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