A Spectral Theory for Certain Operators on a Direct Sum of Hilbert Spaces View Full Text


Ontology type: schema:Chapter     


Chapter Info

DATE

1966

AUTHORS

Nelson Dunford

ABSTRACT

Let ℌp = v ⊕ ... ⊕ ℌ be the direct sum of the Hilbert space ℌ with itself p times. Linear maps y = Ax in ℌp have the form yi = Σ aijxi where aij are linear maps in ℌ and there are many natural problems concerned with the discovery of those properties, enjoyed by the operators aij, that are shared by the operator A. Here we shall discuss, in particular, two such problems; the existence of a resolution of the identity and the existence of an operational calculus. These problems are, of course, closely related and, as is well known, an operator having the former property will have an operational calculus defined on the algebra of bounded Borel functions on its spectrum; but there may be quite a satisfactory operational calculus for an operator which has no resolution of the identity. We consider here only the case where the operators aij are commuting normal operators in ℌ. This is just another way of saying that we assume all of the elements in the matrix representation of A = (aij) to belong to a commutative B*-subalgebra A of the B*-algebra B(ℌ) of bounded linear operators in ℌ. The algebra ep of such operators A is then a non-commutative (in case p > 1) B*-subalgebra of B(ℌp) and a consideration of the most elementary case, where p = 2 and the dimension of ℌ is 1, shows that the algebra Ap contains non-normal operators. Do these non-normal operators in Ap have resolutions of the identity? Unfortunately they need not, but it is easy to state a procedure for determining which ones do have such a spectral reduction and to see therefore that many operators which are not even similar to a normal operator do indeed have resolutions of the identity. More... »

PAGES

294-330

Book

TITLE

Contributions to Functional Analysis

ISBN

978-3-642-85999-1
978-3-642-85997-7

Author Affiliations

Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/978-3-642-85997-7_20

DOI

http://dx.doi.org/10.1007/978-3-642-85997-7_20

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1009301441


Indexing Status Check whether this publication has been indexed by Scopus and Web Of Science using the SN Indexing Status Tool
Incoming Citations Browse incoming citations for this publication using opencitations.net

JSON-LD is the canonical representation for SciGraph data.

TIP: You can open this SciGraph record using an external JSON-LD service: JSON-LD Playground Google SDTT

