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


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