathematics and Statistics Operator Theory |
| title | Canonical Graph Contractions of Linear Relations on Hilbert Spaces |
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