[
  {
    "@context": "https://springernature.github.io/scigraph/jsonld/sgcontext.json", 
    "about": [
      {
        "id": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/01", 
        "inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/", 
        "name": "Mathematical Sciences", 
        "type": "DefinedTerm"
      }, 
      {
        "id": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/0101", 
        "inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/", 
        "name": "Pure Mathematics", 
        "type": "DefinedTerm"
      }
    ], 
    "author": [
      {
        "affiliation": {
          "alternateName": "Florida, USA", 
          "id": "http://www.grid.ac/institutes/None", 
          "name": [
            "Florida, USA"
          ], 
          "type": "Organization"
        }, 
        "familyName": "Dunford", 
        "givenName": "Nelson", 
        "id": "sg:person.010467222741.78", 
        "sameAs": [
          "https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.010467222741.78"
        ], 
        "type": "Person"
      }
    ], 
    "datePublished": "1966", 
    "datePublishedReg": "1966-01-01", 
    "description": "Let \u210cp = v \u2295 ... \u2295 \u210c be the direct sum of the Hilbert space \u210c with itself p times. Linear maps y = Ax in \u210cp have the form yi = \u03a3 aijxi where aij are linear maps in \u210c and there are many natural problems concerned with the discovery of those properties, enjoyed by the operators aij, that are shared by the operator A. Here we shall discuss, in particular, two such problems; the existence of a resolution of the identity and the existence of an operational calculus. These problems are, of course, closely related and, as is well known, an operator having the former property will have an operational calculus defined on the algebra of bounded Borel functions on its spectrum; but there may be quite a satisfactory operational calculus for an operator which has no resolution of the identity. We consider here only the case where the operators aij are commuting normal operators in \u210c. This is just another way of saying that we assume all of the elements in the matrix representation of A = (aij) to belong to a commutative B*-subalgebra A of the B*-algebra B(\u210c) of bounded linear operators in \u210c. The algebra ep of such operators A is then a non-commutative (in case p > 1) B*-subalgebra of B(\u210cp) and a consideration of the most elementary case, where p = 2 and the dimension of \u210c is 1, shows that the algebra Ap contains non-normal operators. Do these non-normal operators in Ap have resolutions of the identity? Unfortunately they need not, but it is easy to state a procedure for determining which ones do have such a spectral reduction and to see therefore that many operators which are not even similar to a normal operator do indeed have resolutions of the identity.", 
    "genre": "chapter", 
    "id": "sg:pub.10.1007/978-3-642-85997-7_20", 
    "inLanguage": "en", 
    "isAccessibleForFree": false, 
    "isPartOf": {
      "isbn": [
        "978-3-642-85999-1", 
        "978-3-642-85997-7"
      ], 
      "name": "Contributions to Functional Analysis", 
      "type": "Book"
    }, 
    "keywords": [
      "non-normal operator", 
      "operational calculus", 
      "Hilbert space", 
      "normal operators", 
      "direct sum", 
      "operator A.", 
      "linear operators", 
      "spectral theory", 
      "operator A", 
      "certain operators", 
      "linear maps", 
      "matrix representation", 
      "form yi", 
      "Borel function", 
      "such problems", 
      "natural problem", 
      "operators", 
      "elementary case", 
      "calculus", 
      "map Y", 
      "problem", 
      "former property", 
      "aij", 
      "spectral reduction", 
      "algebra", 
      "space", 
      "existence", 
      "sum", 
      "theory", 
      "representation", 
      "Yi", 
      "dimensions", 
      "properties", 
      "function", 
      "cases", 
      "maps", 
      "one", 
      "procedure", 
      "consideration", 
      "way", 
      "elements", 
      "axes", 
      "resolution", 
      "time", 
      "identity", 
      "reduction", 
      "spectra", 
      "AP", 
      "A.", 
      "discovery", 
      "EP", 
      "course", 
      "Linear maps y", 
      "aijxi", 
      "operators aij", 
      "satisfactory operational calculus", 
      "algebra ep", 
      "such operators A", 
      "algebra Ap"
    ], 
    "name": "A Spectral Theory for Certain Operators on a Direct Sum of Hilbert Spaces", 
    "pagination": "294-330", 
    "productId": [
      {
        "name": "dimensions_id", 
        "type": "PropertyValue", 
        "value": [
          "pub.1009301441"
        ]
      }, 
      {
        "name": "doi", 
        "type": "PropertyValue", 
        "value": [
          "10.1007/978-3-642-85997-7_20"
        ]
      }
    ], 
    "publisher": {
      "name": "Springer Nature", 
      "type": "Organisation"
    }, 
    "sameAs": [
      "https://doi.org/10.1007/978-3-642-85997-7_20", 
      "https://app.dimensions.ai/details/publication/pub.1009301441"
    ], 
    "sdDataset": "chapters", 
    "sdDatePublished": "2021-11-01T19:02", 
    "sdLicense": "https://scigraph.springernature.com/explorer/license/", 
    "sdPublisher": {
      "name": "Springer Nature - SN SciGraph project", 
      "type": "Organization"
    }, 
    "sdSource": "s3://com-springernature-scigraph/baseset/20211101/entities/gbq_results/chapter/chapter_56.jsonl", 
    "type": "Chapter", 
    "url": "https://doi.org/10.1007/978-3-642-85997-7_20"
  }
]
 

Download the RDF metadata as:  json-ld nt turtle xml License info

HOW TO GET THIS DATA PROGRAMMATICALLY:

JSON-LD is a popular format for linked data which is fully compatible with JSON.

curl -H 'Accept: application/ld+json' 'https://scigraph.springernature.com/pub.10.1007/978-3-642-85997-7_20'

N-Triples is a line-based linked data format ideal for batch operations.

curl -H 'Accept: application/n-triples' 'https://scigraph.springernature.com/pub.10.1007/978-3-642-85997-7_20'

Turtle is a human-readable linked data format.

curl -H 'Accept: text/turtle' 'https://scigraph.springernature.com/pub.10.1007/978-3-642-85997-7_20'

RDF/XML is a standard XML format for linked data.

curl -H 'Accept: application/rdf+xml' 'https://scigraph.springernature.com/pub.10.1007/978-3-642-85997-7_20'


 

This table displays all metadata directly associated to this object as RDF triples.

