_:Na8742ab154264900b747e58fdc0a5344 .
_:N9b49ce6e8d324063b456b7a17f77b078 .
"problem" .
"linear operators" .
"A Spectral Theory for Certain Operators on a Direct Sum of Hilbert Spaces" .
"Borel function" .
"https://doi.org/10.1007/978-3-642-85997-7_20" .
"theory" .
"operators" .
"Dunford" .
"Florida, USA" .
"294-330" .
.
"properties" .
"AP" .
_:Na8742ab154264900b747e58fdc0a5344 "Springer Nature - SN SciGraph project" .
"operator A" .
"form yi" .
_:N21a6e7753f84455ea88147d7025f49bd .
"spectral reduction" .
"representation" .
"Linear maps y" .
"matrix representation" .
_:N85e27f17e2c6449dbc2512feaf322cd0 "978-3-642-85999-1" .
"Florida, USA" .
"Hilbert space" .
"procedure" .
"operators aij" .
_:N9b49ce6e8d324063b456b7a17f77b078 .
"elementary case" .
.
.
"cases" .
"A." .
"axes" .
"aijxi" .
"direct sum" .
"natural problem" .
"identity" .
_:N21a6e7753f84455ea88147d7025f49bd "10.1007/978-3-642-85997-7_20" .
.
.
_:N41cf1ea233884485abb5a8739f196d99 .
"consideration" .
"spectra" .
"spectral theory" .
"algebra ep" .
"way" .
.
.
"elements" .
"false"^^ .
.
"aij" .
"algebra Ap" .
_:N41cf1ea233884485abb5a8739f196d99 .
_:N85e27f17e2c6449dbc2512feaf322cd0 .
"operational calculus" .
"1966-01-01" .
"satisfactory operational calculus" .
"en" .
"certain operators" .
"former property" .
"Yi" .
"https://scigraph.springernature.com/explorer/license/" .
"EP" .
.
"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." .
.
"normal operators" .
"calculus" .
"reduction" .
"Pure Mathematics" .
.
.
_:N85e27f17e2c6449dbc2512feaf322cd0 "Contributions to Functional Analysis" .
"resolution" .
_:N85e27f17e2c6449dbc2512feaf322cd0 .
"chapter" .
"time" .
_:N21a6e7753f84455ea88147d7025f49bd "doi" .
"course" .
"space" .
"1966" .
_:N9b49ce6e8d324063b456b7a17f77b078 .
.
"linear maps" .
_:N41cf1ea233884485abb5a8739f196d99 "pub.1009301441" .
"Mathematical Sciences" .
"such operators A" .
"algebra" .
_:N21a6e7753f84455ea88147d7025f49bd .
"function" .
_:N41cf1ea233884485abb5a8739f196d99 "dimensions_id" .
.
"dimensions" .
_:Ncbf40e3ddee5481281f400ca7debf31f .
"sum" .
"one" .
"discovery" .
"Nelson" .
"chapters" .
_:Ncbf40e3ddee5481281f400ca7debf31f .
"existence" .
_:N85e27f17e2c6449dbc2512feaf322cd0 "978-3-642-85997-7" .
"map Y" .
"non-normal operator" .
"operator A." .
"2022-01-01T19:09" .
_:Ncbf40e3ddee5481281f400ca7debf31f "Springer Nature" .
_:Na8742ab154264900b747e58fdc0a5344 .
"such problems" .
"maps" .