direct sum of bounded operators on Hilbert spaces


direct sum

0.1 Definition

Let {Hi}iI be a family of Hilbert spacesMathworldPlanetmath indexed by a set I. For each iI let Ti:HiHi be a bounded linear operator on Hi such that the family {Ti}iI of bounded linear operators is uniformly bounded, i.e. sup{Ti:iI}<.

Definition - The direct sum of the uniformly bounded family {Ti}iI is the operatorMathworldPlanetmath


on the direct sum of Hilbert spaces iIHi defined by


It can be seen that iITi is well-defined and is in fact a bounded linear operator, whose norm is


0.2 Properties

  • iI(aTi+bSi)=aiITi+biISi, where a,b.

  • (iITi)*=iITi*.

  • (iITi)(iISi)=iITiSi.

Title direct sum of bounded operators on Hilbert spaces
Canonical name DirectSumOfBoundedOperatorsOnHilbertSpaces
Date of creation 2013-03-22 18:00:32
Last modified on 2013-03-22 18:00:32
Owner asteroid (17536)
Last modified by asteroid (17536)
Numerical id 7
Author asteroid (17536)
Entry type Definition
Classification msc 46C05
Classification msc 47A05