113 TRIPLES      22 PREDICATES      84 URIs      77 LITERALS      6 BLANK NODES

Subject Predicate Object
1 sg:pub.10.1007/978-3-642-85997-7_20 schema:about anzsrc-for:01
2 anzsrc-for:0101
3 schema:author N2edb5fe046ea4fd19081a0a5c645531f
4 schema:datePublished 1966
5 schema:datePublishedReg 1966-01-01
6 schema:description Let ℌp = v ⊕ ... ⊕ ℌ be the direct sum of the Hilbert space ℌ with itself p times. Linear maps y = Ax in ℌp have the form yi = Σ aijxi where aij are linear maps in ℌ and there are many natural problems concerned with the discovery of those properties, enjoyed by the operators aij, that are shared by the operator A. Here we shall discuss, in particular, two such problems; the existence of a resolution of the identity and the existence of an operational calculus. These problems are, of course, closely related and, as is well known, an operator having the former property will have an operational calculus defined on the algebra of bounded Borel functions on its spectrum; but there may be quite a satisfactory operational calculus for an operator which has no resolution of the identity. We consider here only the case where the operators aij are commuting normal operators in ℌ. This is just another way of saying that we assume all of the elements in the matrix representation of A = (aij) to belong to a commutative B*-subalgebra A of the B*-algebra B(ℌ) of bounded linear operators in ℌ. The algebra ep of such operators A is then a non-commutative (in case p > 1) B*-subalgebra of B(ℌp) and a consideration of the most elementary case, where p = 2 and the dimension of ℌ is 1, shows that the algebra Ap contains non-normal operators. Do these non-normal operators in Ap have resolutions of the identity? Unfortunately they need not, but it is easy to state a procedure for determining which ones do have such a spectral reduction and to see therefore that many operators which are not even similar to a normal operator do indeed have resolutions of the identity.
7 schema:genre chapter
8 schema:inLanguage en
9 schema:isAccessibleForFree false
10 schema:isPartOf N7c9adf8d072c4c9ca78627f4900db01f
11 schema:keywords A.
12 AP
13 Borel function
14 EP
15 Hilbert space
16 Linear maps y
17 Yi
18 aij
19 aijxi
20 algebra
21 algebra Ap
22 algebra ep
23 axes
24 calculus
25 cases
26 certain operators
27 consideration
28 course
29 dimensions
30 direct sum
31 discovery
32 elementary case
33 elements
34 existence
35 form yi
36 former property
37 function
38 identity
39 linear maps
40 linear operators
41 map Y
42 maps
43 matrix representation
44 natural problem
45 non-normal operator
46 normal operators
47 one
48 operational calculus
49 operator A
50 operator A.
51 operators
52 operators aij
53 problem
54 procedure
55 properties
56 reduction
57 representation
58 resolution
59 satisfactory operational calculus
60 space
61 spectra
62 spectral reduction
63 spectral theory
64 such operators A
65 such problems
66 sum
67 theory
68 time
69 way
70 schema:name A Spectral Theory for Certain Operators on a Direct Sum of Hilbert Spaces
71 schema:pagination 294-330
72 schema:productId N730d773de02f4ea583fa8207bc3e030d
73 Ndc89c912b3074da1ae52966973524264
74 schema:publisher N0d320df6a7f144c8b55512b4818c15e2
75 schema:sameAs https://app.dimensions.ai/details/publication/pub.1009301441
76 https://doi.org/10.1007/978-3-642-85997-7_20
77 schema:sdDatePublished 2021-11-01T19:02
78 schema:sdLicense https://scigraph.springernature.com/explorer/license/
79 schema:sdPublisher N61d363c2736a459085e33f5274db603a
80 schema:url https://doi.org/10.1007/978-3-642-85997-7_20
81 sgo:license sg:explorer/license/
82 sgo:sdDataset chapters
83 rdf:type schema:Chapter
84 N0d320df6a7f144c8b55512b4818c15e2 schema:name Springer Nature
85 rdf:type schema:Organisation
86 N2edb5fe046ea4fd19081a0a5c645531f rdf:first sg:person.010467222741.78
87 rdf:rest rdf:nil
88 N61d363c2736a459085e33f5274db603a schema:name Springer Nature - SN SciGraph project
89 rdf:type schema:Organization
90 N730d773de02f4ea583fa8207bc3e030d schema:name dimensions_id
91 schema:value pub.1009301441
92 rdf:type schema:PropertyValue
93 N7c9adf8d072c4c9ca78627f4900db01f schema:isbn 978-3-642-85997-7
94 978-3-642-85999-1
95 schema:name Contributions to Functional Analysis
96 rdf:type schema:Book
97 Ndc89c912b3074da1ae52966973524264 schema:name doi
98 schema:value 10.1007/978-3-642-85997-7_20
99 rdf:type schema:PropertyValue
100 anzsrc-for:01 schema:inDefinedTermSet anzsrc-for:
101 schema:name Mathematical Sciences
102 rdf:type schema:DefinedTerm
103 anzsrc-for:0101 schema:inDefinedTermSet anzsrc-for:
104 schema:name Pure Mathematics
105 rdf:type schema:DefinedTerm
106 sg:person.010467222741.78 schema:affiliation grid-institutes:None
107 schema:familyName Dunford
108 schema:givenName Nelson
109 schema:sameAs https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.010467222741.78
110 rdf:type schema:Person
111 grid-institutes:None schema:alternateName Florida, USA
112 schema:name Florida, USA
113 rdf:type schema:Organization
 




Preview window. Press ESC to close (or click here)


...