Decompositions

Carlos S. Kubrusly
Additional contact information
Carlos S. Kubrusly: Catholic University — PUC/RJ

Chapter Chapter 5 in An Introduction to Models and Decompositions in Operator Theory, 1997, pp 75-86 from Springer

Abstract: Abstract Decomposition means separation into “parts”. As far as operators are concerned this usually is done by product (factorization) or by sum. For instance, the polar decomposition says that every operator can be factorized as the product of a partial isometry and a nonnegative operator. On the other hand, the Cartesian decomposition is one by (ordinary) sum: every operator T can be written as T = Re(T) + i Im(T) where Re(T) = 1/2(T + T*) and Im(T) = -i/2(T - T* are self-adjoint operators. We shall however not deal with factorization and ordinary sum decomposition here because, in spite of being useful tools in operator theory, they miss a crucial feature: they do not transfer invariant subspaces from the “parts” (factors or ordinary summands) to the original (decomposed) operator. For the lack of a better name let us say that they do not “isolate the parts”. We shall deal with decomposition by direct sums instead, which do “isolate the parts”: an invariant subspace for a direct summand is invariant for the direct sum.

Keywords: Hilbert Space; Unitary Operator; Invariant Subspace; Direct Summand; Partial Isometry (search for similar items in EconPapers)
Date: 1997
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-1-4612-1998-9_6

Ordering information: This item can be ordered from
http://www.springer.com/9781461219989

DOI: 10.1007/978-1-4612-1998-9_6

Access Statistics for this chapter

More chapters in Springer Books from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().